twf_ascii/000075500020410000157000000000001143170434500130175ustar00baezhttp00004600000001twf_ascii/week103000064400020410000157000000157400775436014700141430ustar00baezhttp00004600000001April 26, 1997
This Week's Finds in Mathematical Physics  Week 103
John Baez
As I segue over from the homotopy theory conference at Northwestern
University to the conference on higherdimensional algebra and physics
that took place right after that, it's a good time to mention Ronnie
Brown's web page:
1) Ronald Brown, Higherdimensional group theory,
http://www.bangor.ac.uk/~mas010/home.html
Brown is the one who coined the phrase "higherdimensional algebra", and
for many years he has been developing this subject, primarily as a tool
for doing homotopy theory. I wrote a bit about his ideas two years ago,
in "week53". A lot has happened in higherdimensional algebra since
then, and the web page above is a good place to get an overview of it.
It includes a nice bibliography on the subject. Also, if you find the
math a bit strenuous, you can rest your brain and delight your eyes at
the related site:
2) Symbolic sculptures and mathematics,
http://www.bangor.ac.uk/~mas007/welcome.html
which opens with a striking image of rotating linked tori, and includes
pictures of the mathematical sculpture of John Robinson.
The Workshop on Higher Category Theory and Physics was exciting because
it pulled together a lot of people working on the interface between
these two subjects, many of whom had never before met. It was organized
by Ezra Getzler and Mikhail Kapranov. Getzler is probably most wellknown
for his proof of the AtiyahSinger index theorem. This wonderful
theorem captured the imagination of mathematical physicists for many
years starting in the 1960s. The reason is that it relates the topology
of manifolds to the the solutions of partial differential equations
on these manifolds, and thus ushered in a new age of applications of
topology to physics. In the 1980s, working with ideas that Witten
came up with, Getzler found a nice "supersymmetric proof" of the
AtiyahSinger theorem. Later Getzler turned to other things, such
as the use of "operads" (see "week42") to study conformal field
theory (which shows up naturally in string theory). Kapranov has
also done a lot of work with operads and conformal field theory, and
many other things, but I first learned about him through his paper
with Voevodsky on "braided monoidal 2categories" (see "week4"). This
got me very excited since it turned me on to many of the main themes of
ncategory theory.
Alas, my description of this fascinating conference will be terse
and dry in the extreme, since I am flying to Warsaw in 3 hours for
a quantum gravity workshop. I'll just mention a few papers that
cover some of the themes of this conference. Ross Street gave
two talks on Batanin's definition of weak ncategories (and even
weak omegacategories), which one can get as follows:
3) Ross Street, The role of Michael Batanin's monoidal globular
categories. Lecture I: Globular categories and trees. Lecture II:
Higher operads and weak omegacategories. Available in Postscript form
at wwwmath.mpce.mq.edu.au/~coact/street_nw97.ps
Subsequently Batanin has written a more thorough paper on his
definition:
4) Michael Batanin, Monoidal globular categories as a natural
environment for the theory of weak ncategories, Adv. Math 136
(1998), 39103, also available at
http://wwwmath.mpce.mq.edu.au/~mbatanin/papers.html
I gave a talk on Dolan's and my definition of weak ncategories,
which one can get as follows:
5) John Baez, An introduction to ncategories, to appear in
the proceedings of Category Theory and Computer Science '97,
preprint available as qalg/9705009 or in Postscript form at
http://math.ucr.edu/home/baez/ncat.ps
Unfortunately Tamsamani was not there to present *his* definition
of weak ncategories, but at least I have learned how to get his papers
electronically:
6) Zouhair Tamsamani, Sur des notions de $\infty$categorie et
$\infty$groupoide nonstrictes via des ensembles multisimpliciaux,
preprint available as alggeom/9512006.
Zouhair Tamsamani, Equivalence de la theorie homotopique des
ngroupoides et celle des espaces topologiques ntronques,
preprint available as alggeom/9607010.
Also, Carlos Simpson has written an interesting paper using
Tamsamani's definition:
7) Carlos Simpson, A closed model structure for ncategories, internal
Hom, nstacks and generalized SeifertVan Kampen, preprint available as
alggeom/9704006.
In a different but related direction, Masahico Saito discussed
a paper with Scott Carter and Joachim Rieger in which they come
up with a nice purely combinatorial description of all the ways
to embed 2dimensional surfaces in 4dimensional space:
8) J. Scott Carter, Joachim H. Rieger and Masahico Saito,
A combinatorial description of knotted surfaces and their isotopies,
to appear in Adv. Math., preprint available at
http://www.math.usf.edu/~saito/home.html
My student Laurel Langford has translated their work into
ncategory theory and shown that "unframed unoriented 2tangles
form the free braided monoidal 2category on one unframed
selfdual object":
9) John Baez and Laurel Langford, 2Tangles, preprint available
as qalg/9703033 and in Postscript form at
http://math.ucr.edu/home/baez/2tang.ps
This paper summarizes the results; the proofs will appear later.
While I was there, Carter also gave me a very nice paper
he'd done with Saito and Louis Kauffman. This paper discusses
4manifolds and also 2dimensional surfaces in 3dimensional space,
again getting a purely combinatorial description which is begging
to be translated into ncategory theory:
10) J. Scott Carter, Louis H. Kauffman and Masahico Saito,
Diagrammatics, singularities, and their algebraic interpretations,
preprint available at http://www.math.usf.edu/~saito/home.html
I am sorry not to describe these papers in more detail, but
I've been painfully busy lately. (In fact, I am trying
to figure out how to reform my life to give myself more spare
time. I think the key is to say "no" more often.)
Thanks to Justin Roberts for pointing out an error in "week102".
The phase ambiguity in conformal field theories is not necessarily a
24th root of unity; it's exp(2 pi i c / 24) where c is the central
charge of the associated Virasoro representation. This is a big
hint as far as my puzzle goes.
Also I thank Dan Christensen for helping me understand pi_4(S^2)
in a simpler way, and Scott Carter for a fascinating letter on
the themes of "week102". Alas, I have been too busy to reply
adequately to these nice emails!
Gotta run....

The ftp site at UCR is gone. Previous issues of "This Week's Finds" and
other expository articles on mathematics and physics, as well as some of
my research papers, can be obtained at
http://math.ucr.edu/home/baez/
For a table of contents of all the issues of This Week's Finds, try
http://math.ucr.edu/home/baez/twf.html
A simple jumpingoff point to the old issues is available at
http://math.ucr.edu/home/baez/twfshort.html
If you just want the latest issue, go to
http://math.ucr.edu/home/baez/this.week.html
twf_ascii/week200000064400020410000157000001340571101321643200141200ustar00baezhttp00004600000001Also available at http://math.ucr.edu/home/baez/week200.html
December 31, 2003
This Week's Finds in Mathematical Physics  Week 200
John Baez
Happy New Year!
I'm making some changes in my life. For many years I've dreamt
of writing a book on higherdimensional algebra that will explain
ncategories and their applications to homotopy theory, representation
theory, quantum physics, combinatorics, logic  you name it! It's an
intimidating goal, because every time I learn something new about these
subjects I want to put it in this imaginary book, so it keeps getting
longer and longer in my mind! Actually writing it will require heroic
acts of pruning. But, I want to get started.
It'll be freely available online, and it'll show up here as it
materializes  but so far I've just got a tentative outline:
1) John Baez, HigherDimensional Algebra,
http://math.ucr.edu/home/baez/hda/
Unfortunately, I'm very busy these days. As you get older, duties
accumulate like barnacles on a whale if you're not careful! When I
started writing This Week's Finds a bit more than ten years ago, I
was lonely and bored with plenty of time to spare. My life is very
different now: I've got someone to live with, a house and a garden
that seem to need constant attention, a gaggle of grad students, and
too many invitations to give talks all over the place.
In short, the good news is I'm never bored and there's always something
fun to do. The bad news is there's always TOO MUCH to do! So, a while
ago I decided to shed some duties and make more time for things I consider
really important: thinking, playing the piano, writing this book...
and yes, writing This Week's Finds.
First I quit working for all the journals I helped edit. Then I started
refusing most requests to referee articles. Both these are the sort of
job it's really fun to quit. But doing so didn't free up nearly enough
time.
So now I've also decided to stop moderating the newsgroup
sci.physics.research  and stop posting so many articles there.
This is painful, because I've learned so much from this newsgroup over
the last 10 years, met so many interesting people, and had such fun.
I thank everyone on the group. I'll miss you! I'll probably be back
whenever I get lonely or bored.
Ahem. Before I get weepy and nostalgic, I should talk about some math.
This November in Florence there was a conference in honor of the 40th
anniversary of Bill Lawvere's Ph.D. thesis  a famous thesis called
"Functorial Semantics of Algebraic Theories", which explored the
applications of category theory to algebra, logic and physics.
There are videos of all the talks on the conference website:
2) Ramifications of Category Theory, http://ramcat.scform.unifi.it/
This conference was organized and funded by Michael Wright, a businessman
with a great love of mathematics and philosophy, so it was appropriate
that it was held in the old city of Cosimo de Medici, Renaissance banker
and patron of scholars. And since there were talks both by mathematicians
and philosophers  especially Alberto Peruzzi, a philosopher at the
University of Florence who helped run the show  I couldn't help but
remember Cosimo's "Platonic Academy", which spearheaded the rebirth of
classical learning in Renaissance Italy. When not attending talks, I
spent a lot of time roaming around twisty old streets, talking category
theory at wonderful restaurants, reading The Rise and Fall of the House
of Medici, and desperately trying to soak up the overabundance of incredible
art and architecture: the Ponte Vecchio, the Piazza del Duomo, the Santa
Croce where everyone from Galileo to Dante to Machiavelli is buried....
Ahem. Math!
What was Lawvere's thesis about? It's never been published, so I've
never read it  though I hear it's going to be. So, my impression of
its contents comes from gossip, rumors and later research that refers to
his work.
Lawvere started out as a student of Clifford Truesdell, working
on "continuum mechanics", which is the very practical branch of field
theory that deals with fluids, elastic bodies and the like. In the
process, Lawvere got very interested in the foundations of physics,
particularly the notions of "continuum" and "physical theory".
Somehow he decided that only category theory could give him the tools
to really make progress in understanding these notions. After all, this
was the 1960s, and revolution was in the air. So, he somehow got himself
sent to Columbia University to learn category theory from Sam Eilenberg,
one of the two founders of the subject. He later wrote:
In my own education I was fortunate to have two teachers who used
the term "foundations" in a commonsense way (rather than in the
speculative way of the BolzanoFregePeanoRussell tradition).
This way is exemplified by their work in Foundations of Algebraic
Topology, published in 1952 by Eilenberg (with Steenrod), and
The Mechanical Foundations of Elasticity and Fluid Mechanics,
published in the same year by Truesdell. The orientation of these
works seemed to be "concentrate the essence of practice and in turn
use the result to guide practice".
It may seem like a big jump from the downtoearth world of continuum
mechanics to category theory, but to Lawvere the connection made perfect
sense  and while I've always found his writings inpenetrable, after
hearing him give four long lectures in Florence I think it makes sense
to me too! Let's see if I can explain it.
Lawvere first observes that in the traditional approach to physical
theories, there are two key players. First, there are "concrete
particulars"  like specific ways for a violin string to oscillate,
or specific ways for the planets to move around the sun. Second,
there are "abstract generals": the physical laws that govern the motion
of the violin string or the planets.
In traditional logic, an abstract general is called a "theory", while a
concrete particular is called a "model" of this theory. A theory is
usually presented by giving some mathematical language, some rules of
deduction, and then some axioms. A model is typically some sort of map
that sends everything in the theory to something in the world of sets and
truth values, in such a way that all the axioms get mapped to "true".
Since theories involve playing around with symbols according to fixed
rules, the study of theories is often called "syntax". Since the
meaning of a theory is revealed when you look at its models, the
study of models is called "semantics". The details vary a lot depending
on what you want to do, and physicists rarely bother to formulate their
theories axiomatically, but this general setup has been regarded as the
ideal of rigor ever since the work of Bolzano, Frege, Peano and Russell
around the turn of the 20th century.
And this is what Lawvere wanted to overthrow!
Actually, I'm sort of kidding. He didn't really want to "overthrow" this
setup: he wanted to radically build on it. First, he wanted to free the
notion of "model" from the chains of set theory. In other words, he
wanted to consider models not just in the category of sets, but in other
categories as well. And to do this, he wanted a new way of describing
theories, which is less tied up in the nittygritty details of syntax.
To see what Lawvere did, we need to look at an example. But there
are so many examples that first I should give you a vague sense of the
*range* of examples.
You see, in logic there are many levels of what you might call "strength"
or "expressive power", ranging from wimpy languages that don't let you say
very much and deduction rules that don't let you prove very much, to
ultrapowerful ones that let you do all sorts of marvelous things. Near
the bottom of this hierarchy there's the "propositional calculus" where
we only get to say things like
((P implies Q) and (not Q)) implies (not P)
Further up there's the "firstorder predicate calculus", where we get
to say things like
for all x (for all y ((x = y and P(x)) implies P(y)))
Even further up, there's the "secondorder predicate calculus" where
we get to quantify over predicates and say things like
for all x (for all y (for all P (P(x) iff P(y)) implies x = y))
Etcetera...
And, while you might think it's always best to use the most powerful
form of logic you can afford, this turns out not to be true!
One reason is that the more powerful your logic is, the fewer categories
theories expressed in this logic can have models in. This point may
sound esoteric, but the underlying principle should be familiar. Which
is better: a handoperated drill, an electric drill, or a drill press?
A drill press is the most powerful. But I forgot to mention: you're
using it to board up broken windows after a storm. You can't carry a
drill press around, so now the electric drill sounds best. But another
thing: this is in rural Ghana! With no electricity, now the handoperated
drill is your tool of choice.
In short, there's a tradeoff between power and flexibility. Specialized
tools can be powerful, but they only operate in a limited context.
These days we're all painfully aware of this from using computers: fancy
software only works in a fancy environment!
Lawvere has even come up with a general theory of how this tradeoff works
in mathematical logic... he called this the theory of "doctrines". But
I'm getting way ahead of myself! He came up with "doctrines" in 1969,
and I'm still trying to explain his 1963 thesis.
Just like traditional logic, Lawvere's new approach to logic has been
studied at many different levels in the hierarchy of strength. He began
fairly near the bottom, in a realm traditionally occupied by something
called "universal algebra", developed by Garrett Birkhoff in 1935. The
idea here was that a bunch of basic mathematical gadgets can be defined
using very simple axioms that only involve nary operations on some set
and equations between different ways of composing these operations. A
theory like this is called an "algebraic theory". The axioms for an
algebraic theory aren't even allowed to use words like "and", "or", "not"
or "implies". Just equations.
Okay, now for an example.
A good example is the algebraic theory of "groups". A group is a set
equipped with a binary operation called "multiplication", a unary
operation called "inverse", and a nullary operation (that is, a
constant) called the "unit", satisfying these equational laws:
(gh)k = g(hk) ASSOCIATIVITY
1g = g LEFT UNIT LAW
g1 = g RIGHT UNIT LAW
g^{1}g = 1 LEFT INVERSE LAW
gg^{1} = 1 RIGHT INVERSE LAW
Such a primitive gadget is robust enough to survive in very rugged
environments... it's more like a stone tool than a drill press!
Lawvere noticed that we can talk about models of these axioms not just
in the category of sets, but in any "category with finite products".
The point is that to talk about an nary operation, we just need to be
able to take the product of an object G with itself n times and consider
a morphism
f: G x ... x G > G
 n times 
For example, the category of smooth manifolds has finite products,
so we can talk about a "group object" in this category, which is just
a *Lie group*. The category of topological spaces has finite products,
so we can talk about a group object in this category too: it's a
*topological group*. And so on.
But Lawvere's really big idea was that there's a certain category
with finite products whose only goal in life is to contain a group
object. To build this category, first we put in an object
G
Since our category has finite products this automatically means
it gets objects 1, G, G x G, G x G x G, and so on. Next, we put in
a binary operation called "multiplication", namely a morphism
m: G x G > G
We also put in a unary operation called "inverse":
inv: G > G
and a nullary operation called the "unit":
i: 1 > G
And then we say a bunch of diagrams commute, which express all
the axioms for a group listed above.
Lawvere calls this category the "theory of groups", Th(Grp). The object
G is just like a group  but not any *particular* group, since its
operations only satisfy those equations that hold in *every* group!
By calling this category a "theory", Lawvere is suggesting that like a
theory of the traditional sort, it can have models  and indeed
it can! A "model" of theory of groups in some category X with finite
products is just a productpreserving functor
F: Th(Grp) > X
By the way things are set up, this gives us an object
F(G)
in C, together with morphisms
F(m): F(G) x F(G) > F(G)
F(inv): F(G) > F(G)
F(i): F(1) > F(G)
that serve as the multiplication, inverse and identity element
for F(G)... all making a bunch of diagrams commute, that express
the axioms for a group!
So, a model of the theory of groups in X is just a group object in X.
Whew. So far I've just explained the *title* of Lawvere's PhD thesis:
"Functorial Semantics of Algebraic Theories". In Lawvere's approach,
an "algebraic theory" is given not by writing down a list of axioms,
but by specifying a category C with finite products. And the semantics
of such theories is all about productpreserving functors F: C > X.
Hence the term "functorial semantics".
Lawvere did a lot starting with these ideas. Let me just briefly
summarize, and then move on to his work on topos theory and mathematical
physics.
Wise mathematicians are interested not just in models, but also the
homomorphisms between these. So, given an algebraic theory C,
Lawvere defined its category of models in X, say Mod(C,X), to have
productpreserving functors F: C > X as objects and natural
transformations between these as morphisms. For example, taking
C to be the theory of groups and X to be the category of sets, we get
the usual category of groups:
Mod(Th(Grp),Set) = Grp
That's reassuring, and that's how it always works. What's less obvious,
though, is that one can always recover C from Mod(C,Set) together with
its forgetful functor to the category of sets.
In other words: not only can we get the models from the theory, but we
can also get back the theory from its category of models!
I explained how this works in "week136" so I won't do so again here.
This result actually generalizes an old theorem of Birkhoff on universal
algebra. But fans of the TannakaKrein reconstruction theorem for
quantum groups will recognize this duality between "theories and their
category of models" as just another face of the duality between
"algebras and their category of representations"  the classic
example being the Fourier transform and inverse Fourier transform!
And this gives me an excuse to explain another bit of Lawvere's jargon:
while a theory is an "abstract general", and particular model of it
is a "concrete particular", he calls the category of *all* its models
in some category a "concrete general". For example, Th(Grp) is an
abstract general, and any particular group is a concrete particular, but
Grp is a concrete general. I mention this mainly because Lawvere flings
around this trio of terms quite a bit, and some people find them
offputting. There are lots of reasons to find his work daunting, but
this need not be one.
In short, we have this kind of setup:
ABSTRACT GENERAL CONCRETE GENERAL
theory models
syntax semantics
and a precise duality between the two columns!
I would love to dig deeper in this direction  I've really just
scratched the surface so far, and I'm afraid the experts will be
disappointed... but I'm even more afraid that if I went further,
the rest of you readers would drop like flies. So instead, let me
say a bit about Lawvere's work on topos theory and physics.
Most practical physics makes use of logic that's considerably stronger
than that of "algebraic theories", but still considerably weaker than
what most of us have been brainwashed into accepting as our default
setting, namely ZermeloFraenkel set theory with the axiom of choice.
So if we want, we can do physics in a context less general than an
arbitrary category with finite products, while still not restricting
ourselves to the category of sets. This is where "topoi" come in 
they're a lot like the category of sets, but vastly more general.
Topos theory was born when Grothendieck decided to completely rewrite
algebraic geometry as part of a massive plan to prove the Weil
conjectures. Grothendieck was another revolutionary of the early 1960s,
and he arrived at his concept of "topos" sometime around 1962. In 196970,
Lawvere and Myles Tierney took this concept  now called a "Grothendieck
topos"  and made it both simpler and more general, arriving at the
present definition. Briefly put, a topos is a category with finite
limits, exponentials, and a subobject classifier. But instead of saying
what these words mean, I'll just say that this lets you do most of what
you normally want to do in mathematics, but without the law of excluded
middle or the axiom of choice.
One of the many reasons this middle ground is so attractive is that it
lets you do calculus with infinitesimals the way physicists enjoy doing
it! Lawvere started doing this in 1967  he called it "synthetic
differential geometry". Basically, he cooked up some axioms on a topos
that let you do calculus and differential geometry with infinitesimals.
The most famous topos like this is the topos of "schemes"  algebraic
geometers use this one a lot. The usual category of smooth manifolds is
not even a topos, but there are topoi that can serve as a substitute,
which have infinitesimals.
I won't list the axioms of synthetic differential geometry, but the
main idea is that our topos needs to contain an object T called the
"infinitesimal arrow". This is a rigorous version of those little
arrows physicists like to draw when talking about vectors:
>
The usual problem with these "little arrows" is that they need to be
really tiny, but still point somewhere. In other words, the head
can't be at a finite distance from the tail  but they can't be at the
same place, either! This seems like a paradox, but one can neatly
sidestep it by dropping the law of excluded middle  or in technical
jargon, working with a "nonBoolean topos".
That sounds like a drastic solution  a cure worse than the disease,
perhaps!  but it's really not so bad. Indeed, algebraic geometers
are perfectly comfortable with the topos of schemes, and they don't
even raise an eyebrow over the fact that this topos is nonBoolean 
mainly because you're allowed to use ordinary logic to reason *about*
a topos, even if its internal logic is funny.
But enough logic! Let's do some geometry! Let's say we're in some
topos with an infinitesimal arrow object, T. I'll call the objects
of this topos "smooth spaces" and the morphisms "smooth maps". How
does geometry work in here?
It's very nice. The first nice thing is that given any smooth space X,
a "tangent vector in X" is just a smooth map
f: T > X
that is, a way of drawing an infinitesimal arrow in X. In general, the
maps from any object A of a topos to any other object B form an object
called B^A  this is part of what we mean when we say a topos has
exponentials. So, the space of all tangent vectors in X is X^T.
And this is what people usually call the "tangent bundle" of X!
So, the tangent bundle is pathetically simple in this setup: it's just
a space of maps. This means we can compose a tangent vector f: T > X
with any smooth map g: X > Y to get a tangent vector gf: T > Y. This
is what people usually call "pushing forward tangent vectors". This
trick gives a smooth map between tangent bundles, the "differential of g",
which it makes sense to call
g^T: X^T > Y^T
Moreover, it's pathetically easy to check the chain rule:
(gh)^T = g^T h^T
And so far we haven't used *any* axioms about the object T  just basic
stuff about how maps work!
We can also define higher derivatives using T. For second derivatives
we start with T x T, which looks like an "infinitesimal square". Then
we mod out by the map
S_{T,T}: T x T > T x T
that switches the two factors. You should visualize this map as
"reflection across the diagonal". When we mod out by it, we get
a quotient space that deserves the name
T^2/2!
and if we now use some axioms about T, it turns out that a smooth map
f: T^2/2! > X
picks out what's called a "secondorder jet" in X. This is a concept
familiar from traditional geometry, but not as familiar as it should be.
The information in a secondorder jet consists of a point in X, the
first derivative of a curve through X, and also the *second* derivative
of a curve through X. Or in physics lingo: position, velocity and
acceleration!
We can go ahead and define nthorder jets using T^n/n! in a perfectly
analogous way, and the visual resemblance to Taylor's theorem is by no
means an accident... but let me stick to second derivatives, since I'm
trying to get to Newton's good old F = ma.
Just as the space of all tangent vectors in X is the tangent bundle X^T,
the space of all 2ndorder jets in X is the "2ndorder jet bundle"
X^{T^2/2!}
There's a map called the "diagonal":
diag: T > T^2/2!
and composing this with any 2ndorder jet turns it into a tangent
vector. This defines a smooth map
p_X: X^{T^2/2!} > X^T
from the 2ndorder jet bundle to the tangent bundle. Intuitively
you can think of this as sending any positionvelocityacceleration
triple, say (q,q',q"), to the pair (q,q').
Now for the fun part: Lawvere defines a "dynamical law" to be a smooth
map going the other way:
s_X: X^T > X^{T^2/2!}
such that s_X followed by p_X is the identity. In other words,
it's a way of mapping any positionvelocity pair (q,q') to a triple
(q,q',q"). So, it's a formula for acceleration in terms of position
and velocity!
There is a category where an object is a smooth space equipped
with a dynamical law and a morphism is a "lawful motion": that
is, a smooth map
f: X > Y
that makes the obvious diagram commute:
s_X
X^T > X^{T^2/2!}
 
 
 
f^T   f^{T^2/2!}
 
 
 
V s_Y V
Y^T > Y^{T^2/2!}
In particular, if we take R to be the real numbers  "time"  and equip
it with the law saying
q" = 0
meaning that "time ticks at an unchanging rate", then a lawful motion
f: R > X
is precisely a trajectory in X that "follows the law", meaning that
the acceleration of the trajectory is the desired function of position
and velocity. This example is a setup for the classical mechanics
of a point particle; it's easy to generalize to classical field theory
by replacing R by a higherdimensional space.
In fact, under some mild conditions this category whose objects are
spaces equipped with dynamical law and whose morphisms are lawful
motions is a *topos*! As Lawvere notes, "all the usual smooth
dynamical systems, including the infinitedimensional ones
(elasticity, fluid mechanics, and Maxwellian electrodynamics)
are included as special objects." This topos is an example of
what Lawvere calls a "concrete general". Even better, there is also
a corresponding "abstract general".
I'm sure many of you have the same impression that I had when seeing
this stuff, namely that it's a bit quixotic for a highpowered mathematician
to be reformulating the foundations of classical mechanics here at the turn
of the 21st century, instead of working on something "cuttingedge" like
string theory. Even if Lawvere's approach is better, one can't help but
wonder if it gives truly *new* insights, or just a clearer formulation
of existing ones. And either way, one can't help wonder: does he actually
expect enough people to learn this stuff to make a difference? Does he
really think topos theory can break the Microsoftlike grip that ordinary
set theory has on mathematics?
(Note the software analogy raising its ugly head again. ZermeloFraenkel
set theory is a bit like the Windows operating system: once you're locked
into it, it's hard to imagine breaking out. You use it because everyone
else does and you're too lazy to do anything about it. Topos theory is
more like the "open source" movement: you're welcome and even expected to
keep tinkering with the code.)
I have some sense of the answer to these questions. First of all, Lawvere
wants to do math the right way regardless of whether it's popular. But
secondly, he's been hard at work trying to make the subject accessible
to beginners. He's recently written a couple of textbooks you don't
need a degree in math to read:
3) F. William Lawvere and Steve Schanuel, Conceptual Mathematics: A First
Introduction to Categories, Cambridge U. Press, Cambridge, 1997.
4) F. William Lawvere and Robert Rosebrugh, Sets for Mathematics,
Cambridge U. Press, Cambridge, 2002.
And third, the great thing about topos theory is that you don't
need to "accept it" to profit from it. In math, what really matters
is not "believing the axioms" but coming up with good ideas. Topos
theory is full of good ideas, and these are bound to propagate.
I'll finish off with some references to help you learn more about
this stuff.
Alas, I believe Lawvere's thesis is still lurking in the stacks at
Columbia University:
5) F. W. Lawvere, Functorial semantics of algebraic theories,
Dissertation, Columbia University, 1963.
and so far he's only gotten around to publishing a brief summary:
6) F. William Lawvere, Functorial semantics of algebraic theories,
Proceedings, National Academy of Sciences, U.S.A. 50 (1963), 869872.
But, you can find expositions of his work on algebraic theories here
and there. Here's a gentle one geared towards computer scientists:
7) Roy L. Crole, Categories for Types, Cambridge U. Press, Cambridge,
1993.
A considerably more macho one is available free online:
8) Michael Barr and Charles Wells, Toposes, Triples and Theories,
SpringerVerlag, New York, 1983. Available for free electronically at
http://www.cwru.edu/artsci/math/wells/pub/ttt.html
This book also talks about "sketches", which are a way of syntactically
presenting a category with finite products. It also serves as an
introduction to topoi... umm, or at least toposes. I used to find it
fearsomely difficult and dry. Now I don't, which is sort of scary.
By the way, a "triple" is just another name for a monad.
A really beautiful more advanced treatment of algebraic theories and
also "essentially algebraic theories" can be found here:
9) Maria Cristina Pedicchio, Algebraic Theories, in Textos de Matematica:
School on Category Theory and Applications, Coimbra, July 1317, 1999,
pp. 101159.
Someone should urge her to make this available online  it's already
in TeX, and it deserves to be easier to get!
Shortly after his thesis, Lawvere tackled topoi in this paper:
10) F. William Lawvere, Elementary theory of the category of sets,
Proceedings of the National Academy of Science 52 (1964), 15061511.
He then wrote a number of other papers on algebraic theories and
the like:
11) F. William Lawvere, Algebraic theories, algebraic categories,
and algebraic functors, in Theory of Models, NorthHolland, Amsterdam
(1965), 413418.
12) F. William Lawvere, Functorial semantics of elementary theories,
Journal of Symbolic Logic, Abstract, 31 (1966), 294295.
13) F. William Lawvere, The category of categories as a foundation
for mathematics, in La Jolla Conference on Categorical Algebra,
Springer, Berlin 1966, pp. 120.
14) F. William Lawvere, Some algebraic problems in the context of
functorial semantics of algebraic theories, in Reports of the Midwest
Category Seminar, eds. Jean Benabou et al, Springer Lecture Notes in
Mathematics No. 61, Springer, Berlin 1968, pp. 4161.
Then came his work on "doctrines", which I vaguely alluded to a while
back:
15) F. William Lawvere, Ordinal sums and equational doctrines,
Springer Lecture Notes in Mathematics No. 80, Springer, Berlin,
1969, pp. 141155.
Lawvere started publishing his ideas on mathematical physics in the
late 1970s, though he must have been thinking about them all along:
16) F. William Lawvere, Categorical dynamics, in Proceedings
of Aarhus May 1978 Open House on Topos Theoretic Methods in
Geometry, Aarhus/Denmark (1979).
17) F. William Lawvere, Toward the description in a smooth topos
of the dynamically possible motions and deformations of a continuous
body, Cahiers de Topologie et Geometrie Differentielle Categorique
21 (1980), 337392.
In 1981, Anders Kock came out with a textbook on synthetic differential
geometry:
18) Anders Kock, Synthetic Differential Geometry, Cambridge U. Press,
Cambridge, 1981.
More recently, Lawvere came out with a book on applications of
category theory to physics:
19) F. William Lawvere and S. Schanuel, editors, Categories in
Continuum Physics, Springer Lecture Notes in Mathematics No. 1174,
Springer, Berlin, 1986.
The quote about Lawvere's teachers is from:
20) F. William Lawvere, Foundations and applications: axiomatization and
education, Bulletin of Symbolic Logic 9 (2003), 213224. Also available
at http://www.math.ucla.edu/~asl/bsl/0902/0902006.ps
and this gives a good overview of his ideas, though not easy to read!
He also has some other papers online summarizing his ideas on
differential geometry and physics:
21) F. William Lawvere, Outline of synthetic differential geometry,
available at http://www.acsu.buffalo.edu/~wlawvere/downloadlist.html
22) F. William Lawvere, Toposes of laws of motion, available at
http://www.acsu.buffalo.edu/~wlawvere/downloadlist.html
Finally, Colin McLarty  whom I was delighted to meet in Florence  has
a nice quick introduction to synthetic differential geometry in
his textbook on categories and topos theory:
23) Colin McLarty, Elementary Categories, Elementary Toposes,
Clarendon Press, Oxford, 1995.
Along with Lawvere's books "Conceptual Mathematics" and "Sets for
Mathematics", this is the one reference that's really good for
beginners!
Okay... now that everyone is gone except the people who are absolutely
nuts about category theory, let me say a bit more about doctrines and
theorymodel duality. The nuts who are still reading are probably
disappointed that I kept everything very gentle and expository and
didn't drop any mindblowing bombshells of abstraction, which is what
they like about category theory! So, let's turn up the abstraction a
few notches.
What's a "doctrine"?
Well, in "week89" I described a "monad" in an arbitrary 2category.
But most of the time when people talk about monads they mean monads
in Cat, the 2category of all categories. These are the most important
monads  but I've never really said what they're good for! I need to
come clean and explain this now, since a doctrine is a categorified
version of a monad.
What monads are good for is to describe how objects in one category
can be regarded as objects of some other category "equipped with extra
structure". This theme pervades mathematics, and is of the utmost
importance. For example: groups are sets equipped with extra structure,
abelian groups are groups equipped with extra structure, rings are
abelian groups equipped with extra structure, and so on. We keep building
up fancier gadgets from simpler ones. And pretty much whenever we
do, there's a monad lurking in the background, running the show!
Suppose we've got two categories C and D, and the objects of D are
objects of C equipped with extra structure. Then we get a pair of
adjoint functors:
R: D > C
L: C > D
The right adjoint R sends each Dobject to its "underlying" Cobject,
and the left adjoint L sends each Cobject to the "free" Dobject on
it. Often R is called a "forgetful" functor. For example, if
C = Set
and
D = Grp
then we can take the underlying set of any group, and the
free group on any set.
We get a "monad on C" by letting
T = LR: C > C
Then, we can use facts about adjoint functors to get natural
transformations called "multiplication"
m: TT => T
and the "unit"
i: 1_C => T
Using more facts about adjoint functors, we can check that these
satisfy associativity and the left and right unit laws. I did
all this in "week92" so I won't do it again here. The upshot is
that T is a lot like a monoid  which is why Benabou dubbed it a
"monad".
Now, monoids like to *act* on things, and the same is true for
monads. It turns out that a monad T on C can act on any object
of C. When this happens, we call that object an "algebra" of T,
or a "Talgebra" for short. And when our monad comes from a pair
of adjoint functors as above, the main way we get Talgebras is
from objects of D. And in nice cases, Talgebras are the *same*
as objects of D.
So, for example, we can describe groups as Talgebras where T is
some monad on the category of sets. And we can describe abelian
groups as Talgebras where T is some monad on the category of groups.
And we can describe rings as Talgebras where T is some monad on
the category of abelian groups. And so on!
To really see how this works, we'd need to look at a few examples.
I remember when James Dolan was first teaching me this stuff in a
little coffeeshop here in Riverside, which has since gone out of
business. I considered monads "too abstract" and dug my heels in
like a stubborn mule, refusing to learn about them  until I went
through a bunch of examples and saw that *yes*, this monad business
really *does* capture the essence of what it means to build up
fancy gadgets from simple ones by adding extra structure! And
by now I'm completely sold on it. One reason is the relation to
topology, which I explained in part N of "week118", and also "week174".
But alas, I'm too eager to get to the *really* cool stuff to work
through examples right now. So if you're a complete novice at monads,
you'll have to work out some examples yourself. Right now, I'll just
say a bit of fancier stuff to fill in a couple gaps for the semiexperts.
First, when I said "in nice cases", I really meant that the category of
Talgebras is equivalent to D when the forgetful functor R: D > C is
"monadic". A bit more precisely: for any monad T on C there's a category
of Talgebras, which is usually called C^T for some silly reason.
And, whenever we have a pair of adjoint functors R: D > C and L: C > D,
we get a monad T = LR and a functor from D to C^T. This is just a
careful way of saying that any Dobject gives us a Talgebra. And
finally, we say that R is "monadic" if this functor from D to C^T is
an equivalence of categories. There's a theorem by Beck that says
how to tell when a functor is monadic, just by looking at it.
Second, to make the analogy between monoids and monads precise,
we just need to realize that a monad on C is a monoid object in
the monoidal category hom(C,C). I already explained this in "week92",
in even greater generality than we need here, but we need this now
because I'm about to categorify monads and get "doctrines".
Okay: so, monads are good for describing "objects equipped with extra
structure and properties". But now suppose we want to describe
*categories* equipped with extra structure and properties! For
example, the "categories with finite products" that I was talking
about earlier, or "topoi". There are LOTS of different interesting
kinds of categories equipped with extra structure and properties, and
each of them gives a different kind of *logic*: the logic that works
inside this kind of category! The more structure and properties our
category has, the more powerful logic we can use inside it. This is
what gives the "hierarchy of expressive power" I was talking about.
So, it pays to have a good general way to describe categories equipped
with extra structure and properties.
And this is what Lawvere's "doctrines" do!
I've said how monads on a category C are good for describing
"objects of C equipped with extra structure and properties". But
there's a certain category called Cat whose objects are categories!
So, let's take C = Cat! A monad on Cat will describe categories
equipped with extra structure and properties.
And this is the simplest definition of "doctrine": a monad on Cat.
However, those of you familiar with ncategories will realize that
it's odd to talk about "the category of all categories". Not because
of Russell's paradox  though that's a problem too, forcing us to talk
about the category of *small* categories  but because what's really
important is the 2CATEGORY of all categories. It's best to think
of Cat as a 2category. But this suggests that we should work with
a categorified, *weakened* version of monad when defining doctrines.
For this, we need to categorify and weaken the concept of monad.
People have done this, and the result is sometimes called a "pseudomonad",
but I prefer to call it a weak 2monad, since I have dreams of
categorifying further, and I don't want my notation to become
ridiculous. I'd rather talk about "weak 3monads" than "pseudopseudomonads",
wouldn't you? Furthermore, if you look up "pseudomonad" in the
dictionary you'll get this:
PSEUDOMONAD: bacteria usually producing greenish fluorescent
watersoluble pigment; some pathogenic for plants and animals.
Yuck! So, let's be very general and sketch how to define a weak 2monad
in any weak 3category (aka tricategory).
Given a weak 3category C and an object c of C, a "weak 2monad on c"
is just a weak monoidal category object in hom(c,c).
Huh? Well, hom(c,c) is a weak monoidal 2category, which is precisely
the right environment in which to define a "weak monoidal category
object", and that's what we're doing here. Start with the usual
definition of a weak monoidal category, which is a gadget living in
Cat. Cat is an example of a weak monoidal 2category, and we can
write down the same definition in *any* weak monoidal 2category X,
getting the concept of "weak monoidal category object in X". Then,
take X = hom(c,c).
(Of course I'm lying slightly here: Cat is more strict than your
average weak monoidal 2category, so it may not be immediately obvious
how to generalize the concept of "weak monoidal category" as I'm
suggesting. Still, I claim it's not hard if you know about this stuff.)
Now that you know how to define a weak 2monad on any object c of a
3category C, you can take c to be Cat and C to be 2Cat... and this
is what we really should call a "doctrine".
Unsurprisingly, people often consider stricter versions of the
concept of "2monad" and "doctrine". For example, most people
define their "pseudomonads" not in a weak 3category but just a
semistrict one, also known as a "Graycategory"  since 2Cat is one
of these. For more details, try these papers:
24) R. Blackwell, G. M. Kelly, and A. J. Power, Twodimensional monad
theory, Jour. Pure Appl. Algebra 59 (1989), 141.
23) Brian Day and Ross Street, Monoidal bicategories and Hopf algebroids,
A5v. Math. 129 (1997) 99157.
26) F. Marmolejo, Doctrines whose structure forms a fully faithful
adjoint string, Theory and Applications of Categories 3 (1997), 2344.
Available at http://www.tac.mta.ca/tac/volumes/1997/n2/302abs.html
27) S. Lack, A coherent approach to pseudomonads, Adv. Math. 152 (2000),
179202. Also available at
http://www.maths.usyd.edu.au:8000/u/stevel/papers/psm.ps.gz
Anyway, suppose T is a doctrine. Then we get a 2category of
Talgebras Cat^T, whose objects we should think of as "categories
equipped with extra structure of type T". The classic example would
be "categories with finite products". Just as Lawvere thought of
these as algebraic theories, we can think of *any* Talgebra as a
"theory of type T", and define its category of models: given Talgebras
C and D, the category of models of C in D is hom(C,D), where the hom
is taken in Cat^T.
Depending on what doctrine T we consider, we get many different forms
of logic, and I'll just list a few to whet your appetite:
Cat^T = "categories with finite products" = "algebraic theories"
gives what one might call "algebraic logic"  purely equational
reasoning about nary operations. The theory of groups, or
abelian groups, or rings lives here. The theory of fields does
not since it involves a partially defined operation, division.
(People usually restrict the term "algebraic theories" to the case
of categories with finite products such that every object is of
the form 1, X, X^2, ... for some single object X, but this seems
a bit unnatural to me.)
Cat^T = "symmetric monoidal categories" gives a sort of logic that
allows for theories known as "operads" and "PROPs"  see "week191"
for more. This doctrine is weaker than the previous one, since
we can only use equations where all the same variables appear on both
sides, with no duplications or deletions. The theory of monoids
lives here, as does the theory of commutative monoids; the theory
of groups does not, since the group axioms involve duplication and
deletion of variables. We can think of this doctrine as supporting
a primitive version of quantum logic; stronger doctrines along these
lines are the right context for Graeme Segal's "conformal field
theories" and Michael Atiyah's "topological quantum field theories".
Cat^T = "categories with finite limits" = "essentially algebraic
theories" gives what one might call "essentially algebraic logic".
This doctrine is stronger than that of algebraic theories, since it
allows operations that are defined only when some equations hold.
The theory of categories lives here, since composition of morphisms
is an operation of this sort. The theory of fields does not, since
division is defined only when the denominator satisfies an inequality.
Cat^T = "regular categories" gives "regular logic". This doctrine
is even stronger, since it allows for theories that involve
relations as well as nary operations.
Cat^T = "cartesian closed categories" gives "the typed lambdacalculus".
This allows for operations on operations on operations... etc.
Cat^T = "topoi" gives "topos logic".
The typed lambdacalculus is very popular in theoretical computer
science, and I recommend Crole's book cited above for more about how
it's related to cartesian closed categories. A good introduction to
topos logic is McLarty's book cited above. For an exhaustive study
of many other sorts of logic that should be on this list but aren't,
I recommend part D of this book:
28) Peter Johnstone, Sketches of an Elephant: a Topos Theory
Compendium, Oxford U. Press, Oxford. Volume 1, comprising Part A:
Toposes as Categories, and Part B: 2categorical Aspects of Topos
Theory, 720 pages, 2002. Volume 2, comprising Part C: Toposes as
Spaces, and Part D: Toposes as Theories, 880 pages, 2002.
We can do a lot of fun stuff with all these different forms of logic,
and people have indeed done so... but I think I'll stop here. My
point is merely that higher category theory and logic go handinglove,
and there is plenty of room for exploration here, especially if we keep
categorifying  and also keep trying to craft our logic to realworld
applications, especially in physics and computer science.
I wish you all a Happy New Year, and good luck on all your adventures.
Quote of the week:
"We have had to fight against the myth of the mainstream which says,
for example, that there are cycles during which at one time everybody
is working on general concepts, and at another time anybody of consequence
is doing only particular examples, whereas in fact serious mathematicians
have always been doing both."  F. William Lawvere

Addendum: Micheal Barr wrote me the following email, correcting
some errors in a previous version of this Week's Finds.
Now that I have read it, a few more comments and nitpicks. Lawvere and
Tierney did elementary toposes in 6970. True Bill had looked at toposes
earlier, but had not stated the elementary axioms until he and Myles came
together in Halifax during the years 6971.
The reason Truesdell sent Bill to Columbia was because he and Eilenberg
(and Mac Lane) were all working in the same office in NY doing ballistic
trajectories (or some foolish thing like that) during the years 4245.
When he realized that Bill was really more of a mathematician than
physicist, he thought about what mathematician he knew and came up with
Eilenberg. I heard this version from Truesdell himself.
Mac Lane did not come up with the name "monad". It was Jean Benabou and
it was in the summer of 1966 when there was a category meeting at
Oberwohlfach. We were all trying to come up with something better than
"triple". My contribution was Standard Natural Algebraic Functor with
Unit, but for some reason it was not accepted. Jean was sitting next to
me at lunch one day and came up with that name. I actually liked it,
believe it or not, but Jon Beck disliked it and I was his close friend and
felt obligated to go along. After that it became something of a fetish
with me. Besides TTT was such a nice title.
As for toposes vs. topoi, there I do feel strongly. Whenever we use a
classical plural in English, that plural seems eventually to become a
singular. Need I mention "data" and "media", but I have also heard
"phenomenas". And even "topois" (that from Andre Joyal).

Previous issues of "This Week's Finds" and other expository articles on
mathematics and physics, as well as some of my research papers, can be
obtained at
http://math.ucr.edu/home/baez/
For a table of contents of all the issues of This Week's Finds, try
http://math.ucr.edu/home/baez/twf.html
A simple jumpingoff point to the old issues is available at
http://math.ucr.edu/home/baez/twfshort.html
If you just want the latest issue, go to
http://math.ucr.edu/home/baez/this.week.html
ties". But
there's a certain category called Cat whose objects are categories!
So, let's take C = Cat! A monad on Cat will describe categories
equipped with extra structure and properties.
And this is the simplest definition of "doctrine": a monad on Cat.
However, those of you familiar with ncategories will realize that
it's odd to talk about "the category of all categories". Not because
of Russell's paradox  though that's a problem too, forcing us twf_ascii/week201000064400020410000157000001063541003430673000141230ustar00baezhttp00004600000001Also available at http://math.ucr.edu/home/baez/week201.html
January 10, 2004
This Week's Finds in Mathematical Physics  Week 201
John Baez
Lately James Dolan and I have been studying number theory. I used to
*hate* this subject: it seemed like a massive waste of time. Newspapers,
magazines and even lots of math books seem to celebrate the idea of people
slaving away for centuries on puzzles whose only virtue is that they're
easy to state but hard to solve. For example: are any odd numbers the
sum of all their divisors? Are there infinitely many pairs of primes
that differ by 2? Is every even number bigger than 2 a sum of two primes?
Are there any positive integer solutions to
x^n + y^n = z^n
for n > 2? My response to all these was: WHO CARES?!
Sure, it's noble to seek knowledge for its own sake. But working on a
math problem just because it's *hard* is like trying to drill a hole in
a concrete wall with your nose, just to prove you can! If you succeed,
I'll be impressed  but I'll still wonder why you didn't put all that
energy into something more interesting.
Now my attitude has changed, because I'm beginning to see that behind
these silly hard problems there lurks an actual *theory*, full of deep
ideas and interesting links to other branches of mathematics, including
mathematical physics. It just so happens that now and then this theory
happens to crack another hard nut.
I'd known for a while that something like this must be true: after all,
when Andrew Wiles proved Fermat's Last Theorem, even the newspapers
admitted this was just a spinoff of something more important, namely
a special case of the TaniyamaShimura Conjecture. They said this had
something to do with elliptic curves and modular forms, which are very
nice geometrical things that show up all over in complex analysis and
string theory. Unfortunately, the actual statement of this conjecture
seemed impenetrable  it didn't resonate with things I understood.
In fact, the TaniyamaShimura Conjecture is part of a big *network* of
problems that are more interesting but harder to explain than the flashy
ones I listed above: problems like the Extended Riemann Hypothesis, the
Weil Conjecture (now solved), the BirchSwinnertonDyer Conjecture,
and bigger projects like the Langlands Program and developing the theory
of "motives". And these problems rest on top of a solid foundation of
beautiful stuff that's already known, like Galois theory and class field
theory, and stuff about modular forms and Lfunctions.
As I'm gradually beginning to understand little bits of these things,
I'm getting really excited about number theory... so I'm dying to *explain*
some of it! But where to start? I have to start with something basic that
underlies all the fancy stuff. Hmm, I think I'll start with Galois theory.
As you may have heard, Galois invented group theory in the process of
showing you can't solve the quintic equation
ax^5 + bx^4 + cx^3 + dx^2 + ex + f = 0
by radicals. In other words, he showed you can't solve this equation
by means of some soupedup version of the quadratic formula that just
involves taking the coefficients a,b,c,d,e,f and adding, subtracting,
multiplying, dividing and taking nth roots.
The basic idea is something like this. In general, a quintic equation
has 5 solutions  and there's no "best one", so your formula has got to
be a formula for all five. And there's a puzzle: how do you give one
formula for five things?
Well, think about the quadratic formula! It has that "plus or minus"
in it, which comes from taking a square root. So, it's really a formula
for *both* solutions of the quadratic equation. If there were a formula
for the quintic that worked like this, we'd have to get all 5 solutions
from different choices of nth roots in this formula.
Galois showed this can't happen. And the way he did it used *symmetry*!
Roughly speaking, he showed that the general quintic equation is completely
symmetrical under permuting all 5 solutions, and that this symmetry group 
the group of permutations of 5 things  can't be built up from the symmetry
groups that arise when you take nth roots.
The moral is this: you can't solve a problem if the answer has some
symmetry, and your method of solution doesn't let you write down an
answer that has this symmetry!
An old example of this principle is the medieval puzzle called "Buridan's
Ass". Placed equidistant between two equally good piles of hay, this
donkey starves to death because it can't make up its mind which alternative
is best. The problem has a symmetry, but the donkey can't go to *both*
bales of hay, so the only symmetrical thing he can do is stand there.
Buridan's ass would also get stuck if you asked it for *the* solution
to the quadratic equation. Galois proof of the unsolvability of the
quintic by radicals is just a more sophisticated variation on this theme.
(Of course, you *can* solve the quintic if you strengthen your methods.)
A closely related idea is "Curie's principle", named after Marie's
husband Pierre. This says that if your problem has a symmetry and
it is a unique solution, the solution must be symmetrical.
For example, if some physical system has rotation symmetry and it has a
unique equilibrium state, this state must be rotationally invariant.
Now, in the case of a ferromagnet below its "Curie temperature", the
equilibrium state is *not* rotationally invariant: the little magnetized
electrons line up in some specific direction! But this doesn't
contradict Curie's principle, since there's not a unique equilibrium
state  there are lots, since the electrons can line up in any direction.
Physicists use the term "spontaneous symmetry breaking" when any *one*
solution of a symmetric problem is not symmetrical, but the whole set
of them is. This is precisely what happens with the quintic, or even
the quadratic equation.
While these general ideas about symmetry apply to problems of all sorts,
their application to number theory kicks in when we apply them to *fields*.
A "field" is a gadget where you can add, subtract, multiply and divide
by anything nonzero, and a bunch of familiar laws of arithmetic hold,
which I won't bore you with here. The three most famous fields are the
rational numbers Q, the real numbers R, and the complex numbers C.
However, there are lots of other interesting fields.
Number theorists are especially fond of algebraic number fields. An
"algebraic number" is a solution to a polynomial equation whose coefficients
are rational numbers. You get an "algebraic number field" by taking the
field of rational numbers, throwing in finitely many algebraic numbers, and
then adding, subtracting, multiplying and dividing them to get more numbers
until you've got a field.
For example, we could take the rationals, throw in the square root of 2,
and get a field consisting of all numbers of the form
a + b sqrt(2)
where a and b are rational. Notice: if we add, multiply, subtract or
divide two numbers like this, we get another number of this form.
So this is really a field  and it's called Q(sqrt(2)), since we use
round parentheses to denote the result of taking a field and "extending"
it by throwing in some extra numbers.
More generally, we could throw in the square root of any integer n,
and get an algebraic number field called Q(sqrt(n)), consisting of all
numbers
a + b sqrt(n)
where a and b are rational. If sqrt(n) is rational then this field is
just Q, which is boring. Otherwise, we call it a "quadratic number field".
Even more generally, we could take the rationals and throw in a
solution of any quadratic equation with rational coefficients. But
it's easy to see that this doesn't give anything beyond fields like
Q(sqrt(n)). And that's the real reason we call these the "quadratic
number fields".
There are also "cubic number fields", and "quartic number fields",
and "quintic number fields", and so on. And others, too, where we
throw in solutions to a whole bunch of polynomial equations!
Now, it turns out you can answer lots of classic but rather goofysounding
number theory puzzles like "which integers are a sum of two squares?"
by converting them into questions about algebraic number fields.
And the good part is, the resulting questions are connected to all sorts
of other topics in math  they're not just glorified mental gymnastics!
So, from a modern viewpoint, a bunch of classic number theory puzzles are
secretly just tricks to get certain kinds of people interested in algebraic
number fields.
But right now I *don't* want to explain how we can use algebraic number
fields to solve classic but goofysounding number theory puzzles.
In fact, I want to downplay the whole puzzle aspect of number theory.
Instead, I hope you're reeling with horror at thought of this vast
complicated wilderness of fields containing Q but contained in C.
First there's a huge infinite thicket of algebraic number fields...
and then, there's an ever scarier jungle of fields that contain
transcendental numbers like pi and e! I won't even talk about *that*
jungle, it's so dark and scary. Physicists usually zip straight past
this whole wilderness and work with C.
But in fact, if you stop and carefully examine all the algebraic number
fields and how they sit inside each other, you'll find some incredibly
beautiful patterns. And these patterns are turning out to be related to
Feynman diagrams, topological quantum field theory, and so on...
However, before we can talk about all that, we need to understand the
basic tool for analyzing how one field fits inside another: Galois theory!
A function from a field to itself that preserves addition, subtraction,
multiplication and division is called an "automorphism". It's just
a *symmetry* of the field. But now, suppose we have a field K which
contains some smaller field k. Then we define the "Galois group of K
over k" to be the group of all automorphisms of K that act as the
identity on k. We call this group
Gal(K/k)
for short.
The classic example, familiar to all physicists, is the Galois group of
the complex numbers, C, over the real numbers, R. This group has two
elements: the identity transformation, which leaves everything alone, and
complex conjugation, which switches i and i. Since the only group with 2
elements is Z/2, we have
Gal(C/R) = Z/2
Where does complex conjugation come from? It comes from the fact that
we get C from R by throwing in a solution of the quadratic equation
x^2 = 1.
We say C is a "quadratic extension of R". But as soon as we throw in one
solution of this equation, we inevitably throw in another, namely its
negative  and there's no way to tell which is which. And complex
conjugation is the symmetry that switches them!
Note: we know that i and i are different, but we can't tell which is
which! This sounds a bit odd at first. It's a bit hard to explain
precisely in ordinary language, which is part of why Galois had to invent
group theory. But it's fun to try to explain it in plain English...
so let me try. The complex numbers have two solutions to
x^2 = 1.
By convention, one of them is called "i", and the other is called
"i". Having made this convention, there's never any problem telling
them apart. But we could reverse our convention and nothing would
go wrong. For example, if the ghost of Galois wafted into your office
one moonless night and wrote "i" in all your math and physics books
wherever there had been "i", everything in these books would still be true!
Here's another way to think about it. Suppose we meet some
extraterrestrials and find that they too have developed the complex
numbers by taking the real numbers and adjoining a square root of 1,
only they call it "@". Then there would be no way for us to tell if
their "@" was our "i" or our "i". All we can do is choose an arbitrary
convention as to which is which.
Of course, if they put their "@" in the lower halfplane when drawing the
complex plane, we might feel like calling it "i"... but here we are
secretly making use of a convention for matching their complex plane with
ours, and the *other* convention would work equally well! If they drew
their real line *vertically* in the complex plane, it would be more
obvious that we need a convention to match their complex plane with ours,
and that there are two conventions for doing this, both perfectly
selfconsistent.
If you've studied enough physics, this extraterrestrial scenario
should remind you of those thought experiments where you're trying to
explain to some alien civilization the difference between left and
right... by means of radio, say, where you're *not* allowed to refer
to specific objects you both know  so it's cheating to say "imagine
you're on Earth looking at the Big Dipper and the handle is pointing
down; then Arcturus is to the right."
If the laws of physics didn't distinguish between left and right,
you couldn't explain the difference between left and right without
"cheating" like this, so the laws of physics would have a symmetry
group with two elements: the identity and the transformation that
switches left and right. As it turns out, the laws of physics *do*
distinguish between left and right  see "week73" for more on that.
But that's another story. My point here is that the Galois group of C
over R is a similar sort of thing, but built into the very fabric of
mathematics! And that's why complex conjugation is so important.
I could tell you a nice long story about how complex conjugation is
related to "charge conjugation" (switching matter and antimatter) and
also "time reversal" (switching past and future). But I won't!
Here's another example of a Galois group that physicists should like.
Let C(z) be the field of rational functions in one complex variable z 
in other words, functions like
f(z) = P(z)/Q(z)
where P and Q are polynomials in z with complex coefficients. You can
add, subtract, multiply and divide rational functions and get other
rational functions, so they form a field. And they contain C as a
subfield, because we can think of any complex number as a *constant*
function. So, we can ask about the Galois group of C(z) over C.
What's it like?
It's the Lorentz group!
To see this, it's best to think of rational functions as functions not
on the complex plane but on the "Riemann sphere"  the complex plane
together with one extra point, the "point at infinity". The only
conformal transformations of the Riemann sphere are "fractional linear
transformations":
az + b
T(z) = 
cz + d
So, the only symmetries of the field of rational functions that
act as the identity on constant functions are those coming from
fractional transformations, like this:
f > fT where fT(z) = f(T(z))
If you don't follow my reasoning here, don't worry  the details aren't
hard to fill in, but they'd be distracting here.
The last step is to check that the group of fractional linear
transformations is the same as the Lorentz group. You can do this
algebraically, but you can also do it geometrically by thinking of the
Riemann sphere as the "heavenly sphere": that imaginary sphere the stars
look like they're sitting on. The key step is to check this remarkable fact:
if you shoot past the earth near the speed of light, the constellations will
look distorted by a Lorentz transformation  but if you draw lines connecting
the stars, all the *angles* between these lines will remain the same; only
their *lengths* will get messed up!
Moreover, it's obvious that if you rotate your head, both angles and lengths
on the heavenly sphere are preserved. So, any rotation or Lorentz boost
gives an anglepreserving transformation of the heavenly sphere  that is,
a conformal transformation! And this must be a fractional linear
transformation.
Summarizing, the Galois group of C(z) over C is the Lorentz group, or
more precisely, its connected component, SO_0(3,1):
Gal(C(z)/C) = SO_0(3,1).
We've talked about the Galois group of C(z) over C and the Galois group
of C over R. What about the Galois group of C(z) over R? Unsurprisingly,
this is the group of transformations of the Riemann sphere generated by
fractional linear transformations *and* complex conjugation. And physically,
this corresponds to taking the connected component of the Lorentz group
and throwing in *time reversal*! So you see, complex conjugation is related
to time reversal. But I promised not to go into that....
I've been talking about Galois groups that physicists should like, but
you're probably wondering where the number theory went! Well, it's
all part of the same big story. In number theory we're especially
interested in Galois groups like
Gal(K/k)
where K is some algebraic number field and k is some subfield of K.
For starters, consider this example:
Gal(Q(sqrt(n))/Q)
where sqrt(n) is irrational. I've already hinted at what this group is!
Q(sqrt(n)) has sqrt(n) in it, so it also has sqrt(n) in it, and there's
an automorphism that switches these two while leaving all the rational
numbers alone, namely
a + b sqrt(n) > a  b sqrt(n) (a,b in Q)
So, we have:
Gal(Q(sqrt(n)))/Q) = Z/2
just like the Galois group of C over R.
To get some bigger Galois groups, let's take Q and throw in a "primitive
nth root of unity". Hmm, I may need to explain what that means. There
are n different nth roots of 1  but unlike the two square roots of 1,
these are not all created equal! Only some are "primitive".
For example, among the 4th roots of unity we have 1 and 1, which are
actually square roots of unity, and i and i, which aren't. A "primitive
nth root of unity" is an nth root of 1 that's not an kth root for any
k < n. If you take all the powers of any primitive nth root of unity,
you get *all* the nth roots of unity. So, if we take some primitive nth
root of unity, call it
1^{1/n}
for lack of a better name, and extend the rationals by this number,
we get a field
Q(1^{1/n})
which contains all the nth roots of unity. Since the nth roots of unity
are evenly distributed around the unit circle, this sort of field is called
a "cyclotomic field", for the Greek word for "circle cutting". In fact,
one can apply Galois theory to this field to figure out which regular
ngons one can construct with a ruler and compass!
But what's the Galois group
Gal(Q(1^{1/n})/Q)
like? Any symmetry in this group must map 1^{1/n} to some root of unity,
say 1^{m/n}  and once you know which one, you completely know the
symmetry. But actually, this symmetry must map 1^{1/n} to some *primitive*
root of unity, so m has to be relatively prime to n. Apart from that,
though, anything goes  so the size of
Gal(Q(1^{1/n})/Q)
is just the number of guys m less than n that are relatively prime to n. And
if you think about it, these numbers relatively prime to n are just the
same as elements of Z/n that have multiplicative inverses! So if you think
some more, you'll see that
Gal(Q(1^{1/n})/Q) = (Z/n)*
where (Z/n)* is the "multiplicative group" of Z/n  that is, the
elements of Z/n that have multiplicative inverses, made into a group
via multiplication!
This group can be big, but it's still abelian. Can we get some nonabelian
Galois groups from algebraic number fields?
Sure! Let's say you take some polynomial equation with rational coefficients,
take *all* its solutions, throw them into the rationals  and keep adding,
subtracting, multiplying and dividing until you get some field K. This K
is called the "splitting field" of your polynomial.
But here's the interesting thing: if you pick your polynomial equation at
random, the chances are really good that it has n different solutions if
the polynomial is of degree n, and that *any* permutation of these
solutions comes from a unique symmetry of the field K. In other words:
barring some coincidence, all roots are created equal! So in general we
have
Gal(K/Q) = S_n
where S_n is the group of all permutations of n things.
Sometimes of course the Galois group will be smaller, since our polynomial
could have repeated roots or, more subtly, algebraic relations between
roots  as in the cyclotomic case we just looked at.
But, we can already start to see how to prove the unsolvability of the
general quintic! Pick some random 5thdegree polynomial, let K be its
splitting field, and note
Gal(K/Q) = S_5
Then, show that if we build up an algebraic number field by starting
with Q and repeatedly throwing in nth roots of numbers we've already got,
we just can't get S_5 as its Galois group over the rationals! We've
already seen this in the case where we throw in a square root of n, or
an nth root of 1. The general case is a bit more work. But instead of
giving the details, I'll just mention a good textbook on Galois theory for
beginners:
1) Ian Stewart, Galois Theory, 3rd edition, Chapman and Hall, New York,
2004.
Ian Stewart is famous as a popularizer of mathematics, and it shows
here  he has nice discussions of the history of the famous problems
solved by Galois theory, and a nice demystification of the Galois'
famous duel. But, this is a real math textbook  so you can really
learn Galois theory from it! Make sure to get the 3rd edition, since
it has more examples than the earlier ones.
Having given Ian Stewart the dirty work of explaining Galois theory in
the usual way, let me say some things that few people admit in a first
course on the subject.
So far, we've looked at examples of a field k contained in some bigger
field K, and worked out the group Gal(K/k) consisting of all automorphisms
of K that fix everything in k.
But here's the big secret: this has NOTHING TO DO WITH FIELDS! It works
for ANY sort of mathematical gadget! If you've got a little gadget k
sitting in a big gadget K, you get a "Galois group" Gal(K/k) consisting of
symmetries of the big gadget that fix everything in the little one.
But now here's the cool part, which is also very general. Any subgroup
of Gal(K/k) gives a gadget containing k and contained in K: namely, the
gadget consisting of all the elements of K that are fixed by everything
in this subgroup.
And conversely, any gadget containing k and contained in K gives a
subgroup of Gal(K/k): namely, the group consisting of all the symmetries
of K that fix every element of this gadget.
This was Galois' biggest idea: we call this a GALOIS CORRESPONDENCE.
It lets us use *group theory* to classify gadgets contained in one
and containing another. He applied it to fields, but it turns out
to be useful much more generally.
Now, it would be great if the Galois corresondence were always a perfect
11 correspondence between subgroups of Gal(K/k) and gadgets containing
k and contained in K. But, it ain't true. It ain't even true when we're
talking about fields!
However, that needn't stop us. For example, we can restrict ourselves
to cases when it *is* true. And this is where the Fundamental Theorem of
Galois Theory comes in! It's easiest to state this theorem when k and K
are algebraic number fields, so that's what I'll do. In this case, there's
a 11 correspondence between subgroups of Gal(K/k) and extensions of k
contained in K if:
i) K is a "finite" extension of k. In other words, K is a finitedimensional
vector space over k.
ii) K is a "normal" extension of k. In other words, if a polynomial with
coefficients in k can't be factored at all in k, but has one root in K,
then all its roots are in K.
For general fields we also need another condition, namely that K be a
"separable" extension of k. But this is automatic for algebraic number
fields, so let's not worry about it.
At this point, if we had time, we could work out a bunch of Galois groups
and see a bunch of patterns. Using these, we could see why you can't
solve the general quintic using radicals, why you can't trisect the angle
or double the cube using rulerandcompass constructions, and why you can
draw a regular pentagon using ruler and compass, but not a regular heptagon.
Basically, to prove something is impossible, you just show that some number
can't possibly lie in some particular algebraic number field, because it's
the root of a polynomial whose splitting field has a Galois group that's
"fancier" than the Galois group of that algebraic number field.
For example, rulerandcompass constructions produce distances that lie in
"iterated quadratic extensions" of the rationals  meaning that you just
keep throwing in square roots of stuff you've got. Doubling the cube
requires getting your hands on the cube root of 2. But the Galois group
of the splitting field of
x^3 = 2
has size divisible by 3, while an iterated quadratic extension has a Galois
group whose size is a power of 2. Using the Galois correspondence, we see
there's no way to stuff the former field into the latter.
But you can read about this in any good book on Galois theory, so I'd rather
dive right into that thicket I was hinting at earlier: the field of ALL
algebraic numbers! The roots of any polynomial with coefficients in this
field again lie in this field, so we say this field is "algebraically
closed". And since it's the smallest algebraically closed field containing
Q, it's called the "algebraic closure of Q", or Qbar for short  that is, Q
with a bar over it.
This field Qbar is huge. In particular, it's an infinitedimensional vector
space over Q. So, condition i) in the Fundamental Theorem of Galois Theory
doesn't hold. But that's no disaster: when this happens, we just need to
put a topology on the group Gal(K/k) and set up the Galois correspondence using
*closed* subgroups of Gal(K/k). Using this trick, every algebraic number field
corresponds to some closed subgroup of Gal(Qbar/Q).
So, for people studying algebraic number fields,
Gal(Qbar/Q)
is like the holy grail. It's the symmetry group of the algebraic numbers,
and the key to how all algebraic number fields sit inside each other!
But alas, this group is devilishly complicated. In fact, it has literally
driven men mad. One of my grad students knows someone who had a breakdown
and went to the mental hospital while trying to understand this group!
(There may have been other reasons for his breakdown, too, but as readers
of E. T. Bell's book "Men in Mathematics" know, the facts should never get
in the way of a good anecdote.)
If Gal(Qbar/Q) were just an infinitely tangled thicket, it wouldn't be so
tantalizing. But there are things we can understand about it! To describe
these, I'll have to turn up the math level a notch...
First of all, an extension K of a field k is called "abelian" if Gal(K/k)
is an abelian group. Abelian extensions of algebraic number fields can be
understood using something called class field theory. In particular, the
KroneckerWeber theorem says that every finite abelian extension of Q is
contained in a cyclotomic field. So, they all sit inside a field called
Qcyc, which is gotten by taking the rationals and throwing in *all* nth
roots of unity for *all* n. Since
Gal(Q(1^{1/n})/Q) = (Z/n)*
we know from Galois theory that Gal(Qcyc/Q) must be a big group containing
all the groups (Z/n)* as closed subgroups. It's easy to see that (Z/n)* is
a quotient group of (Z/m)* if m is divisible by n; this lets us take the
"inverse limit" of all the groups (Z/m)*  and that's Gal(Qcyc/Q). This
inverse limit is also the multiplicative group of the ring Z^, the inverse
limit of all the rings Z/n. Z^ is also called the "profinite completion of
the integers", and I urge you to play around with it if you never have!
It's a cute gadget.
In short:
Gal(Qcyc/Q) = Z^*
and if we stay inside Qcyc, we're in a zone where the pattern of algebraic
number fields can be understood. This stuff was worked out by people like
Weber, Kronecker, Hilbert and Takagi, with the final keystone, the Artin
reciprocity theorem, laid in place by Emil Artin in 1927. In a certain
sense Qcyc is to Qbar as homology theory is to homotopy theory: it's all
about *abelian* Galois groups, so it's manageable.
People now use Qcyc as a kind of base camp for further expeditions into
the depths of Qbar. In particular, since
Q is contained in Qcyc and Qcyc is contained in Qbar
we get an exact sequence of Galois groups:
1 > Gal(Qbar/Qcyc) > Gal(Qbar/Q) > Gal(Qcyc/Q) > 1
So, to understand Gal(Qbar/Q) we need to understand Gal(Qcyc/Q),
Gal(Qbar/Qcyc) and how they fit together! The last two steps are not
so easy. Shafarevich has conjectured that Gal(Qbar/Qcyc) is the
profinite completion of a free group, say F^. This would give
1 > F^ > Gal(Qbar/Q) > Z^* > 1
but I have no idea how much evidence there is for Shafarevich's conjecture,
or how much people know or guess about this exact sequence.
More recently, Deligne has turned attention to a certain "motivic" version
of Gal(Qbar/Q), which is a proalgebraic group scheme. This sort of group
has a *Lie algebra*, which makes it more tractable. And there are a bunch
of fascinating conjectures about this Lie algebra is related to the Riemann
zeta function at odd numbers, Connes and Kreimer's work on Feynman diagrams,
Drinfeld's work on the GrothendieckTeichmueller group, and more!
I really want to understand this stuff better  right now, it's a complete
muddle in my mind. When I do, I will report back to you. For now, though,
let me give you some references.
For two very nice but very different introductions to algebraic number
fields, try these:
2) H. P. F. SwinnertonDyer, A Brief Guide to Algebraic Number Theory,
Cambridge U. Press, Cambridge 2001.
3) Juergen Neukirch, Algebraic Number Theory, trans. Norbert Schappacher,
Springer, Berlin, 1986.
Both assume you know some Galois theory or at least can fake it.
Neukirch's book is good for the allimportant analogy between Galois
groups and fundamental groups, which I haven't even touched upon here!
SwinnertonDyer's book has the virtue of brevity, so you can see the
forest for the trees. Both have a friendly, slightly chatty style that
I like.
For Shafarevich's conjecture, try this:
4) K. Iwasawa, On solvable extensions of algebraic number fields,
Ann. Math. 58 (1953) 548572.
For Deligne's motivic analogue, try this:
5) Pierre Deligne, Le groupe fondamental de la droite projective
moins trois points, in Galois Groups over Q, MSRI Publications 16 (1989),
79313.
This stuff has a lot of relationships to 3d topological quantum field
theory, braided monoidal categories, and the like... and it all goes
back to the GrothendieckTeichmueller group. To learn about this group
try this book, and especially this article in it:
6) Leila Schneps, The GrothendieckTeichmuller group: a survey,
in The Grothendieck Theory of Dessins D'Enfants, London Math. Society
Notes 200, Cambridge U. Press, Cambridge 1994, pp. 183204.
To hear and watch some online lectures on this material, try:
7) Leila Schneps, The GrothendieckTeichmuller group and fundamental
groups of moduli spaces, MSRI lecture available at
http://www.msri.org/publications/ln/msri/1999/vonneumann/schneps/1/
GrothendieckTeichmuller group and Hopf algebras,
MSRI lecture available at
http://www.msri.org/publications/ln/msri/1999/vonneumann/schneps/2/
For a quick romp through many mindblowing ideas which touches on this
material near the end:
8) Pierre Cartier, A mad day's work: from Grothendieck to Connes
and Kontsevich  the evolution of concepts of space and symmetry,
Bulletin of the AMS, 38 (2001), 389  408. Also available at
http://www.ams.org/joursearch/index.html
For even more mindblowing ideas along these lines:
9) Jack Morava, The motivic Thom isomorphism, talk at the Newton Institute,
December 2002, also available at math.AT/0306151
Quote of the week:
"Paris, 1 June  A deplorable duel yesterday has deprived the exact
sciences of a young man who gave the highest expectations, but whose
celebrated precosity was lately overshadowed by his political activities.
The young Evariste Galois... was fighting with one of his old friends,
a young man like himself, like himself a member of the Society of
Friends of the People, and who was known to have figured equally in
a political trial. It is said that love was the cause of the combat.
The pistol was the chosen weapon of the adversaries, but because of
their old friendship they could not bear to look at one another and
left their decision to blind fate."  Le Precursor, June 4, 1832

Addendum: I received the following email from Avinoam Mann, which
corrects some mistakes I made:
Dear John,
It's very nice that you've come to appreciate the beauties of number
theory, and I enjoyed reading your description of Galois theory, but
I hope that you would not mind if I ask you not to help spread some
common misunderstandings about it. First, it was not Galois who proved
the impossibility of solving the quintic by radicals. This was attempted
first by Ruffini, I think in 1799, and the proof by Abel, about ten
years before Galois, was the one that the mathematical community
accepted. While I often teach Galois theory (e.g. next semester),
I never studied Ruffini's and Abel's work in detail. What Galois
did was to give a criterion checking for an arbitrary equation whether
it is soluble by radicals or not.
Another point: there is no need for Galois theory to prove that
duplication of the cube and trisection of an angle cannot be done by
ruler and compass. Since ruler and compass constructions are equivalent
to solving a series of quadratics, they can lead only to fields F of
dimension 2^n, for some n, over the rationals. But the two problems
that I mentioned lead to extensions of dimension 3. All this is very
elementary. Similar considerations lead to necessary conditions for the
constructibility of regular polygons, but proving these conditions
sufficient does require more theory (unless, I guess, you provide
directly the relevant system of quadratics; I think that is what
Gauss did  his proof also preceded Galois). Squaring the circle is,
of course, a different matter. Here we need the transcendence of pi.
Best wishes from wet Jerusalem,
Avinoam Mann
It's true that Abel and Ruffini beat Galois when it came to the quintic;
the details of this history are covered pretty well by Ian Stewart's book,
I think. And, it's quite true that one doesn't need of Galois theory to
solve a bunch of these problems: for example, to show one can't duplicate
the cube, we just need to see that Q(2^{1/3}) has dimension 3 as a vector
space over Q, while quadratic extensions have dimension 2n. My use of the
Galois correspondence to express this in terms of the size of certain Galois
groups was overkill! The real point of Galois theory is that it provides a
unified framework for tackling a wide range of problems.

Previous issues of "This Week's Finds" and other expository articles on
mathematics and physics, as well as some of my research papers, can be
obtained at
http://math.ucr.edu/home/baez/
For a table of contents of all the issues of This Week's Finds, try
http://math.ucr.edu/home/baez/twf.html
A simple jumpingoff point to the old issues is available at
http://math.ucr.edu/home/baez/twfshort.html
If you just want the latest issue, go to
http://math.ucr.edu/home/baez/this.week.html
em of Galois Theory
doesn't hold. But that's no disaster: when this happens, we just need to
put a topology on the group Gal(K/k) and set up the Galois correspondence using
*closed* subgroups of Gal(K/k). Using this trick, every algebraic number field
corresponds to some twf_ascii/week1000064400020410000157000000251771013005403300137550ustar00baezhttp00004600000001From galaxy!guitar!baez Mon Jan 18 16:02:39 PST 1993
Article: 43586 of sci.physics
Path: galaxy!guitar!baez
From: baez@guitar.ucr.edu (john baez)
Newsgroups: sci.physics,sci.math
Subject: This Week's Finds in Mathematical Physics
MessageID: <25150@galaxy.ucr.edu>
Date: 19 Jan 93 00:04:23 GMT
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I thought I might try something that may become a regular feature
on sci.physics.research, if that group comes to be. The idea is that
I'll briefly describe the papers I have enjoyed this week in mathematical
physics. I am making no pretense at being exhaustive or objective...
what I review is utterly a function of my own biases and what I happen to
have run into. I am not trying to "rate" papers in any way, just to
entertain some people and perhaps inform them of some papers they hadn't
yet run into. "This week" refers to when I read the papers, not when they
appeared (which may be much earlier or also perhaps later, since some of these
I am getting as preprints).
1) Syzygies among elementary string interactions in 2+1 dimensions,
by J. Scott Carter and Masahico Saito, Lett. Math. Phys. 23 (1991),
287300.
On formulations and solutions of simplex equations, by J. Scott Carter and
Masahico Saito, preprint. (Carter is at
F4T3%USOUTHAL.bitnet@VM.TCS.Tulane.EDU.)
A diagrammatic theory of knotted surfaces, by J. Scott Carter and
Masahico Saito, preprint.
Reidemeister moves for surface isotopies and their interpretations as moves
to movies, by J. Scott Carter and Masahico Saito, preprint.
The idea here is to take what has been done for knots in 3dimensional
space and generalize it to "knotted surfaces," that is, embedded 2manifolds
in 4dimensional space. For knots it is convenient to work with 2dimensional
pictures that indicate over and undercrossings; there is a wellknown
small set of "Reidemeister moves" that enable you to get between any two
pictures of the same knot. One way to visualize knotted surfaces is
to project them down to R^3; there are "Roseman moves" analogous to the
Reidemeister moves that enable to get you between any two projections
of the same knotted surface. Carter and Saito prefer to work with
"movies" that display a knotted surface as the evolution of knots
(actually links) over time. Each step in such a movie consists of one of
the "elementary string interactions." They have developed a set of
"movie moves" that connect any two movies of the same knotted surface.
These papers contain a lot of fascinating pictures! And there does seem
to be more than a minor relation to string theory. For example, one of
the movie moves is very analogous to the 3rd Reidemeister move  which goes
\ /   \ /
\ /   \
\   / \
/ \   / \
/ \  \ / 
 \ / \ / 
 \ = \ 
 / \ \ 
  \ / \ 
\ /   \ /
\ /   \
\   / \
/ \   / \
/ \   / \
I won't try to draw the corresponding movie move, but just as the
3rd Reidemeister move is the basis for the YangBaxter equation
R_{23}R_{13}R_{12} = R_{12}R_{13}R_{23} (the subscripts indicate
which strand is crossing which), the corresponding movie move is the
basis for a variant of the "FrenkelMoore" form of the "Zamolodchikov
tetrahedron equation" which first arose in string theory.
This variant goes like:
S_{124}S_{135}S_{236}S_{456} = S_{456}S_{236}S_{135}S_{124}
and Carter and Saito draw pictures that make this equation almost as
obvious as the YangBaxter equations.
In any event, this is becoming a very hot subject, since topologists
are interested in generalizing the new results on knot theory to
higher dimensions, while some physicists (especially Louis Crane)
are convinced that this is the right way to tackle the "problem of
time" in quantum gravity (which, in the loop variables approach, amounts
to studying the relationship of knot theory to the 4th dimension, time.)
In particular, Carter and Saito are investigating how to construct
solutions of the Zamolodchikov equations from solutions of the YangBaxter
equation  the goal presumably being to find invariants of knotted surfaces
that are closely related to the link invariants coming from quantum groups.
This looks promising, since Crane and Yetter have just constructed a
4dimensional topological quantum field theory from the quantum SU(2).
But apparently nobody has yet done it.
Lovers of category theory will be pleased to learn that the correct framework
for this problem appears to be the theory of 2categories. These are
categories with objects, morphisms between objects, and also
"2morphisms" between objects. The idea is simply that tangles are morphisms
between sets of points (i.e., each of the tangles in the picture above
are morphisms from 3 points to 3 points), while surfaces in R^4 are
2morphisms between tangles. The instigators of the 2categorical
approach here seem to be Kapranov and Voevodsky, whose paper "2categories
and Zamolodhikov tetrahedra equations," to appear in Proc. Symp. in Pure
Math., is something I will have to get ahold of soon by any means possible
(I can probably nab it from Oleg Viro down the hall; he is currently
hosting Kharmalov, who is giving a series of talks on knotted surfaces
at 2categories here at UCR.) But it seems to be Louis Crane who is most
strongly proclaiming the importance of 2categories in *physics*.
2) Knot theory and quantum gravity in loop space: a primer, by Jorge
Pullin, to appear in "Proc. of the Vth Mexican School of Particles and
Fields," ed. J. L. Lucio, World Scientific, Singapore, now available
as hepth/9301028.
This is a review of the new work on knot theory and the loop representation
of quantum gravity. Pullin is among a group who has been carefully
studying the "ChernSimons state" of quantum gravity, so his presentation,
which starts with a nice treatment of the basics, leads towards the
study of the ChernSimons state. This is by far the bestunderstood state of
quantum gravity, and is defined by SU(2) ChernSimons theory in terms of the
connection representation, or by the Kauffman bracket invariant of knots
in the loop representation. It is a state of Euclideanized quantum
gravity with nonzero cosmological constant, and is not invariant under
CP. Ashtekar has recently speculated that it is a kind of "ground
state" for gravity with cosmological constant (evidence for this has
been given by Kodama), and that its CP violation may be a "reason" for
why the cosmological constant is actually zero (this part is extremely
speculative). Louis Crane, on the other hand, seems convinced that the
ChernSimons state (or more generally states arising from modular tensor
categories) is THE WAVEFUNCTION OF THE UNIVERSE. In any event, it's much
nicer to have one state of quantum gravity to play with than none, as
was the case until recently.
3) Time, measurement and information loss in quantum cosmology, by Lee
Smolin, preprint now available as grqc/9301016.
This is, as usual for Smolin, a very ambitious paper. It attempts to
sketch a solution of some aspects of the problem of time in quantum gravity
(in terms of the loop representation). I might as well quote from
the introduction:
Thus, to return to the opening question, if we are, within
a nonperturbative framework, to ask what happens after a black
hole evaporates, we must be able to construct spacetime diffeomorphism
invariant operators that can give physical meaning to the notion of
``after the evaporation.'' Perhaps I can put it in the following way:
the questions about loss of information or breakdown of unitary
evolution rely, implicitly, on a notion of time. Without reference
to time it is impossible to say that something is being lost.
In a quantum theory of gravity, time is a problematic concept which makes
it difficult to even ask such questions at the nonperturbative level,
without reference to a fixed spacetime manifold. [I would prefer
to say "fixed background metric"  JB] The main idea, which
it is the purpose of this paper to develop, is that the problem of time
in the nonperturbative framework is more than an obstacle that
blocks any easy approach to the problem of loss of information in black hole
evaporation. It may be the key to its solution.
As many people have argued, the problem of time is indeed the conceptual
core of the problem of quantum gravity. Time, as it is conceived in
quantum mechanics is a rather different thing than it is from the
point of view of general relativity. The problem of quantum gravity,
especially when put in the cosmological context, requires for
its solution that some single concept of time be invented that is
compatible with both diffeomorphism invariance and the principle
of superposition. However, looking beyond this, what is at stake in
quantum gravity is indeed no less and no more than
the entire and ancient mystery: What is time? For the theory that will
emerge from the search for quantum gravity is likely to be the background
for future discussions about the nature of time, as Newtonian physics
has loomed over any discussion about time from the seventeenth century
to the present.
I certainly do not know the solution to the problem of time. Elsewhere I have
speculated about the direction in which we might search for its
ultimate resolution. In this paper I will take a rather different
point of view, which is based on a retreat to what both Einstein and
Bohr taught us to do when the meaning of a physical concept becomes
confused: reach for an operational definition. Thus, in
this paper I will adopt the point of view that time is precisely
no more and no less than that which is measured by physical clocks. From
this point of view, if we want to understand what
time is in quantum gravity then we must construct a description of
a physical clock living inside a relativistic quantum mechanical
universe.

Technically speaking, what Smolin does is roughly as follows. He
considers quantum gravity coupled to matter, modelled in such a way
that the Hilbert space is spanned by states labelled by isotopy
classes of: any number N loops in a compact 3manifold M ("space") and N
surfaces with boundary in M. (This trick is something I hadn't seen
before, though Smolin gives references to it.) He then introduces
a "clock field," which is just a free scalar field coupled to the gravity,
and does gaugefixing to see what evolution with respect to this
clock field looks like. I will have to read this a number of times!
n to R^3; there are "Roseman moves" analogous to the
Reidemeister moves that enable to get you between any two projections
of the same knotted surface. Carter and Saito prefer to work with
"movies" that display a knotted surface as the evolution of knots
(actually links) over time. Each step in such a movie consists of one of
the "elementary string interactions." They have develtwf_ascii/week10000064400020410000157000000277020774011336300140500ustar00baezhttp00004600000001
Week 10
The most substantial part of this issue is some remarks by
Daniel Ruberman (ruberman@binah.cc.brandeis.edu) on the paper I was
talking about last time by Boguslaw Broda. They apparently show that
Broda's invariant is not as new as it might appear. But they're rather
technical, so I'll put them near the end, and start off with something
on the light side, and then note some interesting progress on the
Vassiliev invariant scene.
1) Beyond Einstein  is space loopy? by Marcia Bartusiak, Discover,
April 1993.
In the airport in Montreal I ran into this article, which was the cover
story, with an upsidedown picture of Einstein worked into a bunch of
linked keyrings. I bought it  how could I resist?  since it is
perhaps the most "pop" exposition of the loop representation of quantum
gravity so far. Those interested in the popularization of modern
physics might want to compare
Gravity quantized? A radical theory of gravity weaves space from tiny
loops, by John Horgan, Scientific American, September 1992.
Given the incredible hype concerning superstring theory, which seems to
have faded out by now, I sort of dread the same thing happening to the
loop representation of quantum gravity. It is intrinsically less
hypeable, since it does not purport to be a theory of everything, and
comes right after superstrings were supposed to have solved all the
mysteries of the universe. Also, its proponents are (so far) a more
cautious breed than the string theorists  note the question marks in
both titles! But we will see....
Marcia Bartusiak is a contributing editor of Discover and the author of
a book on current topics in astronomy and astrophysics, "Thursday's
Universe", which I haven't read. She'll be coming out with a book in
June, "Through a Universe Darkly," that's supposed to be about how
theories of cosmology have changed down through the ages. She does a
decent job of sketching vaguely the outlines of the loop representation
to an audience who must be presumed ignorant of quantum theory and
general relativity. Or course, there is also a certain amount of
humaninterest stuff, with Ashtekar, Rovelli and Smolin (quite rightly)
coming off as the heroes of the story. There are, as usual, little
boxes with geewhiz remarks like
WITH REAMS OF PAPER
SPREAD OUT
OVER THE KITCHEN TABLE
THEY FOUND
SOLUTION AFTER SOLUTION
FOR EQUATIONS
THOUGHT IMPOSSIBLE TO SOLVE
(which is, after all, true  nobody had previously found solutions to
the constraint equations in canonical quantum gravity, and all of a
sudden here were lots of 'em!). And there are some amusing discussions
of personality: "Affable, creative, and easygoing, Rovelli quickly
settled into the role of gobetween, helping mesh the analytic powers of
the quiet, contemplative Ashtekar with the creativity of the brash,
impetuous Smolin." And discussions of how much messier Smolin's office
is than Ashtekar's.
In any event, it's a fun read, and I recommend it. Of course, I'm
biased, so don't trust me.
2) Vassiliev invariants contain more information than all knot
polynomials, by Sergey Piunikhin, preprint. (Piunikhin is at
serguei@math.harvard.edu)
TuraevViro and KauffmanLins invariants for 3manifolds coincide, by
Sergey Piunikhin, Journal of Knot Theory and its Ramifications, 1 (1992)
105  135.
Different presentations of 3manifold invariants arising in rational
conformal field theory, by Sergey Piunikhin, preprint.
Weights of Feynman diagrams, link polynomials and Vassiliev knot
invariants, by Sergey Piunikhin, preprint.
ReshetikhinTuraev and CraneKohnoKontsevich 3manifold invariants
coincide, by Sergey Piunikhin, preprint.
I received a packet of papers by Piunikhin a while ago. The most new
and interesting thing is the first paper listed above. In "week3" I noted
a conjecture of BarNatan that all Vassiliev invariants come from
quantum group knot invariants (or in other words, from Lie algebra
representations.) Piunikhin claims to refute this by showing that there
is a Vassiliev invariant of degree 6 that does not. (However, other
people have told me his claim is misleading!) I have been too busy to
read this paper yet.
3) Bibliography of publications related to classical and quantum
gravity in terms of the Ashtekar variables, by Bernd Bruegmann, 14 pages
(LaTeX, 1 figure), available as grqc/9303015.
Let me just quote the abstract; this should be a handy thing:
This bibliography attempts to give a comprehensive overview of all the
literature related to the Ashtekar variables. The original version was
compiled by Peter Huebner in 1989, and it has been subsequently
updated by Gabriela Gonzalez and Bernd Bruegmann. Information about
additional literature, new preprints, and especially corrections are
always welcome.
4) Surgical invariants of fourmanifolds, by Boguslaw Broda, preprint
available as hepth/9302092. (Revisited  see "week9")
Let me briefly recall the setup: we describe a compact 4manifold by a
handlebody decomposition, and represent this decomposition using a link
in S^3. The 2handles are represented by framed knots, while the
1handles are represented by copies of the unknot (which we may think of
as having the zero framing). The 1handles and 2handles play quite a
different role in constructing the 4manifold  which is why one
normally draws the former as copies of the unknot with a *dot* on them 
but Broda's construction does NOT care about this. Broda simply takes
the link, forgetting the dots, and cooks up a number from it, using
cabling and the Kauffman bracket at an root of unity. Let's call
Broda's invariant by b(M)  actually for each primitive rth root of
unity, we have b_r(M).
Broda shows that this is a 4manifold invariant by showing it doesn't
change under the de Sa moves. One of these consists of adding or
deleting a Hopf link 
/\ /\
/ \ / \
/ \ \
/ / \ \
\ \ / /
\ \ /
\ / \ /
\/ \/
in which both components have the zero framing and one represents a
1handle and the other a 2handle. This move depends on the fact that
we can "cancel" a 1handle and 2handle pair, a special case of a
general result in Morse theory.
But since Broda's invariant doesn't care which circles represent
1handles and which represent 2handles, Broda's invariant is also
invariant under adding two 2handles that go this way. This amounts to
taking a connected sum with S^2 x S^2. I.e., b(M) = b(M#S^2 x S^2).
Now, Ruberman told me a while back that we must also have b(M) =
b(M#CP2#CP2), that is, the invariant doesn't change under taking a
connected sum with a copy of CP2 (complex projective 2space) and an
orientationreversed copy of CP2. This amounts to adding or deleting a
Hopf link in which one component has the zero framing and the other has
framing 1. I didn't understand this, so I pestered Ruberman some more,
and this is what he says (modulo minor edits). I have not had time to
digest it yet:
The first question you asked was about the different framings on a 2handle
which goes geometrically once over a 1handle, i.e. makes a Hopf link in
which one of the circles is special (i.e is really a 1handle, i.e. in
AkbulutKirby's notation is drawn with a dot.) The answer is that the
framing doesn't matter, since the handles cancel. This is explained well
(in the PL case) in RourkeSanderson's book. (Milnor's book on the
hcobordism theorem explains it in terms of Morse functions, in the
smooth case.)
From this, it follows that b(M) = b(M#S^2 x S^2) = b(M#CP2#CP2). For
M is unchanged if you add a cancelling 1,2 pair, independent of the framing
on the 2handle. If you change the special circle to an ordinary one,
b(M) doesn't change. On the other hand, M has been replaced by its sum
with either S2 x S2 or CP2 # CP2, depending on whether the framing on the
2handle is even or odd. (Exercise: why is only the parity relevant?)
Now as I pointed out before, if one replaces all of the 1handles
(special circles) of a 4manifold with 2handles, the invariant doesn't
change. This operation corresponds to doing surgery on the 4manifold,
along the cores of the 1handles. In particular, the manifold has
changed by a cobordism. (This is a basic construction; when you do
surgery you produce a cobordism, in this case it's M x I U 2handles
attached along the circles which you surgered.)
From this, I will now show that Broda's invariant is determined by the
signature. (This is in the orientable case. Actually it seems that his
invariant is really an invariant of an oriented manifold.) The argument
above says that for any M, there is an M', with b(M) = b(M'), where M' has no
1handles, and where M and M' are cobordant. In particular, M' is simply
connected. So it suffices to show that b(N) = b(N') if N and N' are simply
connected.
So now you can assume you have two simply connected manifolds N,N' which are
cobordant via a 5dimensional cobordism W, which you can also assume simply
connnected. By highdimensional handlebody theory, you can get rid of the
1handles and 4handles of W, and assume that all the 2handles are
added, then all of the 3handles. If you add all the 2handles to N,
you get N#k(S^2 x S^2)#l(CP^2#(CP^2)) for some k and l. (Here is where
simple connectivity is relevant; the attaching circle of a 2handle is
nullhomotopic, and therefore isotopic to an unknotted circle. It's a
simple exercise to see what happens when you do surgery on a trivial
circle, ie you add on S2 x S2 or CP2 # CP2. On the other hand you get
the same manifold as the result of adding 2handles to N'. So
N#k(S^2 x S^2)#l(CP^2#(CP^2)) = N'#k'(S^2 x S^2)#l'(CP^2#(CP^2)),
so by previous remarks b(N) = b(N'), i.e b is a cobordism invariant.
Now: b is also multiplicative under connected sum, because connected sum just
takes the union of the link diagrams. The cobordism group is Z, detected by
the signature, so b must be a multiple of the signature, modulo some number.
(Maybe at this point I realize b should be b_r or some such). If you compute
(as a grad student Tianjin Li did for me) b_r(CP^2), you find that b_r
lives in the group of rth (or maybe 4rth; I'm at home and don't have my note)
roots of unity.
My conclusion: this invariant is a rather complicated way to compute the
signature of a 4manifold (modulo r or 4r) from a link diagram of the
manifold.
There is an important moral of the story, which is perhaps not obvious
to someone outside of 4manifolds. Any invariant which purports to go
beyond classical ones (ie invariants of the intersection form) must
treat CP^2 and CP^2 very differently. It seems to be the case that
many manifolds which are different (ie nondiffeomorphic) become
diffeomorphic after you add on CP^2. Thus any useful invariant should
get rather obliterated by adding CP^2. On the other hand,
nondiffeomorphic manifolds seem to stay nondiffeomorphic, no matter
how many CP2's you add on. This phenomenon doesn't seem to be
exhibited by any of the quantumgroup type constructions for
3manifolds; as it shouldn't, since (from the 3manifold point of view)
an unknot with framing + or 1 doesn't change the 3manifold. So if
you're looking for a combinatorial invariant, it seems critical that you
try to build in the asymmetry wrt orientation which 4manifolds seem to
possess.
Exercise: do the nonorientable case. The answer should be that b is
determined by the Euler characteristic, mod 2.

Previous editions of "This Week's Finds," and other expository posts
on mathematical physics, are available by anonymous ftp from
math.princeton.edu, thanks to Francis Fung. They are in the directory
/pub/fycfung/baezpapers. The README file will soon contain lists of the
papers discussed in each week of "This Week's Finds."
Please don't ask me about hepth and grqc; instead, read the
sci.physics FAQ or the file preprint.info in /pub/fycfung/baezpapers.
ompact 4manifold by a
handlebody decomposition, and representtwf_ascii/week100000064400020410000157000000313751077654020200141300ustar00baezhttp00004600000001Also available at http://math.ucr.edu/home/baez/week100.html
March 23, 1997
This Week's Finds in Mathematical Physics  Week 100
John Baez
Pretty much ever since I started writing "This Week's Finds" I've been
trying to get folks interested in ncategories and other aspects of
higherdimensional algebra. There is really an enormous world out
there that only becomes visible when you break out of "linear
thinking"  the mental habits that go along with doing math by writing
strings of symbols in a line. For example, when we write things in a
line, the sums a+b and b+a look very different. Then we introduce
a profound and mysterious equation, the "commutative law":
a + b = b + a
which says that actually they are the same. But in real life,
we prove this equation using higherdimensional reasoning:
a a a
a + b = + = + = + = b + a
b b b
If this seems silly, think about explaining to a kid what 9+17 means,
and how you could prove that 9+17 = 17+9. You might take a pile of 9
rocks and set it to the left of a pile of 17 rocks, and say "this is
9+17 rocks". Alternatively, you might put the pile of 9 rocks to the
right of the pile of 17 rocks, and say "this is 17+9 rocks". Thus to
prove that 9+17=17+9, you would simply need to *switch* the two piles
by moving one around the other.
This is all very simple. Historically, however, it took people a long
to really understand. It's one of those things that's too simple to
take seriously until it turns out to have complicated ramifications.
Now it goes by the name of the "EckmannHilton theorem", which says
that "a monoid object in the category of monoids is a commutative
monoid". You practically need a PhD in math to understand *that*!
However, lest you think that Eckmann and Hilton were merely dressing
up the obvious in fancy jargon, it's important to note that what they
did was to figure out a *framework* that turns the above "picture
proof" that a+b = b+a into an actual rigorous proof! This is one of
the goals of higherdimensional algebra.
The above proof that a+b = b+a uses 2dimensional space, but if you
really think about it also uses a 3rd dimension, namely time: the time
that passes as you move "a" around "b". If we draw this 3rd dimension
as space rather than time we can visualize the process of moving a
around b as follows:
a b
\ /
\ /
\ /
/
/ \
/ \
/ \
b a
This picture is an example of what mathematicians call a "braid".
This particular one is a boring little braid with only two strands and
one place where the two strands cross. It illustrates another major
idea behind higherdimensional algebra: equations are best thought of
as summarizing "processes" (or technically, "isomorphisms"). The
equation a+b = b+a is a summary of the process of switching a and b.
There is more information in the process than in the mere equation a+b
= b+a, because in fact there are two *different* ways to switch a and
b: the above way and
a b
\ /
\ /
\ /
\
/ \
/ \
/ \
b a
If one has a bunch of objects one can switch them around in a lot
of ways, getting lots of different braids.
In fact, the mathematics of braids, and related things like knots, is
crucially important for understanding quantum gravity in 3dimensional
spacetime. Spacetime is really 4dimensional, of course, but quantum
gravity in 4dimensional spacetime is awfully difficult, so in the
late 1980s people got serious about studying 3dimensional quantum
gravity as a kind of warmup exercise. It turned out that the math
required was closely related to some mysterious new mathematics
related to knots and "braidings". At first this must sound bizarre: a
deep relationshiop between knots and 3dimensional quantum gravity!
However, after you fight your way through the sophisticated
mathematical physics that's involved, it becomes clear why they are
related: both rely crucially on "3dimensional algebra", the algebra
describing how you can move things around in 3dimensional spacetime.
However, there is more to the story, because knot theory also seems
deeply related to *4dimensional* quantum gravity. Here the knots
arise as "flux tubes of area" living in 3dimensional space at a given
time. Recent work on quantum gravity suggests that as time passes
these knots (or more generally, "spin networks") move around and
change topology as time passes.
To really understand this, we probably need to understand
"4dimensional algebra". Unfortunately, not enough is known about
4dimensional algebra. The problem is that we don't know much about
4categories! To do ndimensional algebra in a really nice way, you
need to know about ncategories. Roughly speaking, an ncategory is
an algebraic structure that has a bunch of things called "objects", a
bunch of things called "morphisms" that go between objects, and
similarly 2morphisms going between morphisms, 3morphisms going
between 2morphisms, and so on up to the number n. You can think of
the objects as "things" of whatever sort you like, the morphisms as
processes going from one thing to another, the 2morphisms as
metaprocesses going from one process to another, and so on.
Depending on how you play the ncategory game, there are either no
morphisms after level n, or only simple and bland ones playing the
role of "equations". The idea is that in the world of ncategories,
one keeps track of things, processes, metaprocesses, and so on to the
nth level, but after that one calls it quits and uses equations.
So what is the definition of 4categories? Well, Eilenberg and
Mac Lane defined 1categories, or simply "categories", in a paper
that was published in 1945:
1) S. Eilenberg and S. Mac Lane, General theory of natural
equivalences, Trans. Amer. Math. Soc. 58 (1945), 231294.
Benabou defined 2categories  though actually he called them
"bicategories"  in a 1967 paper:
2) J. Benabou, Introduction to bicategories, Springer Lecture Notes in
Mathematics 47, New York, 1967, pp. 177.
Gordon, Power, and Street defined 3categories  or actually
"tricategories"  in a paper that came out in 1995:
3) R. Gordon, A. J. Power, and R. Street, Coherence for tricategories,
Memoirs Amer. Math. Soc. 117 (1995) Number 558.
This step took a long time in part because it took a long time for
people to understand deeply where *braidings* fit into the picture.
But what about 4categories and higher n? Well, the history is
complicated and I won't get it right, but let me say a bit anyway.
First of all, there are some things called "strict ncategories" that
people have known how to define for arbitrarily high n for quite a
while. In fact, people know how to go up to infinity and define
"strict omegacategories"; see for example:
4) S. E. Crans, On combinatorial models for higher dimensional
homotopies, Ph.D. thesis, University of Utrecht, Utrecht, 1991.
Strict ncategories are quite interesting and important, but I'm
mainly mentioning them here to emphasize that they are *not* what I'm
talking about. People sometimes often call strict ncategories simply
"ncategories", and call the more general ncategories I'm talking
about above "weak ncategories". However, I think the weak
ncategories will will eventually be called simply "ncategories",
because they are far more interesting and important than the strict
ones. Anyway, that's what I'm doing here.
Secondly, when you define ncategories you have to make some choice
about the "shapes" of your jmorphisms. In general they should be
some jdimensional things, but they could be simplices, or cubes, or
other shapes. In some ways the simplest shapes are "globes", a
jdimensional globe being a jdimensional ball with its boundary
divided into two hemispheres, the "inface" and "outface", which are
themselves (j1)dimensional globes. This corresponds to a picture
where each "process" has one input and one output, which are themselves
processes having the same input and output. The definitions of
category, bicategory, and tricategory work this way. In fact, Ross
Street came up with a very nice definition of ncategories for all n
using simplices in 1987:
5) Ross Street, The algebra of oriented simplexes, Jour. Pure
Appl. Alg. 49 (1987), 283335.
Since then, however, he and his students and collaborators seem to
have been working to translate this definition into the "globular"
formalism... while also making some other important adjustments too
technical to discuss here. In particular, Dominic Verity and Todd
Trimble have done a lot of work on getting the definition of
ncategory worked out, and a while ago I learned that Trimble came up
with a definition of "tetracategory" (or what I'm calling simply
"4category") in August of 1995. I don't think this has been
published, however.
James Dolan came to U. C. Riverside in the fall of 1993, and ever
since then, he and have been talking about ncategories and their role
in physics. Most of the category theory I know, I learned in this
process. It soon became clear that we needed a nice definition of
ncategory for all n in order to turn our hopes and dreams into
theorems. After a while we started working pretty hard on this. His
job was to come up with all the bright ideas, and mine was to get him
to explain them, to try to poke holes in them, and to figure out
rigorous proofs of all the things that were so obvious to him that he
couldn't figure out how (or why) to prove them. We sent a summarized
version of our definition to Ross Street at the end of 1995:
6) J. Baez and J. Dolan, nCategories  sketch of a definition,
letter to Ross Street, Nov. 29, 1995, available at
http://math.ucr.edu/home/baez/ncat.def.html
and then for a year I worked on trying to write up a longer, clearer
version, while all the meantime Dolan kept coming up with new ways of
looking at everything. I finished in February of this year:
7) J. Baez and J. Dolan, Higherdimensional algebra III: nCategories
and the algebra of opetopes, to appear in Adv. Math., preprint
available as qalg/9702014 and at http://math.ucr.edu/home/baez/op.ps,
or in compressed form as http://math.ucr.edu/home/baez/op.ps.Z
The key feature of this definition is that it uses "jdimensional
opetopes" as the shapes for jmorphisms. These shapes are very handy
because the (j+1)dimensional opetopes describe all the legal ways of
sticking together a bunch of jdimensional opetopes to form another
jdimensional opetope! They are related to the theory of "operads",
which is part of the reason for their name. (By the way, the first
two syllables are pronounced exactly as in "operation".)
In the meantime, Michael Makkai and John Power had begun work using
our definition. Also, other definitions of "ncategory" have appeared
on the scene! Zouhair Tamsamani came up with one in terms of
"multisimplicial sets":
8) Z. Tamsamani, Sur des notions de $\infty$categorie et
$\infty$groupoide nonstrictes via des ensembles multisimpliciaux,
Ph.D. thesis, Universite Paul Sabatier, Toulouse, France, 1995.
Michael Batanin also has a definition of omegacategories, of
the "globular" sort:
9) M. A. Batanin, On the definition of weak omegacategory, Macquarie
Mathematics Report number 96/207.
Now the fun will begin! These different definitions of (weak) ncategory
should be equivalent, albeit in a rather subtle sense, so we should check
to see if they really are. Also, we need to develop many more tools for
working with ncategories. Then we can really start using them as a tool.
When I started writing this I thought I was going to explain the
definition that Dolan and I came up with. Now I'm too tired! It
takes a while to explain, so I think I'll stop here and save that for
some other week or weeks. Perhaps I'll mix it in with my report on
the Workshop on Higher Category Theory and Physics, which is taking
place next weekend at Northwestern University.

Previous issues of "This Week's Finds" and other expository articles on
mathematics and physics, as well as some of my research papers, can be
obtained by anonymous ftp from math.ucr.edu; they are in the
subdirectory pub/baez. The README file lists the contents of all the
papers. On the WorldWide Web, you can get these files by going to
http://math.ucr.edu/home/baez/README.html
For a table of contents of all the issues of This Week's Finds, try
http://math.ucr.edu/home/baez/twf.html
A simple jumpingoff point to the old issues is available at
http://math.ucr.edu/home/baez/twfshort.html
If you just want the latest issue, go to
http://math.ucr.edu/home/baez/this.week.html
twf_ascii/week101000064400020410000157000000251570774011336300141330ustar00baezhttp00004600000001Newsgroups: sci.math.research
Subject: This Week's Finds in Mathematical Physics (Week 101)
From: baez@math.ucr.edu (John Baez)
Keywords: Also available at http://math.ucr.edu/home/baez/week101.html
Date: April 9, 1997
This Week's Finds in Mathematical Physics  Week 101
John Baez
Darwinian evolution through natural selection is an incredibly powerful
way to explain the emergence of complex organized structures. However,
it is not the *only* important process that naturally gives rise to
complex structures. Maybe when we study biology we should also look for
other ways that order can spontaneously arise.
After all, there are plenty of complex structures in the nonbiological
world. When it snows, we see lots of beautiful snowflakes with similar
but different hexagonal structures. Do we conclude that snowflakes
*evolved* to be hexagonal through natural selection? No.
But wait! Maybe in some sense a hexagonal snowflake is "more fit" in
certain weather conditions. Perhaps this shape is more efficient at
getting water molecules to adhere to it than other shapes. We can
think of different snowflakes as engaged in "competition" for water
molecules, and the ones that grow fastest as the "winners". In fact,
the exact shapes of snowflakes in a snowstorm depend crucially on
the temperature, humidity and so on... so who the "winners" are depends
on the environment, just as in Darwinian evolution!
A biologist will reply: fine, but this is still not "Darwinian
evolution". For Darwinian evolution in the strict sense, we require
that there be a "lineage". Darwinian evolution applies only to entities
that reproduce and pass some of their traits down to descendants. The
idea is that over the course of many generations, traits that aid
reproduction will accumulate, while traits that hinder it will be weeded
out. Snowflakes don't have kids. A oneshot competition for resources,
followed by melting into oblivion the next day, is not what Darwinian
evolution is about.
Okay, okay, so it's not Darwinian evolution. But it's still
interesting. It's showing us that Darwinian evolution is just *one*
of various ways that order can arise. So we shouldn't study Darwinian
evolution in isolation. We should study *all* the ways that systems
spontaneously generated complex patterns, and see how they relate. If
we do that, perhaps we'll see a bunch of interesting relationships
between physics and chemistry and biology. Also, maybe we'll get a better
handle on how life arose in the first place... that curious transition
from chemistry to biology.
If I wasn't so hooked on quantum gravity I would love to work on this
stuff. It's obviously cool, and obviously a lot more *practical* than
quantum gravity. The origin of complexity a very hot topic these days.
But alas, I am just an oldfashioned guy in love with simplicity.
Whenever I see a new journal come out with a title like "Complex
Systems" or "Journal of Complexity" or "Santa Fe Institute Studies in
the Science of Complexity", I heave a wistful sigh and dream of starting
a journal entitled "Simplicity".
Actually, the fun lies in the interplay between complexity and
simplicity: how complex phenomena can arise from simple laws, and
sometimes obey new simple laws of their own. I like to hang out on the
simple end of things, but that doesn't stop me from enjoying the new
work on complexity. At one point I got a big kick out of Manfred
Eigen's work on "hypercycles"  systems of chemicals that catalyze
each others formation. (You may remember Eigen as the discoverer of the
"Eigenvalue"... in which case I pity you.) Presumably life started as
some sort of hypercycle, so the mathematical study of the competition
between hypercycles may shed some light on why there is only one genetic
code. There is a lot of nice math of this type in:
1) Manfred Eigen, The Hypercycle, a Principle of Natural
SelfOrganization, SpringerVerlag, Berlin, 1979.
Another name that comes up in this context is Ilya Prigogine, mainly for
his work on nonequilibrium thermodynamics and the spontaneous formation
of patterns in dissipative systems. The following are just a few of his
many books:
2) G. Nicolis and I. Prigogine, SelfOrganization in Nonequilibrium
Systems: from Dissipative Structures to Order Through Fluctuations ,
Wiley, New York, 1977.
Ilya Prigogine, From Being to Becoming: Time and Complexity in the
Physical Sciences, W. H. Freeman, San Francisco, 1980.
Ilya Prigogine, Introduction to Thermodynamics of Irreversible
Processes, 3d ed., Interscience Publishers, New York, 1967.
A bit more recently, the work of Stuart Kauffman has dominated the
subject. It's really him who has pushed for the unified study of the
whole gamut of methods of spontaneous generation of order, particularly
in the context of biological systems. He's written two books. The
latter, in particular, includes a lot of math problems just *waiting*
to be tackled by good mathematicians and physicists.
3) Stuart A. Kauffman, At Home in the Universe: the Search for Laws of
SelfOrganization and Complexity, Oxford University Press, New York,
1995.
Stuart A. Kauffman, The Origins of Order: SelfOrganization and
Selection in Evolution, Oxford University Press, New York, 1993.
If nonDarwinian forms of spontaneous patternformation can be important
in biology, can Darwinian evolution be important in nonbiological
contexts? Well, as I mentioned in "week31" and "week33", the physicist
Lee Smolin has an interesting hypothesis about how the laws of nature
may have evolved to their present point by natural selection. The idea
is that black holes beget new "baby universes" with laws similar but not
necessarily quite the same as their ancestors. Now this is extremely
speculative, but it has the saving virtue of making a lot of testable
predictions: it predicts that all the constants of nature are tuned so
as to maximize black hole production. Smolin has just come out with
a book on this, which also happens to be a good place to learn about
his work on quantum gravity:
4) Lee Smolin, The Life of the Cosmos, Crown Press, 1997.
Interestingly, Stuart Kauffman and Lee Smolin have teamed up to
write a paper on the problem of time in quantum gravity:
5) Stuart Kauffman and Lee Smolin, A possible solution to the problem
of time in quantum cosmology, preprint available as grqc/9703026.
Right now you can also read this paper on John Brockman's website called
"Edge". This website features all sorts of fun interviews and
discussions. For example, if you look now you'll find an intelligent
interview with my favorite musician, Brian Eno. More to the point,
a discussion of Kauffman and Smolin's paper is happening there now.
As a longtime fan of USENET newsgroups and other electronic forms of
chitchat, I'm really pleased to see how Brockman has set up a kind of
modernday version of the French salon.
6) Edge: http://www.edge.org
Okay. Now... what's even more fashionable, trendy, and close to
the cutting edge than complexity theory? You guessed it: homotopy
theory! Currently known only to hippest of the hip, this is bound to
hit the bigtime as soon as they figure out how to make flashy color
graphics illustrating the Adams spectral sequence.
Last week I went to the Workshop on Higher Category Theory and Physics
at Northwestern University, and also, before that, part of a conference
on homotopy theory they had there. Actually these two subjects are
closely related: homotopy theory is a highly algebraic way of studying
the topology of spaces of various dimensions, and lots of what we understand
about "higher dimensional algebra" comes from homotopy theory. So it was
a nice combination.
Lots of the homotopy theory was over my head, alas, but what I
understood I enjoyed. It may seem sort of odd, but the main thing I got
out of the homotopy theory conference was an explanation of why the
number 24 is so important in string theory! In bosonic string theory
spacetime needs to be 26dimensional, but subtracting 2 dimensions for
the surface of the string itself we get 24, and it turns out that it's
really the special properties of the number 24 that make all the magic
happen.
I began to delve into these mysteries in "week95". There, however, I
was mainly reporting on very fancy stuff that I barely understand, stuff
that seems like a pile of complicated coincidences. Now, I am glad to
report, I am beginning to understand the real essence of this 24
business. It turns out that the significance of the number 24 is woven
very deeply into the basic fabric of mathematics. To put it rather
mysteriously, it turns out that every integer has some subtle "hidden
symmetries". These symmetries have symmetries of their own, and in turn
THESE symmetries have symmetries of THEIR own  of which there are
exactly 24.
Hmm, mysterious. Let me put it another way. It probably won't be
obvious why this is another way of saying the same thing, but it has the
advantage of being more concrete. Suppose that the integer n is
sufficiently large  4 or more will do. Then there are 24 essentially
different ways to wrap an (n+3)dimensional sphere around an
ndimensional sphere. More precisely still, given two continuous
functions from an (n+3)sphere to an nsphere, let's say that they lie
in the same "homotopy class" if you can continuously deform one into
another. Then when n is 4 or more, it turns out that there are exactly
24 such homotopy classes.
Now that I have all the ordinary mortals confused and all the homotopy
theorists snickering at me for making such a big deal out of something
everyone knows, I should probably go back and explain what the heck I'm
getting at, and why it has to do with string theory. But I'm getting
worn out, and your attention is probably flagging, so I'll do this next
time. I'll say a bit about homotopy theory, stable homotopy theory, the
sphere spectrum, and why Andre Joyal says we should call the sphere
spectrum the "integers" (thus explaining my mysterious remark above).

Previous issues of "This Week's Finds" and other expository articles on
mathematics and physics, as well as some of my research papers, can be
obtained by anonymous ftp from math.ucr.edu; they are in the
subdirectory pub/baez. The README file lists the contents of all the
papers. On the WorldWide Web, you can get these files by going to
http://math.ucr.edu/home/baez/README.html
For a table of contents of all the issues of This Week's Finds, try
http://math.ucr.edu/home/baez/twf.html
A simple jumpingoff point to the old issues is available at
http://math.ucr.edu/home/baez/twfshort.html
If you just want the latest issue, go to
http://math.ucr.edu/home/baez/this.week.html
twf_ascii/week102000064400020410000157000000627401155545006700141370ustar00baezhttp00004600000001Also available at http://math.ucr.edu/home/baez/week102.html
April 21, 1997
This Week's Finds in Mathematical Physics  Week 102
John Baez
In "week101" I claimed to have figured out the real reason for the
importance of the number 24 in string theory. Now I'm not so sure 
some pieces of the puzzle that I thought would fit together don't seem
to be fitting. Maybe if I explain what I know so far, some experts will
hand me some of the missing pieces, or tell me where the ones I have go.
Most of the puzzle pieces came from a talk at a conference on homotopy
theory that I went to:
1) Ulrike Tillmann, The moduli space of Riemann surfaces  a homotopy
theory approach, talk at Northwestern University Algebraic Topology
Conference, March 27, 1997
However, some conversations with Andre Joyal during this conference
really helped turn my attention towards what might be going on here.
Let's start by recalling some stuff about homotopy groups of spheres.
There are often lots of topologically different ways of wrapping an
mdimensional sphere around a kdimensional sphere. For example, if
m = k = 1, we're talking about the ways of wrapping a circle around a
circle. These are classified by an integer called the "winding number".
We can make this concrete by thinking of the circle as the unit circle
in the complex plane. Take your favorite integer and call it n. Then
the function
f(z) = z^n
maps the unit circle (the complex numbers with z = 1) to itself. If n
is positive, this function wraps the unit circle around itself n times
in the counterclockwise direction. If n is negative, the circle gets
wrapped around in the other direction. If n is zero, f(z) = 1, so we
have a constant function  no "wrapping around" at all!
It turns out that any continuous function from the circle to itself can
be continuously deformed to exactly one of these functions f(z) = z^n.
Homotopy theory is all about such continuous deformations. In the
jargon of homotopy theory, we say two functions from some space to some
other space are "homotopic" if we can continuously deform the first
function to the second. Another way of putting it is that the two
functions lie in the same "homotopy class". Speaking of jargon, real
topologists never say "continuous function": instead, they say "map".
So, using this jargon: we know the homotopy class of a map from the
circle to itself if we know its winding number.
Now: what happens if we go to higher dimensions? What are all the
homotopy classes of maps from the mdimensional sphere to the
kdimensional sphere? Spheres are pretty simple spaces, so one might at
first guess there is some simple answer to this question for all m and
k.
Unfortunately, it's far from simple. In fact, nobody knows the answer
for all m and k! People *do* know the answer for zillions of particular
values of m and k. But there is no simple pattern to it: instead, there
is an incredibly complicated and beautiful weave of subtle patterns,
which we have not gotten to the bottom of... and perhaps never will.
To get a little feel for this, let's bring in some standard notation:
folks use pi_m(X) to denote the set of homotopy classes of maps from an
mdimensional sphere to the space X. When m > 0, this set is actually a
group, called the "mth homotopy group" of X. These groups are of major
importance in algebraic topology.
So, what we are talking about is pi_m(S^k): the set of all homotopy
classes of ways of wrapping an msphere around an ksphere. I already
implicitly said that
pi_1(S^1) = Z
where Z stands for the integers, since the winding number is an
integer. The same thing happens if we go up a dimension:
pi_2(S^2) = Z
In other words: you can wrap a 2sphere (an ordinary sphere) n times around
itself for any integer n. How? Well, say we use spherical coordinates
and describe a point on the sphere using its angle phi from the north
pole, together with the angle theta saying how far east it is from
Greenwich. Then the map
f(phi,theta) = (phi, n theta)
does the job. Any map from S^2 to itself is homotopic to exactly
one of these.
The same basic idea works in any higher dimension, too:
pi_k(S^k) = Z for any k >= 1
In other words, there is always an integer n that plays the role of the
"winding number" of a map from the ksphere to itself  though only
uncouth physicists call it the "winding number"; mathematicians call
it the "degree".
So far, so good. Now, what about mapping a sphere to another sphere of
*higher* dimension? This is nice and simple:
pi_m(S^k) = {0} whenever m < k
The {0} there is just a standard way to write a set with only one
element, which we call "zero". So what we mean is that there's only
*one* homotopy class of ways to map a sphere to a sphere of higher
dimension. There is always enough "room" to wiggle around one map until
it looks like another.
What about mapping a sphere to another sphere of *lower* dimension?
Here is where the trouble starts!  or the fun, depending on your
attitude towards complexity. For example, there is only one homotopy
class of maps from a 2sphere to a circle:
pi_2(S^1) = {0}
There is just no way a 2sphere can get interestingly "stuck" on the
"hole" of the circle. This may seem obvious. But it's not really quite
as obvious at it seems, because if we move up one dimension, we have:
pi_3(S^2) = Z
This came as a big shock when Heinz Hopf first discovered it in the
1930's; before then, people had no idea how sneaky homotopy groups were!
There is a beautiful way to compute an integer called the "Hopf invariant"
that keeps track of the homotopy class of a map from the 3sphere to the
2sphere. There are lots of nice ways to compute it, but alas, I only
have time to briefly sketch one! Suppose that the map f: S^3 >S^2 is
smooth (otherwise we can always smooth it up). Then most points p in
S^2 have the property that the points x in S^3 with f(x) = p form a
"link": a bunch of knots in S^3. If we take two different points in S^2
with this property, we get two links. From these two links we can
compute an integer called the "linking number": for example, we can just
draw these two links and count the times one crosses over or under the
other (with appropriate plus or minus signs for each crossing). This
number turns out not to depend on how we picked the two points!
Moreover, it only depends on the homotopy class of f. It's called the
Hopf invariant of f.
Moving up one dimension, it turns out that
pi_4(S^3) = Z/2
Here Z/2 is the group with two elements, usually written 0 and 1,
with addition mod 2. Why only two homotopy classes of maps from
S^4 to S^3? Well, you can compute something like the Hopf invariant
for these maps, exactly as we did before, but the thing is, links
in 4 dimensions are easy to unlink. You can unlink something like
\ /
\ /
\
/ \
/ \
\ /
\ /
\
/ \
/ \
and make it look like
 
 
 
 
 
 
 
 
 
 
 
so the linking number in 4 dimensions is only defined mod 2. Thus
the "Hopf invariant" is only defined mod 2.
The exact same thing happens in higher dimensions, too, so in fact we
have:
pi_{k+1}(S^k) = Z/2 for any k >= 3
This illustrates an important general fact: when the dimensions get high
enough, there's more room to wiggle things around, and as we keep
jacking up the dimension, homotopy groups simplify a bit and settle down
after a while. This is the idea behind "stable homotopy theory".
Let's look at some more examples. We have
pi_3(S^1) = {0}
pi_4(S^2) = Z/2
pi_5(S^3) = Z/2
pi_6(S^4) = Z/2
and so on:
pi_{k+2}(S^k) = Z/2 for any k >= 2
Sadly, I do *not* understand why this is true. How do you wrap a
4sphere around a 2sphere in an interesting way? Dunno.
(Thanks to Dan Christensen, an answer appears at the end of this post.)
Plunging on undeterred, we have:
pi_4(S^1) = {0}
pi_5(S^2) = Z/2
pi_6(S^3) = Z/12
pi_7(S^4) = Z + Z/12
pi_8(S^5) = Z/24
pi_9(S^6) = Z/24
and so on:
pi_{k+3}(S^k) = Z/24 for any k >= 5.
Here is where the magic number 24 comes in! What the above says
is that if k is large enough, there are exactly 24 different homotopy
class of maps from an (k+3)sphere to an ksphere!
Now I should explain what this has to do with string theory. But first
I should say more about the homotopy groups of spheres. There are some
simple patterns worth knowing about. First,
pi_m(S^1) = {0} for any m >= 2.
Second, there is a nice formula for when the homotopy groups settle
down as we jack up the dimension:
pi_{k+n}(S^k) is independent of k as long as k >= n+2
The homotopy groups can stabilize sooner, as we saw for n = 2, but never
later, and often they stabilize right at k = n+2. There is a simple
reason for this. We saw that pi_{k+1}(S^k) stabilized at k = 3 because
it's easy to unlink links in 4 or more dimensions. Similarly,
pi_{k+n}(S^k) must stabilize by the time k = n+2, because it's easy to
untie knotted ndimensional surfaces in 2n+2 or more dimensions!
For more on stable homotopy groups of spheres, try:
2) Douglas C. Ravenel, Complex cobordism and stable homotopy groups of
spheres, Academic Press, Orlando, 1986.
Douglas C. Ravenel, Nilpotence and periodicity in stable homotopy
theory, Princeton University Press, Princeton, 1992.
Ravenel also spoke at this conference and is a real expert on stable
homotopy groups of spheres. Unfortunately his talk was too highpowered
for me. The 2nd book above is a bit more forgiving to the amateur,
but the first one has lots of nice tables of stable homotopy groups
of spheres.
The relationship between homotopy groups of spheres and higher
dimensional knot theory is a wonderful thing. James Dolan and I are
learning a lot about ncategories by pondering it. When I spoke to
him at the conference at Northwestern, it became clear that Andre
Joyal had also thought about it very deeply. Joyal is famous for his
work relating category theory, combinatorics and topology, and his way
of thinking about the homotopy groups of spheres reflects these
interests. He said a very fascinating thing; he said "really we
should call the sphere spectrum the 'true integers'". I would like to
explain this... but here things get a bit technical, and I am afraid
they will get a lot more technical when I get around to the string
theory stuff.
What's the "sphere spectrum"? Well, roughly it's just the list
of spheres S^0, S^1, S^2, ..., but the word "spectrum" refers to the
way all these spaces are all related, all aspects of one big thing.
Here's a nice way to think of it. Start with the integers. Normally
we think of these as just a set, or actually a group, since we can add
them. But if we avoid the sin of mistaking isomorphism for equality
we can think of them as a category.
I already began to explain this in my parable about the shepherd in
"week99". The shepherd started with the category of finite sets and
"decategorified" it to obtain the set of natural numbers, associating to
each finite set a natural number, its number of elements. Taking
disjoint unions of sets corresponds to addition, the empty set
corresponds to zero, and so on.
Okay. What are the *integers* the decategorification of?
Well, we can imagine finite sets that have both "positive" and
"negative" elements. The "number of elements" of such a set will be the
number of positive elements minus the number of negative elements. This
is a bit weird if we're talking about sheep, but perhaps not so weird if
we talk about positrons and electrons, which can annihilate each other.
(In "week92" I explain what I'm hinting at here: the relation between
antiparticles and adjunctions.)
Topologists prefer to speak of "positively and negatively oriented
points". We can draw a set of positively and negatively oriented points
like this:
 + + + +  
We can add them by setting them side by side. But how do the positively
and negatively oriented points cancel? Well, remember, we're trying to
get a category! If finite lists of positively and negatively oriented
points are our objects, what are our morphisms? How about tangles, like
this:
 + + + +  
\ /   \ / 
\ /   \ / 
\/  \ \/ /
 \ /
 \ /
 \ /
 \ /
/\  \ /
/ \  \ /
/ \  \/
/ \ 
+  +
These let us cancel or create positive and negative points in pairs.
Voila! The categorified integers! Just as the integers form a monoid
under addition, these form a monoidal category (see "week89" for these
concepts). The monoidal structure here is disjoint union, which we can
denote with a plus sign if we like. Similarly, we can write the empty
set as 0.
Above it looks like I'm drawing a 1dimensional tangle in 2dimensional
space. To understand the "commutativity" of the categorified integers
we should work with 1dimensional tangles in higherdimensional space.
If we consider them in 3dimensional space, we have room to switch
things around:
+ +
\ /
\ /
\ /
/
/ \
/ \
/ \
+ +
This gets us commutativity, as I explained in "week100". Technically
speaking, we get a "braided" monoidal category. However, there are two
different ways to switch things around; for example, in addition to the
above way there is
+ +
\ /
\ /
\ /
\
/ \
/ \
/ \
+ +
To get rid of this problem (if you consider it a problem) we can work
with 1dimensional tangles in 4dimensional space, where we can deform
the first way of switching things to the second. We get a "symmetric"
monoidal category. Working in higher dimensions doesn't change
anything: things have stabilized.
If we impose the extra condition that the morphisms
/\
/ \
/ \
/ \
+ 
and
+ 
\ /
\ /
\ /
\/
are inverses, as are
/\
/ \
/ \
/ \
 +
and
 +
\ /
\ /
\ /
\/
then all morphisms become invertible, so we have not just a monoidal
category but a monoidal groupoid  a groupoid being a category with all
morphisms invertible (see "week74"). In fact, not only are morphisms
invertible, so are all objects! By this I mean that every object x
has an object x such that x + x and x + x are isomorphic to 0.
For example, if x is the positively oriented point, x is the negatively
oriented point, and vice versa. So we have not just a monoidal groupoid
but a "groupal groupoid". (I have adopted this charming terminology
from James Dolan.)
Very nice. We seem to have avoided the sin of decategorification, and
are no longer treating the integers as a mere *set* (or group, or
commutative group). We are treating them as a *category* (or groupal
groupoid, or braided groupal groupoid, or symmetric groupal groupoid).
On the other hand, it's a bit odd to say that
/\
/ \
/ \
/ \
+ 
and
+ 
\ /
\ /
\ /
\/
are inverses. This amounts to saying that the morphism:
/\
/ \
/ \
/ \
\ /
+\ /
\ /
\/
is equal to the identity morphism from 0 to 0, which corresponds to the
empty picture:
Hmm. They sure don't *look* equal. We must be doing something wrong.
What are we doing wrong? We're committing the sin of
decategorification: treating isomorphisms as equations! We should treat
the integers not as a mere category, but as a 2category! See "week80"
for the precise definition of this concept; for now, it's enough to say
that a 2category has things called 2morphisms going between morphisms.
If we treat the integers as a 2category, we can say there is a
2morphism going from
/\
/ \
/ \
/ \
\ /
+\ /
\ /
\/
to the identity morphism. This 2morphism has a nice geometrical
description in terms of a 2dimensional surface: the surface in
3d space that's traced out as the above picture shrinks down to the
empty picture. It's hard to draw, but let me try:
/\
/ \ /\
/ \ / \ /\
/ \ => / \ => / \ => /\ =>
\ / \ / \ / +\/
+\ / +\ / +\/
\ / \/
\/
Okay, say we do this: treat the integers as a 2category. We again are
faced with a question: do we make all the 2morphisms invertible? If we
do, we get a "2groupoid", or actually a "groupal 2groupoid". But
again, to do so amounts to committing the sin of decategorification. To
avoid this sin, we should tread the integers as a 3category. Etc, etc!
You may have noted how worlds of ever higherdimensional topology are
automatically unfolding from our attempt to avoid the sin of
decategorification. This is what's so neat about ncategories. I
haven't told you how it all works, but let me summarize what's actually
happening here. Normally we treat the integers as the free group on
one generator, or else the free commutative group on one generator.
But groups and commutative groups are just the tip of the iceberg!
The following picture is similar to that in "week74":
ktuply groupal ngroupoids
n = 0 n = 1 n = 2
k = 0 sets groupoids 2groupoids
k = 1 groups groupal groupal
groupoids 2groupoids
k = 2 commutative braided braided
groups groupal groupal
groupoids 2groupoids
k = 3 `' symmetric weakly
groupal involutory
groupoids groupal
2groupoids
k = 4 `' `' strongly
involutory
groupal
2groupoids
k = 5 `' `' `'
What are all these things? Well, an ngroupoid is an ncategory with
all morphisms invertible, at least up to equivalence. An (k+n)groupoid
with only trivial jmorphisms for j < k can be seen as a special sort of
ngroupoid, which we call a "ktuply groupal ngroupoid".
Part of Joyal's point was that we should really think of the integers as
the "free ktuply monoidal ngroupoid on one object". Here the idea is
not to fix n and k once and for all  this would only prevent us from
understanding the subtleties that show up when we increase n and k!
Instead, we should think of them as variable, or perhaps consider the
limit as they become large.
The other part of his point was that there's a correspondence between
ngroupoids and the information left in topological spaces when we
ignore all homotopy groups above dimension n  socalled "homotopy
ntypes". Using this correspondence, the "free ktuply monoidal
ngroupoid on one object" corresponds to the homotopy (k+n)type of the
ksphere. Moreover, if we keep jacking up k, this stabilizes when k >=
n+2. Actually, as the dittos in the above chart suggest, it's a quite
general fact that the notion of ktuply monoidal ngroupoid stabilizes
for k >= n+2.
Yet another point is that the pictures above explain the relation
between higherdimensional knot theory and the homotopy groups of
spheres in a very vivid, direct way.
Okay. What about string theory and the magic number 24? Well, notice
that the pictures above started looking a bit like strings! Hmm....
Here's the idea, as far as I understand it. Presumably all but the
hardy have stopped reading this article by now, so I will pull out all
the stops. The string worldsheet is a Riemann surface so we'll need
some stuff about Riemann surfaces from "week28". Let M(g,n) be the
moduli space of Riemann surfaces with genus g and n punctures, and let
G(g,n) be the corresponding mapping class group. Since M(g,n) is the
quotient of Teichmueller space by G(g,n) and Teichmueller space is
contractible, we have
M(g,n) = BG(g,n)
where "B" means "classifying space". There's a natural inclusion
G(g,n) > G(g+1,n)
defined by sewing an torus with two punctures onto your genusg surface
with n punctures, which increases the genus by 1. Let's define
G(infinity,n) to be direct limit as g > infinity, and let
M(infinity,n) = BG(infinity,n).
Now it turns out M(infinity,1) has a kind of product on it.
The reason is that there are products
M(g,1) x M(h,1) > M(g+h,1)
given sewing two surfaces together with a 3punctured sphere.
Using this product we can define the group completion
M(infinity,1)+
and the result Tillmann stated which got me so excited was
that
pi_3(M(infinity,1)+) = Z/24 + H
for some unknown group H. Since this is really a result about the
mapping class groups of surfaces, it *must* have something to do with
how conformal field theories always give projective representations of
these mapping class groups, with the "phase ambiguity" always being a
24th root of unity. No? I just don't quite see why. Maybe someone
will enlighten me.
Anyway, the way she proved this definitely ties right into the stuff
about stable homotopy groups of spheres. She used explicit maps between
the third stable homotopy group of spheres
pi_{k+3}(S^k) = Z/24 for k >= 5
and pi_3(M(infinity,1)+)! And the way she got the map from the latter
to the former amounts to working with pictures I was drawing above.
To put it more precisely, in
3) Higherdimensional algebra and topological quantum field theory, by
John Baez and James Dolan, Jour. Math. Phys. 36 (1995), 60736105.
we argue that framed ndimensional surfaces embedded in (n+k)dimensions
should be described by the free ktuply monoidal ncategory with duals
on one object. This should map down to the free ktuply groupal
ngroupoid on one object, by the usual yoga of "freeness". Taking n = 3
and k sufficiently large, we should obtain a homomorphism from the
mapping class group of any Riemann surface to the third stable homotopy
group of spheres! Presumably the idea is that in the limit of
large genus this homomorphism is onto!
Of course, Tillmann doesn't prove her result using the stillnascent
formalism of ncategories, but I think it will eventually be possible.
(Also, my rough argument applies to Riemann surfaces with no punctures,
while she considers those with one puncture, but various things she said
make me think this might not be such a big deal.) The real puzzle is
this: what does
pi_3(M(infinity,n)+)
have to do with central extensions of G(g,n) for finite g? If I could
figure this out I'd be very happy.

Quote of the Week:
Think of one and minus one. Together they add up to zero, nothing,
nada, niente, right? Picture them together, then picture them
separating, peeling part.... Now you have something, you have two
somethings, where you once had nothing.  John Updike, Roger's Version
/\
/ \
/ \
/ \
+ 

Addendum  Dan Christensen answered a puzzle above. Here's how to get
a nontrivial element of
pi_4(S^2)
Take the map f: S^3 > S^2 generating
pi_3(S^2)
and compose it with the map g: S^4 > S^3 generating
pi_4(S^2)
(which, by the way, is obtained from f by "suspension") to obtain the
desired map from S^4 to S^2. This is an instance of a very general
trick: composing elements of homotopy groups of spheres to get new
ones!

Previous issues of "This Week's Finds" and other expository articles on
mathematics and physics, as well as some of my research papers, can be
obtained at
http://math.ucr.edu/home/baez/
For a table of contents of all the issues of This Week's Finds, try
http://math.ucr.edu/home/baez/twfcontents.html
A simple jumpingoff point to the old issues is available at
http://math.ucr.edu/home/baez/twfshort.html
If you just want the latest issue, go to
http://math.ucr.edu/home/baez/this.week.html
*set* (or group, or
commutativetwf_ascii/week202000064400020410000157000000754761137742704300141520ustar00baezhttp00004600000001Also available at http://math.ucr.edu/home/baez/week202.html
February 21, 2004
This Week's Finds in Mathematical Physics  Week 202
John Baez
This week I'll deviate from my plan of discussing number theory, and
instead say a bit about something else that's been on my mind lately:
structure types. But, you'll see my fascination with Galois theory
lurking beneath the surface.
Andre Joyal invented structure types in 1981  he called them "especes
de structure", and lots of people call them "species". Basically, a
structure type is just any sort of structure we can put on finite sets:
an ordering, a coloring, a partition, or whatever. In combinatorics
we count such structures using "generating functions". A generating
function is a power series where the coefficient of x^n keeps track of
how many structures of the given kind we can put on an nelement set.
By playing around with these functions, we can often figure out the
coefficients and get explicit formulas  or at least asymptotic formulas 
that count the structures in question.
The reason this works is that operations on generating functions come
from operations on structure types. For example, in "week190", I
described how addition, multiplication and composition of generating
functions correspond to different ways to get new structure types from
old.
Joyal's great contribution was to give structure types a rigorous
definition, and use this to show that many calculations involving
generating functions can be done directly with structure types. It
turns out that just as generating functions form a *set* equipped with
various operations, structure types form a *category* with a bunch of
completely analogous operations. This means that instead of merely
proving *equations* between generating functions, we can construct
*isomorphisms* between their underlying structure types  which imply
such equations, but are worth much more. It's like the difference
between knowing two things are equal and knowing a specific reason WHY
they're equal!
Of course, this business of replacing equations by isomorphisms is called
"categorification". In this lingo, structure types are categorified power
series, just as finite sets are categorified natural numbers.
A while back, James Dolan and I noticed that since you can use power
series to describe states of the quantum harmonic oscillator, you can
think of structure types as states of a categorified version of this
physical system! This gives new insights into the combinatorial
underpinnings of quantum physics.
For example, the discrete spectrum of the harmonic oscillator
Hamiltonian can be traced back to the discreteness of finite sets!
The commutation relations between annihilation and creation operators
boil down to a very simple fact: there's one more way to put a ball in
a box and then take one out, than to take one out and then put one in.
Even better, the whole theory of Feynman diagrams gets a simple
combinatorial interpretation. But for this, one really needs to go
beyond structure types and work with a generalization called "stuff
types".
I've been thinking about this business for a while now, so last fall
I decided to start giving a yearlong course on categorification and
quantization. The idea is to explain bunches of quantum theory,
quantum field theory and combinatorics all from this new point of view.
It's fun! Derek Wise has been scanning in his notes, and a bunch of
people have been putting their homework online. So, you can follow
along if you want:
1) John Baez and Derek Wise, Categorification and Quantization.
Fall 2003 notes: http://math.ucr.edu/home/baez/qgfall2003/
Winter 2004 notes: http://math.ucr.edu/home/baez/qgwinter2004/
Spring 2004 notes: http://math.ucr.edu/home/baez/qgspring2004/
I'd like to give you a little taste of this subject now. But, instead
of explaining it in detail, I'll just give some examples of how structure
types yield some farout generalizations of the concept of "cardinality".
This stuff is a continuation of some themes developed in "week144",
"week185", "week190", so I'll start with a review.
Suppose F is a structure type. Let F_n be the *set* of ways we can put
this structure on a nelement set, and let F_n be the *number* of ways
to do it. In combinatorics, people take all these numbers F_n and pack
them into a single power series. It's called the generating function of
F, and it's defined like this:
F_n
F(x) = sum  x^n
n!
It may not converge, so in general it's just a "formal" power series 
but for interesting structure types it often converges to an interesting
function.
What's good about generating functions is that simple operations on them
correspond to simple operations on structure types. We can use this to count
structures on finite sets. Let me remind you how it works for binary trees!
There's a structure type T where a Tstructure on a set is a way of
making it into the leaves of a binary tree drawn in the plane. For example,
here's one Tstructure on the set {a,b,c,d}:
b d a c
\ \ / /
\ \/ /
\ / /
\/ /
\ /
\/
Thanks to the choice of different orderings, the number of Tstructures
on an nelement set is n! times the number of binary trees with n
leaves. Annoyingly, the latter number is traditionally called the (n1)st
Catalan number, C_{n1}. So, we have:
T(x) = sum C_{n1} x^n
where the sum starts at n = 1.
There's a nice recursive definition of T:
"To put a Tstructure on a set, either note that it has one element,
in which case there's just one Tstructure on it, or chop it into
two subsets and put a Tstructure on each one."
In other words, any binary tree is either a degenerate tree with just one
leaf:
X
or a pair of binary trees stuck together at the root:
 
   
 T   T 
   
 
\ /
\ /
\ /
\/
We can write this symbolically as
T = X + T^2
Here's why: X is a structure type called "being the oneelement set",
+ means "exclusive or", and squaring a structure type means you chop your set
in two parts and put that structure on each part. (I explained these rules
more carefully in "week190".)
I should emphasize that the equals sign here is really an *isomorphism*
between structure types  I'm only using "equals" because the isomorphism
key on my keyboard is stuck. But if we take the generating function of
both sides we get an actual equation, and the notation is set up to make
this really easy:
T = x + T^2
In "week144" I showed how you can solve this using the quadratic equation:
T = (1  sqrt(1  4x))/2.
and then do a Taylor expansion to get
T = x + x^2 + 2x^3 + 5x^4 + 14x^5 + 42x^6 + ...
Lo and behold! The coefficient of x^n is the number of binary trees
with n leaves!
There's also another approach where we work directly with the
structure types themselves, instead of taking generating functions.
This is harder because we can't subtract structure types, or divide
them by 2, or take square roots of them  at least, not without
stretching the rules of this game. All we can do is use the
isomorphism
T = X + T^2
and the basic rules of category theory. It's not as efficient, but it's
illuminating. It's also incredibly simple: we just keep sticking in
"X + T^2" wherever we see "T" on the righthand side, over and over again.
Like this:
T = X + T^2
T = X + (X + T^2)^2
T = X + (X + (X + T^2)^2)^2
and so on. You might not think we're getting anywhere, but if
you stop at the nth stage and expand out what we've got, you'll
get the first n terms of the Taylor expansion we had before!
At least, you will if you count "stages" and "terms" correctly.
I won't actually do this, because it's better if you do it yourself.
When you do, you'll see it captures the recursive process of building
a binary tree from lots of smaller binary trees. Each time you see a "T"
and replace it with an "X + T^2", you're really taking a little binary
tree:

 
 T 
 

and replacing it with either a degenerate tree with just a single leaf:
X
or a pair of binary trees:
 
   
 T   T 
   
 
\ /
\ /
\ /
\/
So, each term in the final result actually corresponds to a specific
tree! This is a good example of categorification: when we calculate the
coefficient of x^n this way, we're not just getting the *number* of binary
planar trees with n leaves  we're getting an actual explicit description
of the *set* of such trees.
Now, what happens if we take the generating function T(x) and
evaluate it at x = 1? On the one hand, we get a divergent series:
T(1) = 1 + 1 + 2 + 5 + 14 + 42 + ...
This is the sum of all Catalan numbers  or in other words, the number
of binary planar trees. On the other hand, we can use the formula
T = (1  sqrt(1  4x))/2
to get
T(1) = (1  sqrt(3))/2
It may seem insane to conclude
1 + 1 + 2 + 5 + 14 + 42 + ... = (1  sqrt(3))/2
but Lawvere noticed that there's a kind of strange sense to it.
The trick is to work not with generating function T but with the structure
type T itself. Since T(1) is equal to the *number* of planar binary trees,
T(1) should be naturally isomorphic to the *set* of planar binary trees.
And it is  it's obvious, once you think about what it really means.
The number of binary planar trees is not very interesting, but the set of
them is. In particular, if we take the isomorphism
T = X + T^2
and set X = 1, we get an isomorphism
T(1) = 1 + T(1)^2
which says
"a planar binary tree is either the tree with one leaf or
a pair of planar binary trees."
Starting from this, we can derive lots of other isomorphisms involving
the set T(1), which turn out to be categorified versions of equations
satisfied by the number
T(1) = (1  sqrt(3))/2
For example, this number is a sixth root of unity. While there's no
onetoone correspondence between 6tuples of trees and the 1 element
set, which would categorify the formula
T(1)^6 = 1
there *is* a very nice onetocorrespondence between 7tuples of
trees and trees, which categorifies the formula
T(1)^7 = T(1)
Of course the set of binary trees is countably infinite, and so is the
set of 7tuples of binary trees, so they can be placed in onetoone
correspondence  but that's boring. When I say "very nice", I mean
something more interesting: starting with the isomorphism
T = x + T^2
we get a onetoone correspondence
T(1) = 1 + T(1)^2
which says that any binary planar tree is either degenerate or a
pair of binary planar trees... and using this we can *construct*
a onetoone correspondence
T(1)^7 = T(1)
The construction is remarkably complicated. Even if you do it
as efficiently as possible, I think it takes 18 steps, like this:
T(1)^7 = T(1)^6 + T(1)^8
= T(1)^5 + T(1)^7 + T(1)^8
.
.
.
= 1 + T(1) + T(1)^2 + T(1)^4
= 1 + T(1) + T(1)^3
= 1 + T(1)^2
= T(1)
I'll let you fill in the missing steps  it's actually quite fun if you
like puzzles.
If you get stuck, you can find the answer online in a couple of places:
2) Andreas Blass, Seven trees in one, Jour. Pure Appl. Alg. 103 (1995),
121. Also available at http://www.math.lsa.umich.edu/~ablass/cat.html
3) Marcelo Fiore, Isomorphisms of generic recursive polynomial
types, to appear in 31st Symposium on Principles of Programming
Languages (POPL04). Also available at
http://www.cl.cam.ac.uk/~mpf23/papers/Types/recisos.ps.gz
Or, take a peek at the "Addenda" down below.
Robbie Gates, Marcelo Fiore and Tom Leinster have also proved some very
general theorems about this sort of thing. Gates focused on "distributive
categories" (categories with products and coproducts, the former distributing
over the latter), while the work of Fiore and Leinster applies to more
general "rig categories":
4) Robbie Gates, On the generic solution to P(X) = X in distributive
categories, Jour. Pure Appl. Alg. 125 (1998), 191212.
5) Marcelo Fiore and Tom Leinster, Objects of categories as complex numbers,
available as arXiv:math.CT/0212377.
A rig category is basically the most general sort of category in which
we can "add" and "multiply" as we do in a ring  but without negatives,
hence the missing letter "n". It turns out that whenever we have an object
Z in a rig category and it's equipped with an isomorphism
Z = P(Z)
where P is a polynomial with natural number coefficients, we can associate
to it a "cardinality" Z, namely any complex solution of the equation
Z = P(Z)
Which solution should we use? Well, for simplicity let's consider the
case where P has degree at least 2 and the relevant Galois group acts
transitively on the solutions of this equation, so "all roots are created
equal". Then we can pick *any* solution as the cardinality Z. Any
polynomial equation with natural number coefficients satisfied by one
solution will be satisfied by *all* solutions, so it doesn't matter
which one we choose.
Now suppose the cardinality Z satisfies such an equation:
Q(Z) = R(Z)
where neither Q nor R is constant. Then the results of Fiore and
Leinster say we can construct an isomorphism
Q(Z) = R(Z)
in our distributive category! In other words, a bunch of equations satisfied
by the object's cardinality automatically come from isomorphisms involving the
object itself.
This explains why the set T(1) of binary trees acts like it has cardinality
T(1) = (1  sqrt(3))/2
or equally well,
T(1) = (1 + sqrt(3))/2
(Since the relevant Galois group interchanges these two numbers, we can
use either one.) More generally, the set T(n) consisting of binary trees
with ncolored leaves acts a lot like the number T(n).
This has gotten me interested in trying to find a nice example of a
"Golden Object": an object G in some distributive category that's
equipped with an isomorphism
G^2 = G + 1
The Golden Object doesn't fit into Fiore and Leinster's formalism,
since this isomorphism is not of the form G = P(G) where P has natural
number coefficients. But, it still seems that such an object deserves
to have a "cardinality" equal to the golden number:
G = (1 + sqrt(5))/2 = 1.618033988749894848204586834365...
James Propp came up with an interesting idea related to the Golden Object:
consider what happens when we evaluate the generating function for binary
trees at 1. On the one hand we get an alternating sum of Catalan numbers:
T(1) = 1 + 1  2 + 5  14 + 42 + ...
On the other hand, we can use the formula
T = (1  sqrt(1  4x))/2
to get
T(1) = (1  sqrt(5))/2
which is 1 divided by the golden number. Of course, it's possible we
should use the other sign of the square root, and get
T(1) = (1 + sqrt(5))/2
which is just the golden number! Galois theory says these two roots are
created equal. Either way, we get a bizarre and fascinating formula:
 1 + 1  2 + 5  14 + 42 + ... = (1 + sqrt(5))/2
Can we fit this into some clear and rigorous framework, or is it just nuts?
We'd like some generalization of cardinality for which "the set of binary
trees with 1colored leaves" has cardinality equal to the golden number.
James Propp suggested one avenue. A while back, Steve Schanuel made an
incredibly provocative observation: if we treat "Euler measure" as a
generalization of cardinality, it makes sense to treat the real line
as a "space of cardinality 1":
6) Stephen H. Schanuel, What is the length of a potato?: an introduction
to geometric measure theory, in Categories in Continuum Physics, Spring
Lecture Notes in Mathematics 1174, Springer, Berlin, 1986, pp. 118126.
7) Stephen H. Schanuel, Negative sets have Euler characteristic and
dimension, Lecture Notes in Mathematics 1488, Springer Verlag, Berlin,
1991, pp. 379385.
James Propp has developed this idea in a couple of fascinating papers:
8) James Propp, Euler measure as generalized cardinality, available as
arXiv:math/0203289.
9) James Propp, Exponentiation and Euler measure, available as
arXiv:math/0204009.
Using this idea, it seems reasonable to consider the space of binary
trees with leaves labelled by real numbers as a rigorous version of
"the set of binary trees with 1colored leaves". So, we just need
to figure out what generalization of Euler characteristic gives this
space an Euler characteristic equal to the golden number. It would
be great if we could make this space into a Golden Object in some
rig category, but that may be asking too much.
Whew! There's obviously a lot of work left to be done here. Here's
something easier: a riddle. What's this sequence?
un, dos, tres, quatre, cinc, sis, set, vuit, nou, deu,...
The answer is at the end of this article.
Now I'd like to mention some important papers on ncategories. You
may think I'd lost interest in this topic, because I've been talking
about other things. But it's not true!
Most importantly, Tom Leinster has come out with a big book on ncategories
and operads:
5) Tom Leinster, Higher Operads, Higher Categories, Cambridge U. Press,
Cambridge, 2003. Also available as arXiv:math.CT/0305049.
As you'll note, he managed to talk the press into letting him keep his book
freely available online! We should all do this. Nobody will ever make much
cash writing esoteric scientific tomes  it takes so long, you could earn
more per hour digging ditches. The only *financial* benefit of writing such
a book is that people will read it, think you're smart, and want to hire you,
promote you, or invite you to give talks in cool places. So, maximize your
chances of having people read your books by keeping them free online! People
will still buy the paper version if it's any good....
And indeed, Leinster's book has many virtues besides being free. He
gracefully leads the reader from the very basics of category theory
straight to the current battle front of weak ncategories, emphasizing
throughout how operads automatically take care of the otherwise mindnumbing
thicket of "coherence laws" that inevitably infest the subject. He doesn't
take wellestablished notions like "monoidal category" and "bicategory"
for granted  instead, he dives in, takes their definitions apart, and
compares alternatives to see what makes these concepts tick. It's this sort
of careful thinking that we desperately need if we're ever going to reach the
dream of a clear and powerful theory of higherdimensional algebra. He
does a similar careful analysis of "operads" and "multicategories" before
presenting a generalized theory of operads that's powerful enough to support
various different approaches to weak ncategories. And then he describes
and compares some of these different approaches!
In short: if you want to learn more about operads and ncategories, this is
*the* book to read.
Leinster doesn't say too much about what ncategories are good for, except for
a nice clear introduction entitled "Motivation for Topologists", where he
sketches their relevance to homology theory, homotopy theory, and cobordism
theory. But this is understandable, since a thorough treatment of their
applications would vastly expand an already hefty 380page book, and diffuse
its focus. It would also steal sales from *my* forthcoming book on higher
dimensional algebra  which would be really bad, since I plan to retire on
the fortune I'll make from this.
Secondly, Michael Batanin has worked out a beautiful extension of his ideas
on ncategories which sheds new light on their applications to homotopy
theory:
6) Michael A. Batanin, The EckmannHilton argument, higher operads and
E_n spaces, available as math.CT/0207281.
Michael A. Batanin, the combinatorics of iterated loop spaces, available as
arXiv:math.CT/0301221.
Getting a manageable combinatorial understanding of the space of loops
in the spaces of loops in the space of loops... in some space has
always been part of the dream of higherdimensional algebra. These
"kfold loop spaces" or have been important in homotopy theory since
the 1970s  see the end of "week199" for a little bit about them.
People know that kfold loop spaces have k different products that
commute up to homotopy in a certain way that can be summarized by
saying they are algebras of the E_k operad, also called the "little
kcubes operad". However, their wealth of structure is still a bit
mindboggling. James Dolan and I made some conjectures about their
relation to ktuply monoidal categories in our paper "Categorification"
(see "week121"), and now Batanin is making this more precise using
his approach to ncategories  which is one of the ones described in
Leinster's book.
There's also been a lot of work applying higherdimensional algebra to
topological quantum field theory  that's what got me interested in
ncategories in the first place, but a lot has happened since then.
For a highly readable introduction to the subject, with tons of great
pictures, try:
7) Joachim Kock, Frobenius Algebras and 2D Topological Quantum Field
Theories, Cambridge U. Press, Cambridge, 2003.
This is mainly about 2d TQFTs, where the concept of "Frobenius algebra"
reigns supreme, and everything is very easy to visualize.
When we go up to 3dimensional spacetime life gets harder, but also more
interesting. This book isn't so easy, but it's packed with beautiful math
and wonderfully drawn pictures:
8) Thomas Kerler and Volodymyr L. Lyubashenko, NonSemisimple Topological
Quantum Field Theories for 3Manifolds with Corners, Lecture Notes in
Mathematics 1765, Springer, Berlin, 2001.
The idea is that if we can extend the definition of a quantum field
theory to spacetimes that have not just boundaries but *corners*, we
can try to build up the theory for arbitrary spacetimes from its
behavior on simple building blocks  since it's easier to chop
manifolds up into a few basic shapes if we let those shapes have
corners. However, it takes higherdimensional algebra to describe all
the ways we can stick together manifolds with corners! Here Kerler
and Lyubashenko make 3dimensional manifolds going between 2manifolds
with boundary into a "double category"... and make a bunch of famous
3d TQFTs into "double functors".
Closely related is this paper by Kerler:
9) Thomas Kerler, Towards an algebraic characterization of 3dimensional
cobordisms, Contemp. Math. 318 (2003) 141173. Also available as
arXiv:math/0106253.
It relates the category whose objects are 2manifolds with a circle as
boundary, and whose morphisms are 3manifolds with corners going
between these, to a braided monoidal category "freely generated by a
Hopf algebra object". (I'm leaving out some fine print here, but
probably putting in more than most people want!) It comes close to
showing these categories are the same, but suggests that they're not
quite  so the perfect connection between topology and higher categories
remains elusive in this important example.
Answer to the riddle: these are the Catalan numbers  i.e., the natural
numbers as written in Catalan. This riddle was taken from the second
volume of Stanley's book on enumerative combinatorics (see "week144").

Addenda: Long after this issue was written, we had a discussion on the
nCategory Cafe about the "seven trees in one" problem. Let B be the
set of binary planar trees  the set I was calling T(1) above.
Starting from the isomorphism
B = B^2 + 1
we want to construct an isomorphism
B = B^7
Here is a picture of the proof in Marcelo Fiore's paper:
http://math.ucr.edu/home/baez/seven_trees_in_one_fiore.jpg
At each step he either replaces B^n by B^{n1} + B^{n+1}, The
underlined portion shows where this will be done. Over at the nCafe,
Stuart Presnell made a beautiful picture of this proof:
http://math.ucr.edu/home/baez/seven_trees_in_one_presnell_fiore.gif
He also made a picture of another proof, which is on page
29 of Pierre Ageron's book Logiques, Ensembles, Catégories:
Le Point de Vue Constructif:
http://math.ucr.edu/home/baez/seven_trees_in_one_presnell_bell.gif
You can watch a *movie* of a proof here:
15) Dan Piponi, Arboreal isomorphisms from nuclear pennies, A
Neighborhood of Infinity, September 30, 2007. Available at
http://blog.sigfpe.com/2007/09/arborealisomorphismsfromnuclear.html
It was in the ensuing discussion on this blog that George Bell came
up with his more efficient proof. For a bit more discussion, see:
16) John Baez, Searching for a video proof of "seven trees in one",
nCategory Cafe, July 16, 2009. Available at
http://golem.ph.utexas.edu/category/2009/07/searching_for_a_video_proof_of.html
Now, on to some older addenda!
My pal Squark pointed out that if we try to compute the
generating function for binary trees by making an initial guess for
T(x), say t, and repeatedly improving this guess via
t > x + t^2
the guess will converge to the right answer if t is small  but the
process will fail miserably, with t approaching infinity, if and only
if the complex number x lies outside the Mandelbrot set!
After an earlier version of this Week appeared on the category theory
mailing list, Steve Schanuel posted some corrections. I've tried to
correct the text above as much as possible without making it too
technical  for example, by citing the important work of Robbie Gates,
and distinguishing more clearly between his work on distributive
categories and the paper by Fiore and Leinster, which applies to rig
categories. I tend to talk about 3 different sorts of ringlike
categories in This Week's Finds:
* Rig categories. A "rig category" is one equipped with a symmetric
monoidal structure called + and a monoidal structure called x, with
all the usual rig axioms holding up to natural isomorphism, and these
isomorphisms satisfying a set of coherence laws worked out by Laplaza
and Kelly:
17) M. Laplaza, Coherence for distributivity, Lecture Notes in
Mathematics 281, Springer Verlag, Berlin, 1972, pp. 2972.
18) G. Kelly, Coherence theorems for lax algebras and distributive
laws, Lecture Notes in Mathematics 420, Springer Verlag, Berlin,
1974, pp. 281375.
(These authors spoke of "ring categories", but the term "rig category"
is more appropriate since, as in a rig, there need be no additive inverses.)
* 2Rigs. A "2rig" is a symmetric monoidal cocomplete category where
the monoidal structure, which we call x, distributes over the colimits,
which we think of as a generalized form of addition. For more on rigs,
2rigs and structure types see "week191". In a 2rig, distributivity
is just a property of the monoidal structure, rather than a structure,
as it is in a rig category. However, by choosing a particular coproduct
for each pair of objects, and a particular initial object, we can promote
any 2rig to a rig category. To get an example of a rig category that's
not a 2rig, just take any rig and think of it as a discrete category
(a category with only identity morphisms). Another example would be
the category of finitedimensional vector spaces, since this only has
finite colimits. (Of course, we could make up some sort of "finitary
2rig" that only had finite colimits, but the profusion of terminology
is already annoying.)
* Distributive categories. A "distributive category" is a category
with finite products and coproducts, the products distributing over
the coproducts. Here again, distributivity is just a property. But,
by choosing specified products and coproducts for every pair of objects,
and choosing terminal and initial objects, we can promote any distributive
category into a rig category. A good example of a 2rig that is not a
distributive category is the category Vect, with direct sum and tensor
product as + and x. Another example is the discrete category on a rig.
By not distinguishing these, the original version of "week202" made it
sound as if Fiore and Leinster had simply redone Gates' work on distributive
categories. I hope this is a bit clearer now. Schanuel's remarks are
still worth reading for their description of what Gates actually did:
Dear colleagues,
For those who read the most recent long discursion of John Baez, a few
of the errors in the section on distributive categories merit correction:
(1) J. B. suggests that Blass published what Lawvere had already worked
out. In fact, Lawvere (partly to counteract some incorrect uses of
infinite series in analyses of 'data types' in computer science) had
worked out the algebra of the rig presented by one generator X and one
relation X=1+X^2, roughly by the method in (3) below, and conjectured that
this rig could be realized as the isomorphism classes in a distributive
(even extensive) category, which conjecture Blass then proved (and a bit
more) in "Seven Trees...".
(2) The generalization of Blass's theorem to one generator ond one
polynomial relation of the 'fixedpoint' form X=p(X), where p is a
polynomial with natural number coefficients and nonzero constant term is
not, as J. B. seems to suggest, due to Fiore and Leinster; it was part of
the prizewinning doctoral thesis of Robbie Gates, who (using a calculus
of fractions) described explicitly the free distributive category on one
object X together with an isomorphism from p(X) to X, proving that this
category is extensive and that its rig of isomorphism classes satisfies no
further relations, i.e. is the rig R presented by one generator and the
one relation above.
(3) If p is as in (2) and of degree at least 2, the algebra of the rig R
is made by J. B. to seem mysterious. It is more easily understood in the
way the X=2X+1 case was treated in my "Negative Sets..." paper; just show
that the Euler and dimension homomorphisms, tensoring with Z and with 2
(the rig true/false) respectively, are jointly injective. In this case the
dimension rig has only three elements, which explains why the Euler
characteristic captures almost, but not quite, everything.
Greetings to all,
Steve Schanuel

Quote of the week:
"A traveller who refuses to pass over a bridge until he personally tests
the soundness of every part of it is not likely to go far; something must
be risked, even in mathematics."  Horace Lamb

Previous issues of "This Week's Finds" and other expository articles on
mathematics and physics, as well as some of my research papers, can be
obtained at
http://math.ucr.edu/home/baez/
For a table of contents of all the issues of This Week's Finds, try
http://math.ucr.edu/home/baez/twf.html
A simple jumpingoff point to the old issues is available at
http://math.ucr.edu/home/baez/twfshort.html
If you just want the latest issue, go to
http://math.ucr.edu/home/baez/this.week.html
at's what got me interested in
ncategories in the first place, but a lot has happened since then.
For a highly readable introduction to the subject, with tons of great
pictures, try:
7) Joachtwf_ascii/week104000064400020410000157000000501261046132003700141200ustar00baezhttp00004600000001Also available at http://math.ucr.edu/home/baez/week104.html
June 8, 1997
This Week's Finds in Mathematical Physics  Week 104
John Baez
A couple of months ago I flew up to Corvallis, Oregon to an AMS
meeting. The AMS, in case you're unfamiliar with it, is the American
Mathematical Society. They have lots of regional meetings with special
sessions on various topics. One reason I went to this one is that there
was a special session on octonions, organized by Tevian Dray and Corinne
Manogue.
After the real numbers come the complex numbers, and after the complex
numbers come the quaternions, and after the quaternions come the
octonions, the most mysterious of all. The real numbers, complex
numbers, and quaternions have lots of applications to physics. What
about the octonions? Aren't they good for something too? This
question has been bugging me for a while now.
In fact, it bugs me so much that I decided to go to Corvallis to look
for clues. After all, in addition to Tevian Dray and Corinne Manogue 
the former a mathematician, the latter a physicist, both deeply
interested in octonions  a bunch of other octonion experts were going
to be there. One was my friend Geoffrey Dixon. I told you about him in
"week59". He wrote a book on the complex numbers, quaternions and
octonions and their role in physics. He has a theory of physics in
which these are related to electromagnetism, the weak force, and the
strong force, respectively. It's a bit far out, but far from crazy!
In fact, it's fascinating.
After writing about his book I got in touch with him in Cambridge,
Massachusetts. I found out that his other main interest in life,
besides the octonions, is the game Myst. This is probably not a
coincidence. In both the main question is "What the heck is really
going on here?" He has Myst all figured out, but he loves watching
people play it, so he got me to play it for a while. Someday I will buy
a CDROM drive and waste a few weeks on that game. Anyway, I got to
know him back in the summer of 1995, so it was nice to see him again in
Corvallis.
Another octonion expert is Tony Smith. He too has a farout but
fascinating theory of physics involving octonions! I wrote about his
stuff in "week91". I had never met him before the Corvallis conference,
but I instantly recognized him when I met him, because there's a picture
of him wearing a cowboy hat on his homepage. It turns out he always
wears that hat. He is a wonderful repository of information concerning
octonions and other interesting things. He is also a very friendly and
laidback sort of guy. He lives in Atlanta, Georgia.
I also met another octonion expert I hadn't known about, Tony Sudbery,
from York. (The original York, not the "new" one.) He gave a talk on
"The Exceptions that Prove the Rule". The octonions are related to a
host of other mathematical structures in a very spooky way. In all
sorts of contexts, you can classify algebraic structures and get a nice
systematic infinite list together with a finite number of exceptions.
What's spooky is how the exceptions in one context turn out to be
related to the exceptions in some other context. These relationships
are complicated and mysterious in themselves. It's as if there were a
hand underneath the water and all we see is the fingers poking out here
and there. There seems to be some "unified theory of exceptions"
waiting to be discovered, and the octonions must have something to do
with it. I figure that to really understand what the octonions are good
for, we need to understand this "unified theory of exceptions".
Let's start by recalling what the octonions are!
I presume you know the real numbers. The complex numbers are things
like
a + bi
where a and b are real. We can multiply them using the rule
i^2 = 1
They may seem mysterious when you first meet them, but they lose their
mystery when you see they are just a nice way of keeping track of
rotations in the plane.
Similarly, the quaternions are guys like
a + bi + cj + dk
which we can multiply using the rules
i^2 = j^2 = k^2 = 1
and
ij = k, jk = i, ki = j
ji = k, kj = i, ik = j
They aren't commutative, but they are still associative. Again they may
seem mysterious at first, but they lose their mystery when you see that
they are just a nice way of keeping track of rotations in 3 and 4
dimensions. Rotations in more than 2 dimensions don't commute in
general, so the quaternions had *better* not commute. In fact, Hamilton
didn't invent the quaternions to study rotations  his goal was merely
to cook up a "division algebra", where you could divide by any nonzero
element (see "week82"). However, after he discovered the quaternions,
he used them to study rotations and angular momentum. Nowadays people
tend instead to use the vector cross product, which was invented later
by Gibbs. The reason is that in the late 1800s there was a big battle
between the fans of quaternions and the fans of vectors, and the
quaternion crowd lost. For more on the history of this stuff, see:
1) Michael J. Crowe, A History of Vector Analysis, University of Notre
Dame, Notre Dame, 1967.
Octonions were invented by Cayley later on in the 1800s. For these,
we start with *seven* square roots of 1, say e1 up to e7. To learn
how multiply these, draw the following diagram:
e6
e4 e1
e7
e3 e2 e5
Draw a triangle, draw a line from each vertex to the midpoint of the
opposite edge, and inscribe a circle in the triangle. Label the
7 points shown with e1 through e7  it doesn't matter how, I've just
drawn my favorite way. Draw arrows on the edges of the triangle
going around clockwise, draw arrows on the circle also going around
clockwise, and draw arrows on the three lines pointing from each vertex
of the triangle to the midpoint of the opposite edge. Come on, DO it!
I'm doing all this work for you... you should do some, too.
Okay. Now you have your very own octonion multiplication table. Notice
that there are six lines and a circle in your picture. Each one of
these gives us a copy of the quaternions inside the octonions. For
example, say you want to multiply e6 and e7. You notice that the
the vertical line says "e6, e7, e2" on it as we follow the arrow down.
Thus, just as for i, j, and k in the quaternions, we have
e6 e7 = e2, e7 e2 = e6, e2 e6 = e7
e7 e6 = e2, e2 e7 = e6, e6 e2 = e7
So in particular we have e6 e7 = e2.
In case you lose your octonion table, don't worry: you don't really need
to remember the *names* of those 7 square roots of 1 and their
positions on the chart. You just need to remember the geometry of the
chart itself. Names are arbitrary and don't really matter, unless
you're talking to someone else, in which case you have to agree on them.
If you want to see spiffy hightech octonion multiplication tables,
check out the following websites:
2) Tony Smith, http://galaxy.cau.edu/tsmith/TShome.html
3) Geoffrey Dixon, http://www.7stones.com (Warning: to really get into
this you need to have at least Netscape 3.0 with Java and Shockwave
stuff.)
What's so great about the octonions? They are not commutative, and
worse, they are not even *associative*. What's great about them is that
they form a division algebra, meaning you can divide by any nonzero
octonion. Better still, they form a "normed" division algebra. Just as
with the reals, complexes, and quaternions, we can define the norm of
the octonion
x = a0 + a1 e1 + a2 e2 + a3 e3 + a4 e4 + a5 e5 + a6 e6 + a7 e7
to be
x = sqrt(a0^2 + a1^2 + a2^2 + a3^2 + a4^2 + a5^2 + a6^2 + a7^2).
What makes them a "normed division algebra" is that
xy = xy.
It's a wonderful fact about the world that the reals, complexes,
quaternions and octonions are the *only* normed division algebras.
That's it!
However, the octonions remain mysterious, at least to me. They are
related to rotations in 7 and 8 dimensions, but not as simply as one
might hope. After all, rotations in *any* number of dimensions are
still associative. Where is this nonassociative business coming from?
I don't really know. This question really bugs me.
A while ago, in "week93", I summarized a paper by John Schwarz on
supersymmetric YangMills theory and why it works best in dimensions 3,
4, 6, and 10. Basically, only in these dimensions can you cook up
spin1/2 particles that have as many physical degrees of freedom as
massless spin1 particles. I sort of explained why. This in turn
allows a symmetry between fermions and gauge bosons. I didn't explain
how *this* works... it seems pretty tricky to me... but anyway, it works.
So far, so good. But Schwarz wondered: is it a coincidence that the
numbers 3, 4, 6, and 10 are just two more than the numbers 1, 2, 4, and
8  the dimensions of the reals, complexes, quaternions, and octonions?
Apparently not! The following papers explain what's going on:
4) Corinne A. Manogue and Joerg Schray, Finite Lorentz transformations
automorphisms, and division algebras, Jour. Math. Phys. 34 (1993),
37463767.
Corinne A. Manogue and Joerg Schray, Octonionic representations of
Clifford algebras and triality, preprint available as hepth/9407179.
5) Anthony Sudbery, Division algebras, (pseudo)orthogonal groups and
spinors, Jour. Phys. A 17 (1984), 939955.
Anthony Sudbery, Seven types of incongruity, handwritten notes.
Here's the basic idea. Let
R = real numbers
C = complex numbers
H = quaternions
O = octonions
Let SO(n,1) denote the Lorentz group in n+1 dimensions. Roughly
speaking, this is the symmetry group of (n+1)dimensional Minkowski
spacetime. Let so(n,1) be the corresponding Lie algebra (see "week63"
for a lightning introduction to Lie algebras). Then it turns out that:
sl(2,R) = so(2,1)
sl(2,C) = so(3,1)
sl(2,H) = so(5,1)
sl(2,O) = so(9,1)
This relates reals, complexes, quaternions and octonions to the Lorentz
group in dimensions 3, 4, 6, and 10, and explains the "coincidence"
noted by Schwarz! But it requires some explanation. Roughly speaking,
if SL(2,K) is the group of 2x2 matrices with determinant 1 whose entries
lie in the division algebra K = R, C, H, O, then sl(2,K) is defined to
be the Lie algebra of this group. This is simple enough for R or C.
However, one needs to be careful when defining the determinant of a 2x2
quaternionic matrix, since quaternions don't commute. One needs to be
even more careful in the octonionic case. Since octonions aren't even
associative, it's far from obvious what the group SL(2,O) would be, so
defining the Lie algebra "sl(2,O)" requires a certain amount of finesse.
For the details, read the papers.
As Corinne Manogue explained to me, this relation between the octonions
and Lorentz transformations in 10 dimensions suggests some interesting
ways to use octonions in 10dimensional physics. As we all know, the
10th dimension is where string theorists live. There is also a nice
relation to Geoffrey Dixon's theory. This theory relates the
electromagnetic force to the complex numbers, the weak force to the
quaternions, and the strong force to octonions. How? Well, the gauge
group of electromagnetism is U(1), the unit complex numbers. The gauge
group of the weak force is SU(2), the unit quaternions. The gauge group
of the strong force is SU(3)....
Alas, the group SU(3) is *not* the unit octonions. The unit octonions
do not form a group since they aren't associative. SU(3) is related to
the octonions more indirectly. The group of symmetries (or technically,
"automorphisms") of the octonions is the exceptional group G2, which
contains SU(3). To get SU(3), we can take the subgroup of G2 that
preserves a given unit imaginary octonion... say e1. This is how Dixon
relates SU(3) to the octonions.
However, why should one unit imaginary octonion be different from the
rest? Some sort of "symmetry breaking", presumably? It seems a bit ad
hoc. However, as Manogue explained, there is a nice way to kill two
birds with one stone. If we pick a particular unit imaginary octonion,
we get a copy of the complex numbers sitting inside the octonions, so we
get a copy of sl(2,C) sitting inside sl(2,O), so we get a copy of
so(3,1) sitting inside so(9,1)! In other words, we get a particular
copy of the good old 4dimensional Lorentz group sitting inside the
10dimensional Lorentz group. So fixing a unit imaginary octonion not
only breaks the octonion symmetry group G2 down to the strong force
symmetry group SU(3), it might also get us from 10dimensional physics
down to 4dimensional physics.
Cool, no? There are obviously a lot of major issues involved in turning
this into a fullfledged theory, and they might not work out. The whole
idea could be completely misguided! But it takes guts to do physics, so
it's good that Tevian Dray and Corinne Manogue are bravely pursuing this
idea.
Upon learning that there is a deep relation between R, C, H, O and the
Lorentz group in dimensions 3, 4, 6, 10, one is naturally emboldened to
take seriously a few more "coincidences". For example, in "week82" I
described the Clifford algebras C_n  i.e., the algebras generated by
n anticommuting square roots of 1. These Clifford algebras are
relevant to ndimensional *Euclidean* geometry, as opposed to the
Clifford algebras relevant to ndimensional *Lorentzian* geometry, which
appeared in "week93". They go like this:
C_0 R
C_1 C
C_2 H
C_3 H + H
C_4 H(2)
C_5 C(4)
C_6 R(8)
C_7 R(8) + R(8)
C_8 R(16)
where K(n) stands for n x n matrices with entries taken from K = R, C,
or H, and "+" stands for "direct sum". Note that C_8 is the same as 16
x 16 matrices with entries taken from C_0. That's part of a general
pattern called "Bott periodicity": in general, C_{n+8} is the same as 16
x 16 matrices with entries taken from C_n.
Now consider the dimension of the smallest real representation of C_n.
It's easy to work this out if you keep in mind that the smallest
representation of K(n) or K(n) + K(n) is on K^n  the vector space
consisting of ntuples of elements of K. We get
Dimension of smallest real representation
C_0 1
C_1 2
C_2 4
C_3 4
C_4 8
C_5 8
C_6 8
C_7 8
C_8 16
Note that it increases at n = 1, 2, 4, and 8. These are the dimensions
of R, C, H, and O. Coincidence?
No! Indeed, C_n has a representation on a kdimensional real
vector space if and only if the unit sphere in that vector space,
S^{k1}, admits n linearly independent smooth vector fields. So the
above table implies that:
The sphere S^0 admits 0 linearly independent vector fields.
The sphere S^1 admits 1 linearly independent vector fields.
The sphere S^3 admits 3 linearly independent vector fields.
The sphere S^7 admits 7 linearly independent vector fields.
These spheres are the unit real numbers, the unit complex numbers, the
unit quaternions, and the unit octonions, respectively! If you know
about normed division algebras, it's obvious that these sphere admit the
maximum possible number of linear independent vector fields: you can
just take a basis of vectors at one point and "left translate" it to get
a bunch of linearly independent vector fields.
Now  Bott periodicity has period 8, and the octonions have dimension
8. And as we've seen, both have a lot to do with Clifford algebras. So
maybe there is a deep relation between the octonions and Bott
periodicity. Could this be true? If so, it would be good news, because
while octonions are often seen as weird exceptional creatures, Bott
periodicity is bigtime, mainstream stuff!
And in fact it *is* true. More on Bott periodicity and the octonions
coming up next Week.

Addendum: Robert Helling provided some interesting further comments on
supersymmetric gauge theories and the division algebras, which I quote
below. He recommends the following reference:
6) J. M. Evans, Supersymmetric YangMills theories and division
algebras, Nucl. Phys. B298 (1988), 92108.
and he writes:
Let me add a technical remark that I extract from Green, Schwarz,
and Witten, Vol 1, Appendix 4A.
The appearance of dimensions 3,4,6, and 10 can most easily been seen
when one tries to write down a supersymmetric gauge theory in arbitrary
dimension. This means we're looking for a way to throw in some spinors
to the Lagrangian of a pure gauge theory:
1/4 F^2
in a way that the new Lagrangian is invariant (up to a total derivative)
under some infinitesimal variations. These describe supersymmetry if
their commutator is a derivative (a generator of spacetime
translations). As usual, we parameterize this variation by a parameter
epsilon, but now epsilon is a spinor.
From people that have been doing this for their whole life we learn
that the following Ansatz is common:
variation of A_m = i/2 epsilonbar Gamma_m psi
variation of psi = 1/4 F_{mn} Gamma^{mn} epsilon
Here A is the connection, F its field strength and psi a spinor of a
type to be determined. I suppressed group indices on all these
fields. They are all in the adjoint representation. Gamma are the
generators of the Clifford algebra described by John Baez before.
For the Lagrangian we try the usual YangMills term and add a minimally
coupled kinetic term for the fermions:
1/4 F^2 + ig/2 psibar Gamma^m D_m psi
Here D_m is the gauge covariant derivative and g is some number that we
can tune to to make this vanish under the above variations. When we vary
the first term we find g = 1. In fact everything cancels without
considering a special dimension except for the term that is trilinear in
psi that comes from varying the connection in the covariant derivative
in the fermionic term. This reads something like
f_{abc} epsilonbar Gamma_m psi^a psibar^b Gamma^m psi^c
where I put in the group indices and the structure constants f_{abc}. This
has to vanish for other reasons since there is no other trilinear term
in the fermions available. And indeed, after you've written out the
antisymmetry of f explicitly and take out the spinors since this should
vanish for all choices of psi and epsilon. We are left with an
expression that is only made of gammas. And in fact, this expression
exactly vanishes in dimensions 3, 4, 6, and 10 due to a Fierz
identity. (Sorry, I don't have time to work this out more explicitly.)
This is related to the division algebra as follows (as explained in
the papers pointed out by John Baez): Take for concreteness d = 10.
Here we go to a lightcone frame by using coordinates
x+ = x^0 + x^9 and
x = x^0  x^1.
Then we write the Gamma_m as block matrices where Gamma+ and Gamma have
the +/ unit matrix as blocks and the others have gamma_i as blocks
where gamma_i are the SO(8) Dirac matrices (i=1,...,9). But they are
intimately related to the octonions. Remember there is triality in SO(8)
which means that we can treat lefthanded spinors, righthanded spinors
and vectors on an equal basis (see "week61", "week90", and "week91").
Now I write out all three indices of gamma_i. Because of triality I can
use i,j,k for spinor, dotted spinor and vector indices. Then it is
known that
gamma_{ijk} = c_{ijk} for i,j,k < 8
delta_{ij }for k=8 (and ijk permuted)
0 for more than 2 of ijk equal 8.
is a representation of Cliff(8) if c_ijk} are the structure constants of
the octonions (i.e. e_i e_j = c_{ijk} ek for the 7 roots of 1 in the
octonions).
When plug this representation of the Gammas in the above mentioned Gamma
expression you will will find that it vanishes due to the antisymmetry
of the associator
[a,b,c] = a(bc)  (ab)c
in the division algebras. This is my understanding of the relation of
supersymmetry to the divison algebras.
Robert

Previous issues of "This Week's Finds" and other expository articles on
mathematics and physics, as well as some of my research papers, can be
obtained at
http://math.ucr.edu/home/baez/
For a table of contents of all the issues of This Week's Finds, try
http://math.ucr.edu/home/baez/twfcontents.html
A simple jumpingoff point to the old issues is available at
http://math.ucr.edu/home/baez/twfshort.html
If you just want the latest issue, go to
http://math.ucr.edu/home/baez/this.week.html
ned to
be the Lie algebra of this group. This is simple enough for R or C.
However, one needs to be careful when defining the determinant of a 2x2
quaternionic matrix, since quaternions don't commute. One needs to be
even more careful in the octonionic case. Since octonions aren't even
associative, it's far from obvious what the group SL(2,O) would be, so
defining the Lie algebra "sl(2,O)" requires a certain amount of ftwf_ascii/week105000064400020410000157000000574361117105522600141400ustar00baezhttp00004600000001Also available at http://math.ucr.edu/home/baez/week105.html
June 21, 1997
This Week's Finds in Mathematical Physics  Week 105
John Baez
There are some spooky facts in mathematics that you'd never guess in a
million years... only when someone carefully works them out do they
become clear. One of them is called "Bott periodicity".
A 0dimensional manifold is pretty dull: just a bunch of points.
1dimensional manifolds are not much more varied: the only
possibilities are the circle and the line, and things you get by
taking a union of a bunch of circles and lines. 2dimensional
manifolds are more interesting, but still pretty tame: you've got your
nholed tori, your projective plane, your Klein bottle, variations on
these with extra handles, and some more related things if you allow
your manifold to go on forever, like the plane, or the plane with a
bunch of handles added (possibly infinitely many!), and so on....
You can classify all these things. 3dimensional manifolds are a lot
more complicated: nobody knows how to classify them. 4dimensional
manifolds are a *lot* more complicated: you can *lot* that it's
*impossible* to classify them  that's called Markov's Theorem.
Now, you probably wouldn't have guessed that a lot of things start
getting simpler when you get up around dimension 5. Not everything,
just some things. You still can't classify manifolds in these high
dimensions, but if you make a bunch of simplifying assumptions you
sort of can, in ways that don't work in lower dimensions. Weird, huh?
But that's another story. Bott periodicity is different. It says
that when you get up to 8 dimensions, a bunch of things are a whole
lot like in 0 dimensions! And when you get up to dimension 9, a bunch
of things are a lot like they were in dimension 1. And so on  a
bunch of stuff keeps repeating with period 8 as you climb the ladder
of dimensions.
(Actually, I have this kooky theory that perhaps part of the reason
topology reaches a certain peak of complexity in dimension 4 is that
the number 4 is halfway between 0 and 8, topology being simplest in
dimension 0. Maybe this is even why physics likes to be in 4
dimensions! But this is a whole other crazy digression and I will
restrain myself here.)
Bott periodicity takes many guises, and I already described one in
"week104". Let's start with the real numbers, and then throw in n
square roots of 1, say e_1,...,e_n. Let's make them "anticommute",
so
e_i e_j =  e_j e_i
when i is different from j. What we get is called the "Clifford
algebra" C_n. For example, when n = 1 we get the complex numbers,
which we call C. When n = 2 we get the quaternions, which we call H,
for Hamilton. When n = 3 we get... the octonions?? No, not the
octonions, since we always demand that multiplication be associative!
We get the algebra consisting of *pairs* of quaternions! We call that
H + H. When n = 4 we get the algebra consisting of 2x2 *matrices* of
quaternions! We call that H(2). And it goes on, like this:
C_0 R
C_1 C
C_2 H
C_3 H + H
C_4 H(2)
C_5 C(4)
C_6 R(8)
C_7 R(8) + R(8)
C_8 R(16)
Note that by the time we get to n = 8 we just have 16x16 matrices of
real numbers. And that's how it keeps going: C_{n+8} is just 16x16
matrices of guys in C_n! That's Bott periodicity in its simplest form.
Actually right now I'm in Vienna, at the Schroedinger Institute, and
one of the other people visiting is Andrzej Trautman, who gave a talk
the other day on "Complex Structures in Physics", where he mentioned a
nice way to remember the above table. Imagine the day is only 8 hours
long, and draw a clock with 8 hours. Then label it like this:
R
R+R C
R H
C H+H
H
The idea here is that as the dimension of space goes up, you go around
the clock. One nice thing about the clock is that it has a reflection
symmetry about the axis from 3 o'clock to 7 o'clock. To use the
clock, you need to know that the dimension of the Clifford algebra
doubles each time you go up a dimension. This lets you figure out,
for example, that the Clifford algebra in 4 dimensions is not really
H, but H(2), since the latter has dimension 16 = 2^4.
Now let's completely change the subject and talk about rotations
in infinitedimensional space! What's a rotation in infinitedimensional
space like? Well, let's start from the bottom and work our way up.
You can't really rotate in 0dimensional space. In 1dimensional
space you can't really rotate, you can only *reflect* things... but we
will count reflections together with rotations, and say that the
operations of multiplying by 1 or 1 count as "rotations" in
1dimensional space. In 2dimensional space we describe rotations by
2x2 matrices like
cos t sin t
sin t cos t
and since we're generously including reflections, also matrices like
cos t sin t
sin t cos t
These are just the matrices whose columns are orthonormal vectors. In
3dimensional space we describe rotations by 3x3 matrices whose
columns are orthonormal, and so on. In ndimensional space we call
the set of nxn matrices with orthonormal columns the "orthogonal
group" O(n).
Note that we can think of a rotation in 2 dimensions
cos t sin t
sin t cos t
as being a rotation in 3 dimensions if we just stick one more row and one
column like this:
cos t sin t 0
sin t cos t 0
0 0 1
This is just a rotation around the z axis. Using the same trick
we can think of any rotation in n dimensions as a rotation in n+1
dimensions. So we can think of O(0) as sitting inside O(1), which
sits inside O(2), which sits inside O(3), which sits inside O(4),
and so on! Let's do that. Then let's just take the *union* of
all these guys, and we get... O(infinity)! This is the group of
rotations, together with reflections, in infinite dimensions.
(Now if you know your math, or you read "week82", you'll realize that I
didn't really change the subject, since the Clifford algebra C_n is
really just a handy way to study rotations in n dimensions. But never
mind.)
Now O(infinity) is a very big group, but it elegantly summarizes a lot
of information about rotations in all dimensions, so it's not
surprising that topologists have studied it. One of the thing
topologists do when studying a space is to work out its "homotopy
groups". If you hand them a space X, and choose a point x in this
space, they will work out all the topologically distinct ways you can
stick an ndimensional sphere in this space, where we require that the
north pole of the sphere be at x. This is what they are paid to do.
We call the set of all such ways the homotopy group pi_n(X). For a
more precise description, try "week102"  but this will do for now.
So, what are the homotopy groups of O(infinity)? Well, they start out
looking like this:
n pi_n(O(infinity))
0 Z/2
1 Z/2
2 0
3 Z
4 0
5 0
6 0
7 Z
And then they repeat, modulo 8. Bott periodicity strikes again!
But what do they mean?
Well, luckily Jim Dolan has thought about this a lot. Discussing
it repeatedly in the little cafe we tend to hang out at, we came up
with the following story. Most of it is known to various people
already, but it came as sort of a revelation to us.
The zeroth entry in the table is easy to understand. pi_0 keeps track
of how many connected components your space has. The rotation group
O(infinity) has two connected components: the guys that are rotations,
and the guys that are rotations followed by a reflection. So pi_0 of
O(infinity) is Z/2, the group with two elements. Actually this is
also true for O(n) whenever n is higher enough, namely 1 or more.
So the zeroth entry is all about "reflecting".
The first entry is a bit subtler but very important in physics. It
means that there is a loop in O(infinity) that you can't pull tight,
but if you go around that loop *twice*, you trace out a loop that you
*can* pull tight. In fact this is true for O(n) whenever n is 3 or
more. This is how there can be spin1/2 particles when space is
3dimensional or higher. There are lots of nice tricks for seeing
that this is true, which I hope the reader already knows and loves.
In short, the first entry is all about "rotating 360 degrees and not
getting back to where you started".
The second entry is zero.
The third entry is even subtler but also very important in modern
physics. It means that the ways to stick a 3sphere into O(infinity)
are classified by the integers, Z. Actually this is true for O(n)
whenever n is 5 or more. It's even true for all sorts of other
groups, like all "compact simple groups". But can I summarize this
entry in a snappy phrase like the previous nonzero entries? Not
really. Actually a lot of applications of topology to quantum field
theory rely on this pi_3 business. For example, it's the key to stuff
like "instantons" in YangMills theory, which are in turn crucial for
understanding how the pion gets its mass. It's also the basis of
stuff like "ChernSimons theory" and "BF theory". Alas, all this
takes a while to explain, but let's just say the third entry is about
"topological field theory in 4 dimensions".
The fourth entry is zero.
The fifth entry is zero.
The sixth entry is zero.
The seventh entry is probably the most mysterious of all. From one
point of view it is the subtlest, but from another point of view it is
perfectly trivial. If we think of it as being about pi_7 it's very
subtle: it says that the ways to stick a 7sphere into O(infinity) are
classified by the integers. (Actually this is true for O(n) whenever
n is 7 or more.) But if we keep Bott periodicity in mind, there is
another way to think of it: we can think of it as being about pi_{1},
since 7 = 1 mod 8.
But wait a minute! Since when can we talk about pi_n when n is
*negative*?! What's a 1dimensional sphere, for example?
Well, the idea here is to use a trick. There is a space very related
to O(infinity), called kO. As with O(infinity), the homotopy
groups of this space repeat modulo 8. Moreover we have:
pi_n(O(infinity)) = pi_{n+1}(kO)
Combining these facts, we see that the very subtle pi_7 of O(infinity)
is nothing but the very unsubtle pi_0 of kO, which just keeps track
of how many connected components kO has.
But what *is* kO?
Hmm. The answer is very important and interesting, but it would
take a while to explain, and I want to postpone doing it for a while,
so I can get to the punchline. Let me just say that when we work
it all out, we wind up seeing that the seventh entry in the table
is all about *dimension*.
To summarize:
pi_0(O(infinity)) = Z/2 is about REFLECTING
pi_1(O(infinity)) = Z/2 is about ROTATING 360 DEGREES
pi_3(O(infinity)) = Z is about TOPOLOGICAL FIELD THEORY IN 4 DIMENSIONS
pi_7(O(infinity)) = Z is about DIMENSION
But wait! What do those numbers 0, 1, 3, and 7 remind you of?
Well, after I stared at them for a few weeks, they started to remind
me of the numbers 1, 2, 4, and 8. And *that* immediately reminded me
of the reals, the complexes, the quaternions, and the octonions!
And indeed, there is an obvious relationship. Let n be 1, 2, 4, or 8,
and correspondingly let A stand for either the reals R, the complex
numbers C, the quaternions H, or the octonions O. These guys are
precisely all the "normed division algebras", meaning that the obvious
sort of absolute value satisfies
xy = xy.
Thus if we take any guy x in A with x = 1, the operation
of multiplying by x is lengthpreserving, so it's a reflection or
rotation in A. This gives us a function from the unit sphere in
A to O(n), or in other words from the (n1)sphere to O(n). We thus
get nice elements of
pi_0(O(1))
pi_1(O(2))
pi_3(O(4))
pi_7(O(8))
which turn out to be precisely why these particular homotopy groups
of O(infinity) are nontrivial.
So now we have the following fancier chart:
pi_0(O(infinity)) is about REFLECTING and the REAL NUMBERS
pi_1(O(infinity)) is about ROTATING 360 DEGREES and the COMPLEX NUMBERS
pi_3(O(infinity)) is about TOPOLOGICAL FIELD THEORY IN 4 DIMENSIONS and the
QUATERNIONS
pi_7(O(infinity)) is about DIMENSION and the OCTONIONS
Now this is pretty weird. It's not so surprising that reflections and the
real numbers are related: after all, the only "rotations" in the real
line are the reflections. That's sort of what 1 and 1 are all about.
It's also not so surprising that rotations by 360 degrees are related to
the complex numbers. That's sort of what the unit circle is all
about. While far more subtle, it's also not so surprising that
topological field theory in 4 dimensions is related to the quaternions.
The shocking part is that something so basicsounding as "dimension"
should be related to something so eruditesounding as the "octonions"!
But this is what Bott periodicity does, somehow: it wraps things around
so the most complicated thing is also the least complicated.
That's more or less the end of what I have to say, except for some
references and some remarks of a more technical nature.
Bott periodicity for O(infinity) was first proved by Raoul Bott in
1959. Bott is a wonderful explainer of mathematics and one of the
main driving forces behind applications of topology to physics, and
a lot of his papers have now been collected in book form:
1) The Collected Papers of Raoul Bott, ed. R. D. MacPherson. Vol. 1:
Topology and Lie Groups (the 1950s). Vol. 2: Differential Operators
(the 1960s). Vol. 3: Foliations (the 1970s). Vol. 4: Mathematics
Related to Physics (the 1980s). Birkhauser, Boston, 1994, 2355 pages
total.
A good paper on the relation between O(infinity) and Clifford algebras
is:
2) M. F. Atiyah, R. Bott, and A. Shapiro, Clifford modules, Topology
(3) 1964, 338.
For more stuff on division algebras and Bott periodicity try Dave Rusin's
web page, especially his answer to "Q5. What's the question with the
answer n = 1, 2, 4, or 8?"
3) Dave Rusin, Binary products, algebras, and division rings,
http://www.math.niu.edu/~rusin/papers/knownmath/products/division.alg
Let me briefly explain this kO business. The space kO is related to a
simpler space called BO(infinity) by means of the equation
kO = BO(infinity) x Z,
so let me first describe BO(infinity). For any topological group G
you can cook up a space BG whose loop space is homotopy equivalent
to G. In other words, the space of (basepointpreserving) maps
from S^1 to BG is homotopic to G. It follows that
pi_n(G) = pi_{n+1}(BG).
This space BG is called the classifying space of G because it has a
principal Gbundle over it, and given *any* decent topological space X
(say a CW complex) you can get all principal Gbundles over X (up to
isomorphism) by taking a map f: X > BG and pulling back this
principal Gbundle over BG. Moreover, homotopic maps to BG give
isomorphic Gbundles over X this way.
Now a principal O(n)bundle is basically the same thing as an
ndimensional real vector bundle  there are obvious ways to go
back and forth between these concepts. A principal O(infinity)bundle
is thus very much like a real vector bundle of *arbitrary* dimension,
but where we don't care about adding on arbitrarily many 1dimensional
trivial bundles. If we take the collection of isomorphism classes of
real vector bundles over X and decree two to be equivalent if they
become isomorphic after adding on trivial bundles, we get something
called KX, the "real Ktheory of X". It's not hard to see that this
is a group. Taking what I've said and working a bit, it follows that
KX = [X, BO(infinity)]
where the righthand side means "homotopy classes of maps from X to
BO(infinity). If we take X to be S^{n+1}, we see
KS^{n+1} = pi_{n+1}(BO(infinity)) = pi_n(O(infinity))
It follows that we can get all elements of pi_n of O(infinity)
from real vector bundles over S^{n+1}.
Of course, the above equations are true only for nonnegative n, since
it doesn't make sense to talk about pi_{1} of a space. However,
to make Bott periodicity work out smoothly, it would be nice if we could
pretend that
KS^{1} = pi_0(BO(infinity)) = pi_{1}(O(infinity)) = pi_7(O(infinity)) = Z
Alas, the equations don't make sense, and BO(infinity) is connected,
so we don't have pi_0(BO(infinity)) = Z. However, we can cook up a
slightly improved space kO, which has
pi_n(kO) = pi_n(BO(infinity))
when n > 0, but also has
pi_0(kO) = Z
as desired. It's easy  we just let
kO = BO(infinity) x Z.
So, let's use this instead of BO(infinity) from now on.
Taking n = 0, we can think of S^1 as RP^1, the real projective line,
i.e. the space of 1dimensional real subspaces of R^2. This has a
"canonical line bundle" over it, that is, a 1dimensional real vector
bundle which to each point of RP^1 assigns the 1dimensional subspace of
R^2 that *is* that point. This vector bundle over S^1 gives the generator
of KS^1, or in other words, pi_0(O(infinity)).
Taking n = 1, we can think of S^2 as the "Riemann sphere", or in other
words CP^1, the space of 1dimensional complex subspaces of C^2. This
too has a "canonical line bundle" over it, which is a 1dimensional
complex vector bundle, or 2dimensional real vector bundle. This
bundle over S^2 gives the generator of KS^2, or in other words,
pi_1(O(infinity)).
Taking n = 3, we can think of S^4 as HP^1, the space of 1dimensional
quaternionic subspaces of H^2. The "canonical line bundle" over this
gives the generator of KS^4, or in other words, pi_3(O(infinity)).
Taking n = 7, we can think of S^8 as OP^1, the space of 1dimensional
octonionic subspaces of O^2. The "canonical line bundle" over this
gives the generator of KS^8, or in other words, pi_7(O(infinity)).
By Bott periodicity,
pi_7(O(infinity)) = pi_8(kO) = pi_0(kO)
so the canonical line bundle over OP^1 also defines an element of
pi_0(kO). But
pi_0(kO) = [S^0,kO] = KS^0
and KS^0 simply records the *difference in dimension* between
the two fibers of a vector bundle over S^0, which can be any
integer. This is why the octonions are related to dimension.
If for any pointed space we define
K^n(X) = K(S^n smash X)
we get a cohomology theory called Ktheory, and it turns out that
K^{n+8}(X) = K(X)
which is another say of stating Bott periodicity. Now if * denotes
a single point, K(*) is a ring (this is quite common for cohomology
theories), and it is generated by elements of degrees 1, 2, 4, and 8.
The generator of degree 8 is just the canonical line bundle over OP^1,
and multiplication by this generator gives a map
K^n(*) > K^{n+8}(*)
which is an isomorphism of groups  namely, Bott periodicity!
In this sense the octonions are responsible for Bott periodicity.

Addendum: The Clifford algebra clock is even better than I described
above, because it lets you work out the fancier Clifford algebras
C_{p,q}, which are generated by p square roots of 1 and q square
roots of 1, which all anticommute with each other. These Clifford
algebras are good when you have p dimensions of "space" and q
dimensions of "time", and I described the physically important case
where q = 1 in "week93". To figure them out, you just work out p  q
mod 8, look at what the clock says for that hour, and then take NxN
matrices of what you see, with N chosen so that C_{p,q} gets the right
dimension, namely 2^{p+q}. So say you're a string theorist and you
think there are 9 space dimensions and 1 time dimension. You say:
"Okay, 9  1 = 8, so I look and see what's at 8 o'clock. Okay, that's
R, the real numbers. But my Clifford algebra C_{9,1} is supposed to have
dimension 2^{9 + 1} = 1024 = 32^2, so my Clifford algebra must consist
of 32x32 *matrices* with real entries."
By the way, it's not so easy to see that the canonical line bundle
over OP^1 is the generator of KS^8  or equivalently, that left
multiplication by unit octonions defines a map from S^7 into SO(8)
corresponding to the generator of pi_7(O(infinity)). I claimed it's
true above, but when someone asked me why this was true, I realized
I couldn't prove it! That made me nervous. So I asked on
sci.math.research if it was really true, and I got this reply:
From: Linus Kramer
Newsgroups: sci.math.research
Subject: pi_7(O) and octonions
Date: Tue, 09 Nov 1999 12:44:33 +0100
John Baez asked if pi_7(O) is generated by
the (multiplication by) unit octonions.
View this as a question in KOtheory: the claim is
that H^8 generates the reduced real Ktheory
\tilde KO(S^8) of the 8sphere; the bundle
H^8 over S^8 is obtained by the standard glueing
process along the equator S^7, using the octonion
multiplication. So H^8 is the octonion Hopf bundle.
Its Thom space is the projective Cayley plane
OP^2. Using this and Hirzebruch's signature theorem,
one sees that the Pontrjagin class of H^8 is
p_8(H^8)=6x, for a generator x of the 8dimensional
integral cohomology of S^8 [a reference for this
calulation is my paper 'The topology of smooth
projective planes', Arch. Math 63 (1994)].
We have a diagram
cplx ch
KO(S^8) > K(S^8) > H(S^8)
the left arrow is complexification, the second arrow
is the Chern character. In dimension 8, these maps form
an isomorphism. Now ch(cplx(H^8))=8+x (see the formula
in the last paragraph in Husemoller's "Fibre bundles",
the chapter on "Bott periodicity and integrality
theorems". The constant factor is unimportant, so the
answer is yes, pi_7(O) is generated by the map
S^7> O which sends a unit octonion A to the
map l_A:x > Ax in SO(8).
Linus Kramer
More recently I got an email from Todd Trimble which cites another
reference to this fact:
From: Todd Trimble
Subject: Hopf bundles
To: John Baez
Date: Fri, 25 Mar 2005 16:37:11 0500
John,
In the book Numbers (GTM 123), there is an article by
Hirzebruch where the Bott periodicity result is formulated
as saying that the generators of \tilde KO(S^n) in the cases
n = 1, 2, 4, 8 are given by [eta]  1 where eta is the Hopf
bundle corresponding to R, C, H, O and 1 is the trivial
line bundle over these scalar "fields" (of real dimension
1, 2, 4, 8), and is 0 for n = 3, 5, 6, 7 [p. 294]. Also that
the Bott periodicity isomorphism
\tilde KO(S^n) > \tilde KO(S^{n+8})
is induced by [eta(O)]  1 [p. 295]. I know you are aware
of this already (courtesy of the response of Linus Kramers
to your sci.math.research query), but I thought you might
find a published reference, on the authority of no less than
Hirzebruch, handier (should you need it) than referring to a
sci.math.research exchange.
Unfortunately no proof is given. Hirzebruch says (p. 295),
Remark. Our formulation of the Bott periodicity theorem
will be found, in essentials, in [reference to Bott's Lectures
on K(X), without proofs]. A detailed proof within the
framework of Ktheory is given in the textbook [reference
to Karoubi's Ktheory]. The reader will have a certain amount
of difficulty, however, in extracting the results used here from
Karoubi's formulation.
Todd

"...for geometry, you know, is the gate of science, and the
gate is so low and small that one can only enter it as a little
child.  William Clifford

Previous issues of "This Week's Finds" and other expository articles
on mathematics and physics, as well as some of my research papers,
can be obtained at
http://math.ucr.edu/home/baez/README.html
For a table of contents of all the issues of This Week's Finds, try
http://math.ucr.edu/home/baez/twf.html
A simple jumpingoff point to the old issues is available at
http://math.ucr.edu/home/baez/twfshort.html
If you just want the latest issue, go to
http://math.ucr.edu/home/baez/this.week.html
ford modules, Topology
(3) 1964, 338.
For more stuff on division algebras and Bott periodicity try Dave Rusin's
web page, especially his answer to "Q5. What's the question with the
answer n = 1, 2, 4, or 8?"
3) Dave Rustwf_ascii/week106000064400020410000157000000736411061746651600141510ustar00baezhttp00004600000001Also available at http://math.ucr.edu/home/baez/week106.html
July 23, 1997
This Week's Finds in Mathematical Physics  Week 106
John Baez
Well, it seems I want to talk one more time about octonions before
moving on to other stuff. I'm a bit afraid this obsession with
octonions will mislead the nonexperts, fooling them into thinking
octonions are more central to mainstream mathematical physics than they
actually are. I'm also worried that the experts will think I'm spend
all my time brooding about octonions when I should be working on
practical stuff like quantum gravity. But darn it, this is summer
vacation! The only way I'm going to keep on cranking out "This Week's
Finds" is if I write about whatever I feel like, no matter how
frivolous. So here goes.
First of all, let's make sure everyone here knows what projective space
is. If you don't, I'd better explain it. This is honest mainstream
stuff that everyone should know, good nutritious mathematics, so I
won't need to feel too guilty about serving the extravagant octonionic
dessert which follows.
Start with R^n, good old ndimensional Euclidean space. We can imagine
wanting to "compactify" this so that if you go sailing off to infinity
in some direction you'll come sailing back from the other side like
Magellan. There are different ways to do this. A wellknown one is to
take R^n and add on one extra "point at infinity", obtaining the
ndimensional sphere S^n. Here the idea is that start anywhere in R^n
and start sailing in any direction, you are sailing towards this "point
at infinity".
But there is a sneakier way to compactify R^n, which gives us not the
ndimensional sphere but "projective nspace". Here we add on a lot of
points, one for each line through the origin. Now there are *lots* of
points at infinity, one for every direction! The idea here is that if
you start at the origin and start sailing along any straight line, you
are sailing towards the point at infinity corresponding to that line.
Sailing along any parallel line takes you twoards the same point at
infinity. It's a bit like a perspective picture where different
families of parallel lines converge to different points on the horizon
 the points on the horizon being points at infinity.
Projective nspace is also called RP^n. The R is for "real", since
this is actually "real projective nspace". Later we'll see what
happens if we replace the real numbers by the complex numbers,
quaternions, or octonions.
There are some other ways to think about RP^n that are useful either
for visualizing it or doing calculations. First a nice way to visualize
it. First take R^n and squash it down so it's just the ball of radius 1,
or more precisely, the "open ball" consisting of all vectors of length
less than 1. We can do this using a coordinate transformation like:
x > x' = x/sqrt(1 + x^2)
Here x stands for a vector in R^n and x is its length. Dividing the
vector x by sqrt(1 + x^2) gives us a vector x' whose length never
quite gets to 1, though it can get as close at it likes. So we have
squashed R^n down to the open ball of radius 1.
Now say you start at the origin in this squashed version of R^n and
sail off in any direction in a straight line. Then you are secretly
heading towards the boundary of the open ball. So the points an the
boundary of the open ball are like "points at infinity".
We can now compactify R^n by including these points at infinity. In
other words, we can work not with the open ball but with the "closed
ball" consisting of all vectors x' whose length is less than or equal
to 1.
However, to get projective nspace we also have to decree that antipodal
points x' and x' with x' = 1 are to be regarded as the same. In
other words, we need to "identify each point on the boundary of the
closed ball with its antipodal point". The reason is that we said that
when you sail off to infinity along a particular straight line, you are
approaching a particular point in projective nspace. Implicit in this
is that it doesn't matter which *way* you sail along that straight line.
Either direction takes you towards the same point in projective nspace!
This may seem weird: in this world, when the cowboy says "he went
thataway" and points at a particular point on the horizon, you gotta
remember that his finger points both ways, and the villian could equally
well have gone in the opposite direction. The reason this is good is
that it makes projective space into a kind of geometer's paradise: any
two lines in projective space intersect in a *single* point. No more
annoying exceptions: even "parallel" lines intersect in a single point,
which just happens to be a point at infinity. This simplifies life
enormously.
Okay, so RP^n is the space formed by taking a closed ndimensional ball
and identifying pairs of antipodal points on its boundary.
A more abstract way to think of RP^n, which is incredibly useful in
computations, is as the set of all lines through the origin in R^{n+1}.
Why is this the same thing? Well, let me illustrate it in an example.
What's the space of lines through the origin in R^3? To keep track of
these lines, draw a sphere around the origin. Each line through the
origin intersects this sphere in two points. Either one point is in the
northern hemisphere and the other is in the southern hemisphere, or
both are on the equator. So we can keep track of all our lines using
points on the northern hemisphere and the equator, but identifying
antipodal points on the equator. This is just the same as taking the
closed 2dimensional ball and identifying antipodal points on the
boundary! QED. The same argument works in higher dimensions too.
Now that we know a point in RP^n is just a line through the origin in
R^{n+1}, it's easy to put coordinates on RP^n. There's one line through
the origin passing through any point in R^{n+1}, but if we multiply the
coordinates (x_1,...,x_{n+1}) of this point by any nonzero number we
get the same line. Thus we can use a list of n+1 real numbers
to describe a point in RP^n, with the proviso that we get the same
point in RP^n if someone comes along and multiplies them all by some
nonzero number! These are called "homogeneous coordinates".
If you don't like the ambiguity of homogeneous coordinates, you can go
right ahead and divide all the coordinates by the real number x_1,
getting
(1, x_2/x_1, ..., x_{n+1}/x_1)
which lets us describe a point in RP^n by n real numbers, as befits an
ndimensional real manifold. Of course, this won't work if x_1
happens to be zero! But we can divide by x_2 if x_2 is nonzero, and
so on. *One* of them has to be nonzero, so we can cover RP^n with n+1
different coordinate patches corresponding to the regions where
different x_i's are nonzero. It's easy to change coordinates, too.
This makes everything very algebraic, which makes it easy to
generalize RP^n by replacing the real numbers with other number
systems. For example, to define "complex projective nspace" or CP^n,
just replace the word "real" by the word "complex" in the last two
paragraphs, and replace "R" by "C". CP^n is even more of a geometer's
paradise than RP^n, because when you work with complex numbers you can
solve all polynomial equations. Also, now there's no big difference
between an ellipse and a hyperbola! This sort of thing is why CP^n is
so widely used as a context for "algebraic geometry".
We can go even further and replace the real numbers by the
quaternions, H, defining the "quaternionic projective nspace" HP^n.
If we are careful about writing things in the right order, it's no
problem that the quaternions are noncommutative... we can still divide
by any nonzero quaternion, so we can cover HP^n with n+1 different
coordinate charts and freely change coordinates as desired.
We can try to go even further and use the octonions, O. Can we define
"octonionic projective nspace", OP^n? Well, now things get tricky!
Remember, the octonions are nonassociative. There's no problem
defining OP^1; we can cover it with two coordinate charts,
corresponding to homogeneous coordinates of the form
(x, 1)
and
(1, y),
and we can change coordinates back and forth with no problem. This
amounts to taking O and adding a single point at infinity, getting the
8dimensional sphere S^8. This is part of a pattern:
RP^1 = S^1
CP^1 = S^2
HP^1 = S^4
OP^1 = S^8
I discussed the implications of this pattern for Bott periodicity in
"week105".
We can also define OP^2. Here we have 3 coordinate charts corresponding
to homogeneous coordinates of the form
(1, y, z),
(x, 1, z),
and
(x, y, 1).
We can change back and forth between coordinate systems, but now we
have to *check* that if we start with the first coordinate system,
change to the second coordinate system, and then change back to the
first, we wind up where we started! This is not obvious, since
multiplication is not associative. But it works, thanks to a couple
of identities that are not automatic in the nonassociative context,
but hold for the octonions:
(xy)^{1} = y^{1} x^{1}
and
(xy)y^{1} = x.
Checking these equations is a good exercise for anyone who wants to
understand the octonions.
Now for the cool part: OP^2 is where it ends!
We can't define OP^n for n greater than 2, because the
nonassociativity keeps us from being able to change coordinates a
bunch of times and get back where we started! You might hope that we
could weasel out of this somehow, but it seems that there is a real
sense in which the higherdimensional octonionic projective spaces
don't exist.
So we have a fascinating situation: an infinite tower of RP^n's, an
infinite tower of CP^n's, an infinite tower of HP^n's, but an abortive
tower of OP^n's going only up to n = 2 and then fizzling out. This
means that while all sorts of geometry and group theory relating to
the reals, complexes and quaternions fits into infinite systematic
patterns, the geometry and group theory relating to the octonions is
quirky and mysterious.
We often associate mathematics with "classical" beauty, patterns
continuing ad infinitum with the ineluctable logic of a composition by
some divine Bach. But when we study OP^2 and its implications, we see
that mathematics also has room for "exceptional" beauty, beauty that
flares into being and quickly subsides into silence like a piece by
Webern. Are the fundamental laws of physics based on "classical"
mathematics or "exceptional" mathematics? Since our universe seems
unique and special  don't ask me how would we know if it weren't 
Witten has suggested the latter. Indeed, it crops up a lot in
string theory. This is why I'm trying to learn about the octonions:
a lot of exceptional objects in mathematics are tied to them.
I already discussed this a bit in "week64", where I sketched how there
are 3 infinite sequences of "classical" simple Lie groups corresponding
to rotations in R^n, C^n, and H^n, and 5 "exceptional" simple Lie groups
related to the octonions. After studying it all a bit more, I can now
go into some more detail.
In order of increasing dimension, the 5 exceptional Lie groups are
called G2, F4, E6, E7, and E8. The smallest, G2, is easy to
understand in terms of the octonions: it's just the group of
symmetries of the octonions as an algebra. It's a marvelous fact that
all the bigger ones are related to OP^2. This was discovered by
people like Freudenthal and Tits and Vinberg, but a great place to
read about it is the following fascinating book:
1) Boris Rosenfeld, Geometry of Lie Groups, Kluwer Academic Publishers,
1997.
The space OP^2 has a natural metric on it, which allows us to measure
distances between points. This allows us to define a certain symmetry
group OP^2, the group of all its "isometries", which are
transformations preserving the metric. This symmetry group is F4!
However, there is another bigger symmetry group of OP^2. As in real
projective nspace, the notion of a "line" makes sense in OP^2. One
has to be careful: these are octonionic "lines", which have 8 real
dimensions. Nonetheless, this lets us define the group of all
"collineations" of OP^2, that is, transformations that take lines to
lines. This symmetry group is E6! (Technically speaking, this is a
"noncompact real form" of E6; the rest of the time I'll be talking
about compact real forms.)
To get up to E7 and E8, we need to take a different viewpoint, which
also gives us another way to get E6. The key here is that the tensor
product of two algebras is an algebra, so we can tensor the octonions
with R, C, H, or O and get various algebras:
The algebra (R tensor O) is just the octonions.
The algebra (C tensor O) is called the "bioctonions".
The algebra (H tensor O) is called the "quateroctonions".
Finally, the algebra (O tensor O) is called the "octooctonions".
I'm not making this up: it's all in Rosenfeld's book! The poet Lisa
Raphals suggested calling the octooctonions the "highoctane
octonions", which I sort of like. But compared to Rosenfeld, I'm a
model of restraint: I won't even mention the dyoctonions, duoctonions,
split octonions, semioctonions, split semioctonions, 1/4octonions or
1/8octonions  for the definitions of these, you'll have to read his
book.
Apparently one can define projective planes for all of these algebras,
and all these projective planes have natural metrics on them, all of
them same general form. So each of these projective planes has a group
of isometries. And, lo and behold:
The group of isometries of the octonionic projective plane is F4.
The group of isometries of the bioctonionic projective plane is E6.
The group of isometries of the quateroctonionic projective plane is E7.
The group of isometries of the octooctonionic projective plane is E8.
Now I still don't understand this as well as I'd like to  I'm not
sure how to define projective planes for all these algebras (though I
have my guesses), and Rosenfeld is unfortunately a tad reticent on
this issue. But it looks like a cool way to systematize the study of
the expectional groups! That's what I want: a systematic theory of
exceptions.
I want to say a bit more about the above, but first let me note that
there are lots of other ways of thinking about the exceptional groups.
A great source of information about them is the following posthumously
published book by the great topologist Adams:
2) John Frank Adams, Lectures on Exceptional Lie Groups, eds. Zafer
Mahmud and Mamoru Mimura, University of Chicago Press, Chicago, 1996.
He has a bit about octonionic constructions of G2 and F4, but mostly he
concentrates on constructions of the exceptional groups using classical
groups and spinors.
In "week90" I explained Kostant's constructions of F4 and E8 using
spinors in 8 dimensions and triality  which, as noted in "week61",
is just another way of talking about the octonions. Unfortunately I
don't yet see quite how this relates to the above stuff, nor do I see
how to get E6 and E7 in a beautiful way using Kostant's setup.
There's also a neat construction of E8 using spinors in 16 dimensions!
Adams gives a nice explanation of this, and it's also discussed in the
classic tome on string theory:
3) Michael B. Green, John H. Schwarz, and Edward Witten, Superstring
Theory, two volumes, Cambridge U. Press, Cambridge, 1987.
The idea here is to take the direct sum of the Lie algebra so(16) and
its 16dimensional lefthanded spinor representation Spin+ to get the
Lie algebra of E8. The bracket of two guys in so(16) is defined as
usual, the bracket of a guy in so(16) and a guy in Spin+ is defined to
be the result of acting on the latter by the former, and the bracket
of two guys in Spin+ is defined to be a guy in Spin+ by dualizing the
map
so(16) x Spin+ > Spin+
to get a map
Spin+ x Spin+ > so(16).
This is a complete description of the Lie algebra of E8!
Anyway, there are lots of different ways of thinking about exceptional
groups, and a challenge for the octonionic approach is to systematize
all these ways.
Now I want to wrap up by saying a bit about how the exceptional Jordan
algebra fits into the above story. Jordan algebras were invented as a
way to study the selfadjoint operators on a Hilbert space, which
represent observables in quantum mechanics. If you multiply two
selfadjoint operators A and B the result needn't be selfadjoint, but
the "Jordan product"
A o B = (AB + BA)/2
is selfadjoint. This suggests seeing what identities the Jordan
product satisfies, cooking up a general notion of "Jordan algebra",
seeing how much quantum mechanics you can do with an arbitrary Jordan
algebra of observables, and classifying Jordan algebras if possible.
We can define a "projection" in a Jordan algebra to be an element A
with A o A = A. If our Jordan algebra consists of selfadjoint
operators on the complex Hilbert space C^n, a projection is a
selfadjoint operator whose only eigenvalues are zero and one.
Physically speaking, this corresponds to a "yesorno question" about
our quantum system. Geometrically speaking, such an operator is a
projection onto some subspace of our Hilbert space. All this stuff also
works if we start with the real Hilbert space R^n or the quaternionic
Hilbert space H^n.
In these special cases, one can define a "minimal projection" to be a
projection on a 1dimensional subspace of our Hilbert space.
Physically, minimal projections correspond to "pure states"  states of
affairs in which the answer to some maximally informative question is
"yes", like "is the z component of the angular momentum of this spin1/2
particle equal to 1/2?" Geometrically, the space of minimal projections
is just the space of "lines" in our Hilbert space. This is either
RP^{n1}, or CP^{n1}, or HP^{n1}, depending on whether we're working
with the reals, complexes or quaternions. So: the space of pure states
of this sort of quantum system is also a projective space! The relation
between quantum theory and "projective geometry" has been intensively
explored for many years. You can read about it in:
4) V. S. Varadarajan, Geometry of Quantum Theory, SpringerVerlag,
Berlin, 2nd ed., 1985.
Most people do quantum mechanics with complex Hilbert spaces. Real
Hilbert spaces are apparently too boring, but some people have
considered the quaternionic case:
5) Stephen L. Adler, Quaternionic Quantum Mechanics and Quantum Fields,
Oxford U. Press, Oxford, 1995.
If our Hilbert space is the complex Hilbert space C^n, its group of
symmetries is usually thought of as U(n)  the group of nxn unitary
matrices. This group also acts as symmetries on the Jordan algebra of
selfadjoint nxn complex matrices, and also on the space CP^{n1}.
Similarly, if we start with R^n, we get the group of orthogonal nxn
matrices O(n), which acts on the Jordan algebra of real selfadjoint
nxn matrices and on RP^{n1}. Likewise, if we start with H^n, we get
the group Sp(n), which acts on the Jordan algebra of quaternionic
selfadjoint nxn matrices and on HP^{n1}. This pretty much explains
how the classical groups are related to different flavors of quantum
mechanics.
Now what about the octonions? Well, here we can only go up to n = 3,
basically for the reasons explained before: the same stuff that keeps
us from defining octonionic projective spaces past a certain point
keeps us from getting Jordan algebras! The interesting case is the
Jordan algebra of 3 x 3 selfadjoint octonionic matrices. This is
called the "exceptional Jordan algebra", J. The group of symmetries
of this is  you guessed it, F4. One can also define a "minimal
projection" in J and the space of these is OP^2.
Is it possible that octonionic quantum mechanics plays some role in
physics?
I don't know.
Anyway, here is my hunch about the bioctonionic, quateroctonionic, and
octooctonionic projective planes. I think to define them you should
probably tensor the exceptional Jordan algebra with C, H, and O,
respectively, and take the space of minimal projections in the
resulting algebra. Rosenfeld seems to suggest this is the way to go.
However, I'm vague about some important details, and it bugs me,
because the special identities I needed above to define OP^2 are
related to O being an alternative algebra, but C tensor O, H tensor O
and O tensor O are not alternative.
I should add that in addition to octonionic projective geometry, one
can do octonionic hyperbolic geometry. One can read about this in
Rosenfeld and also in the following:
6) Daniel Allcock, Reflection groups on the octave hyperbolic plane,
University of Utah Mathematics Department preprint.
Quote of the week:
"Mainstream mathematics" is a name given to mathematics that more
fittingly belongs on Sunset Boulevard  GianCarlo Rota, Indiscrete Thoughts

Addenda: Here's an email from David Broadhurst, followed by various
remarks:
John:
Shortly before his death I spent a charming afternoon with Paul Dirac.
Contrary to his reputation, he was most forthcoming.
Among many things, I recall this: Dirac explained that while trained
as an engineer and known as a physicist, his aesthetics were mathematical.
He said (as I can best recall, nearly 20 years on): At a young age,
I fell in love with projective geometry. I always wanted to use to
use it in physics, but never found a place for it.
Then someone told him that the difference between complex and
quaternionic QM had been characterized as the failure of theorem in
classical projective geometry.
Dirac's face beamed a lovely smile: Ah he said, it was just such a
thing that I hoped to do.
I was reminded of this when bactracking to your "week106", today.
Best
David
The theorem that fails for quaternions but holds for R and C is the
"Pappus theorem", discussed in "week145".
Next, a bit about OP^n. There are different senses in which we can't
define OP^n for n greater than 2. One is that if we try to define
coordinates on OP^n in a similar way to how we did it for OP^2,
nonassociativity keeps us from being able to change coordinates a
bunch of times and get back where we started! It's definitely
enlightening to see how the desired transition functions g_{ij} fail
to satisfy the necessary cocycle condition g_{ij} g_{jk} = g_{ik}
when we get up to OP^3, which would require 4 charts.
But, a deeper way to think about this emerged in conversations
I've had with James Dolan. Stasheff invented a notion of
"A_infinity space", which is a pointed topological space with
a product that is associative up to homotopy which satisfies the
pentagon identity up to... etc. Any A_infinity space G has a
classifying space BG such that
Loops(BG) ~ G.
In other words, BG is a pointed space such that the space of loops
based at this point is homotopy equivalent to G. One can form this
space BG by the Milnor construction: sticking in one 0simplex, one
1simplex for every point of G, one 2simplex for every triple (g,h,k)
with gh = k, one 3simplex for every associator, and so on. If we do
this where G is the group of lengthone elements of R (i.e. Z/2) we get
RP^infinity, as we expect, since
RP^infinity = B(Z/2).
Even better, at the nth stage of the Milnor construction we get a
space homeomorphic to RP^n. Similarly, if we do this where G is
the group of lengthone elements of C or H we get CP^infinity or
HP^infinity. But if we take G to be the units of O, which has a
product but is not even homotopyassociative, we get OP^1 = S^7
at the first step, OP^2 at the second step, ... but there's no way
to perform the third step!
Next: here's a little more information on the octonionic, bioctonionic,
quateroctonionic and octooctonionic projective planes. Rosenfeld
claims that the groups of isometries of these planes are F4, E6, E7,
and E8, respectively. The problem is, I can't quite understand
how he constructs these spaces, except for the plain octonionic
case.
It appears that these spaces can also be constructed using the
ideas in Adams' book. Here's how it goes.
The Lie algebra F4 has a subalgebra of maximal rank isomorphic to
so(9). The quotient space is 16dimensional  twice the dimension
of the octonions. It follows that the Lie group F4 mod the subgroup
generated by this subalgebra is a 16dimensional Riemannian manifold
on which F4 acts by isometries.
The Lie algebra E6 has a subalgebra of maximal rank isomorphic to
so(10) x u(1). The quotient space is 32dimensional  twice the
dimension of the bioctonions. It follows that the Lie group E6 mod
the subgroup generated by this subalgegra is a 32dimensional
Riemannian manifold on which E6 acts by isometries.
The Lie algebra E7 has a subalgebra of maximal rank isomorphic to
so(12) x su(2). The quotient space is 64dimensional  twice the
dimension of the quateroctonions. It follows that the Lie group E6
mod the subgroup generated by this subalgegra is a 64dimensional
Riemannian manifold on which E7 acts by isometries.
The Lie algebra E8 has a subalgebra of maximal rank isomorphic to so(16).
The quotient space is 128dimensional  twice the dimension of
the octooctonions. It follows that the Lie group E6 mod the
subgroup generated by this subalgegra is a 128dimensional Riemannian
manifold on which E8 acts by isometries.
According to:
6) Arthur L. Besse, Einstein Manifolds, Springer, Berlin, 1987, pp.
313316.
the above spaces are the octonionic, bioctonionic, quateroctonionic
and octooctonionic projective planes, respectively. However, I don't
yet fully understand the connection.
I thank Tony Smith for pointing out the reference to Besse
(who, by the way, is apparently a cousin of the famous Bourbaki).
Thanks also go to Allen Knutson for showing me a trick for finding
the maximal rank subalgebras of a simple Lie algebra.
Next, here's some more stuff about the biquaternions, bioctonions,
quaterquaternions, quateroctonions and octooctonions! I wrote this
extra stuff as part of a post to sci.physics.research on November 8,
1999....
One reason people like these algebras is that some of them  the
associative ones  are also Clifford algebras. I talked a bit about
Clifford algebras in "week105", but just remember that we define the
Clifford algebra C_{p,q} to be the associative algebra you get by taking
the real numbers and throwing in p square roots of 1 and q square
roots of 1, all of which anticommute with each other. This algebra is
very important for understanding spinors in spacetimes with p space
and q time dimensions. (It's also good for studying things in other
dimensions, so things can get a bit tricky, but I don't want to talk
about that now.)
For example: if you just thrown in one square root of 1 and no square
roots of 1, you get C_{1,0}  the complex numbers!
Similarly, one reason people like the quaternions is because they
are C_{2,0}. Start with the real numbers, throw in two square roots
of 1 called I and J, make sure they anticommute (IJ = JI) and voila 
you've got the quaternions!
Similarly, one reason people like the biquaternions is because
they are C_{2,1}. You take the quaternions and complexify them 
this amounts to throwing in an extra number i that's a square root
of 1 and commutes with the quaternionic I and J  and you get
an algebra which is also generated by I, J, and K = iI. Note
that I, J, and K all anticommute, and K is a square root of 1.
Thus the biquaternions are C_{2,1}!
Similarly, one reason people like the quaterquaternions is because
they are C_{2,2}. You take the quaternions and quaternionify them 
this amounts to throwing in two square roots of 1, say i and j,
which anticommute but which commute with the quaternionic I and J 
and you get an algebra which is also generated by I, J, K = iI, and
L = jI. Note that I, J, K, and L all anticommute, and K and L are
square roots of 1. Thus the quaterquaternions are C_{2,2}!
Now, as soon as we thrown the octonions into the mix we don't get
Clifford algebras anymore, since octonions aren't associative, while
Clifford algebras are. However, there are still relationships to
Clifford algebras. For example, suppose we look at all the linear
transformations of the octonions generated by the left multiplication
operations
x > ax
This is an associative algebra, and it turns out to be all linear
transformations of the octonions, regarded as an 8dimensional real
vector space. In short, it's just the algebra of 8x8 real matrices.
And this is C_{6,0}.
If you do the same trick for the bioctonions, quateroctonions and
octooctonions, you get other Clifford algebras... but I'll leave the
question of which ones as a puzzle for the reader. If you need some
help, look at the "Footnote" in "week105".
Perhaps the fanciest example of this trick concerns the
biquateroctonions. Now actually, I've never heard anyone use this
term for the algebra C tensor H tensor O! The main person interested
in this algebra is Geoffrey Dixon, and he just calls it T. But anyway,
if we look at the algebra of linear transformations of C tensor H tensor
O generated by left multiplications, we get something isomorphic to
the algebra of 16 x 16 complex matrices. And this in turn is isomorphic
to C_{9,0}.
The biquateroctonions play an important role in Dixon's grand unified
theory of the electromagnetic, weak and strong forces. There are lots
of nice things about this theory  for example, it gets the right
relationships between weak isospin and hypercharge for the fermions
in any one generation of the Standard Model (though, as in the Standard
Model, the existence of 3 generations needs to be put in "by hand").
It may or may not be right, but at least it comes within shooting distance!
You can read a bit more about his work in "week59".

Previous issues of "This Week's Finds" and other expository articles
on mathematics and physics, as well as some of my research papers,
can be obtained at
http://math.ucr.edu/home/baez/
For a table of contents of all the issues of This Week's Finds, try
http://math.ucr.edu/home/baez/twf.html
A simple jumpingoff point to the old issues is available at
http://math.ucr.edu/home/baez/twfshort.html
If you just want the latest issue, go to
http://math.ucr.edu/home/baez/this.week.html
but C tensor O, H tensor O
and O tensor O are not alternative.
I should add that in addition ttwf_ascii/week107000064400020410000157000000306321025632404500141300ustar00baezhttp00004600000001Also available at http://math.ucr.edu/home/baez/week107.html
August 19, 1997
This Week's Finds in Mathematical Physics  Week 107
John Baez
This summer I've been hanging out in Cambridge Massachusetts, working
on quantum gravity and also having some fun. Not so long ago I gave a
talk on cellular automata at Boston University, thanks to a kind
invitation from Bruce Boghosian, who is using cellular automata to
model cool stuff like emulsions:
1) Florian W. J. Weig, Peter V. Coveney, and Bruce M. Boghosian, Lattice
gas simulations of minorityphase domain growth in binary immiscible and
ternary amphiphilic fluid, preprint available as condmat/9705248.
As you add more and more of an amphiphilic molecule (e.g. soap) to a
binary immiscible fluid (e.g. oil and water), the boundary layer likes
to grow in area. This is why you wash your hands with soap. There
are various phases depending on the concentrations of the three
substances  a "spongy" phase, a "droplet phase", and so on  and
it is very hard to figure out what is going on quantitatively using
analytical methods.
Luckily, one can simulate this stuff using a cellular automaton!
Standard numerical methods for solving the NavierStokes equation tend
to outrun cellular automata when it comes to plain old hydrodynamics,
but with these fancy "ternary amphiphilic fluids", cellular automata
really seem to be the most practical way to study things  apart
from experiments, of course. This is very heartwarming to me, since
like many people I've been fond of cellular automata ever after
learning of John Conway's game of Life, and I've always hoped they
could serve some practical purpose.
I spoke about the thesis of my student James Gilliam and a paper
we wrote together:
2) James Gilliam, Lagrangian and Symplectic Techniques in Discrete
Mechanics, Ph.D. thesis, Department of Mathematics, University of
Riverside, 1996.
John Baez and James Gilliam, An algebraic approach to discrete mechanics,
Lett. Math. Phys. 31 (1994), 205212.
Here the idea was to set up as much as possible of the machinery of
classical mechanics in a purely discrete context, where time proceeds
in integer steps and the space of states is also discrete. The most
famous examples of this "discrete mechanics" are cellular automata,
which are the discrete analogs of classical field theories, but there
are also simpler systems more reminiscent of elementary classical
mechanics, like a particle moving on a line  where in this case the
"line" is the integers rather than the real numbers. It turns out
that with a little skullduggery one can apply the techniques of
calculus to some of these situations, and do all sorts of stuff like
prove a discrete version of Noether's theorem  the famous theorem
which gives conserved quantities from symmetries.
After giving this talk, I visited my friend Robert Kotiuga in the
Functorial Electromagnetic Analysis Lab in the Photonics Building at
Boston University. "Photonics" is the currently fashionable term for
certain aspects of optics, particularly quantum optics. As befits its
flashy name, the Photonics Building is brand new and full of gadgets
like a device that displays Maxwell's equations in moving lights when
you speak the words "Maxwell's equations" into an inconspicuous
microphone. (It also knows other tricks.) Robert told me about what
he's been doing lately with topology and finiteelement methods for
solving magnetostatics problems  this blend of higbrow math and
practical engineering being the reason for the somewhat
tongueincheek name of his office, inscribed soberly on a plaque
outside the door.
Like the topologist Raoul Bott, Kotiuga started in electrical
engineering at McGill University, and gradually realized how much
topology there is lurking in electrical circuit theory and Maxwell's
equations. Apparently a paper of his was the first to cite Witten's
famous work on ChernSimons theory  though presumably this is
merely a testament to the superiority of engineers over mathematicians
and physicists when it comes to rapid publication. In fluid dynamics,
the integral of the following quantity
v . curl(v)
(where v is the velocity vector field) is known as the "helicity
functional". Kotiuga been studying applications of the same mathematical
object in the context of magnetostatics, namely
A . curl(A)
where A is the magnetic vector potential. It shows up in impedance
tomography, for example. But in quantum field theory, a
generalization of this quantity to other forces is known as the
"ChernSimons functional", and Witten's work on the 3dimensional
field theory having this as its Lagrangian turned out to revolutionize
knot theory. Personally, I'm mainly interested in the applications to
quantum gravity  see "week56" for a bit about this. Here are some
papers Kotiuga has written on the helicity functional, or what we
mathematicians would call "U(1) ChernSimons theory":
3) P. R. Kotiuga, Metric dependent aspects of inverse problems and
functionals based helicity, Journal of Applied Physics, 70 (1993),
54375439.
Analysis of finite element matrices arising from discretizations of
helicity functionals, Journal of Applied Physics, 67 (1990),
58155817.
Helicity functionals and metric invariance in three dimensions, IEEE
Transactions on Magnetics, MAG25 (1989), 28132815.
Variational principles for threedimensional magnetostatics based on
helicity, Journal of Applied Physics, 63 (1988), 33603362.
Later Jon Doyle, a computer scientist at M.I.T. who had been to my
talk, invited me to a seminar at M.I.T. where I met Gerald Sussman,
who with Jack Wisdom has run the best longterm simulations of the
solar system, trying to settle the old question of whether the darn
thing is stable! It turns out that the system is afflicted with
chaos and can only be predicted with any certainty for about 4
million years... though their simulation went out to 100 million.
Here are some fun facts: 1) They need to take general relativity into
account even for the orbit of Jupiter, which precesses about one
radian per billion years. 2) They take the asteroid belt into account
only as modification of the sun's quadrupole moment (which they also
use to model its oblateness). 3) The most worrisome thing about the
whole simulation  the most complicated and unpredictable aspect of
the whole solar system in terms of its gravitational effects on
everything else  is the EarthMoon system, with its big tidal
effects. 4) The sun loses one Earth mass per 100 million years due to
radiation, and another quarter Earth mass due to solar wind. 5) The
first planet to go is Mercury! In their simulations, it eventually
picks up energy through a resonance and drifts away.
For more, try:
4) Gerald Jay Sussman and Jack Wisdom, Chaotic evolution of the solar system,
Science, 257, 3 July 1992.
Gerald Jay Sussman and Jack Wisdom, Numerical evidence that the motion
of Pluto is chaotic, Science, 241, 22 July 1988.
James Applegate, M. Douglas, Y. Gursel, Gerald Jay Sussman, Jack Wisdom,
The outer solar system for 200 million years, Astronomical Journal, 92,
pp 176194, July 1986, reprinted in Lecture Notes in Physics #267 
Use of Supercomputers in Stellar Dynamics, Springer Verlag, 1986.
James Applegate, M. Douglas, Y. Gursel, P Hunter, C. Seitz, Gerald Jay
Sussman, A digital orrery, in IEEE Transactions on Computers, C34,
No. 9, pp. 822831, September 1985, reprinted in Lecture Notes in
Physics #267, Springer Verlag, 1986.
Meanwhile, I've also been trying to keep up with recent developments
in ncategory theory. Some readers of "This Week's Finds" have expressed
frustration with how I keep tantalizing all of you with the concept of
ncategory without ever quite defining it. The reason is that it's a
lot of work to write a nice exposition of this concept!
However, I eventually got around to taking a shot at it, so now you can
read this:
5) John Baez, An introduction to ncategories, to appear in
7th Conference on Category Theory and Computer Science, eds.
E. Moggi and G. Rosolini, Springer Lecture Notes in
Computer Science vol. 1290, Springer, Berlin. Preprint available
as qalg/9708005 or at http://math.ucr.edu/home/baez/ncat.ps
There are different definitions of "weak ncategory" out there now and
it will take a while of sorting through them to show a bunch are
equivalent and get the whole machinery running smoothly. In the
above paper I mainly talk about the definition that James Dolan
and I came up with. Here are some other new papers on this sort of
thing... I'll just list them with abstracts.
6) Claudio Hermida, Michael Makkai and John Power, On weak higher
dimensional categories, 104 pages, preprint available at
http://hypatia.dcs.qmw.ac.uk/authors/M/MakkaiM/papers/multitopicsets/
Inspired by the concept of opetopic set introduced in a recent
paper by John C. Baez and James Dolan, we give a modified notion
called multitopic set. The name reflects the fact that, whereas
the Baez/Dolan concept is based on operads, the one in this paper
is based on multicategories. The concept of multicategory used here
is a mild generalization of the samenamed notion introduced by
Joachim Lambek in 1969. Opetopic sets and multitopic sets are both
intended as vehicles for concepts of weak higher dimensional category.
Baez and Dolan define weak ncategories as (n+1)dimensional opetopic
sets satisfying certain properties. The version intended here,
multitopic ncategory, is similarly related to multitopic sets.
Multitopic ncategories are not described in the present paper;
they are to follow in a sequel. The present paper gives complete details
of the definitions and basic properties of the concepts involved with
multitopic sets. The category of multitopes, analogs of the opetopes
of Baez and Dolan, is presented in full, and it is shown that the
category of multitopic sets is equivalent to the category of set
valued functors on the category of multitopes.
7) Michael Batanin, Finitary monads on globular sets and notions of
computad they generate, available as postscript files at
http://wwwmath.mpce.mq.edu.au/~mbatanin/papers.html
Consider a finitary monad on the category of globular sets. We prove
that the category of its algebras is isomorphic to the category of
algebras of an appropriate monad on the special category (of
computads) constructed from the data of the initial monad. In the case
of the free ncategory monad this definition coincides with R.
Street's definition of ncomputad. In the case of a monad generated
by a higher operad this allows us to define a pasting operation in a
weak ncategory. It may be also considered as the first step toward
the proof of equivalence of the different definitions of weak
ncategories.
8) Carlos Simpson, Limits in ncategories, approximately 90 pages,
preprint available as alggeom/9708010.
We define notions of direct and inverse limits in an ncategory. We
prove that the (n+1)category nCAT' of fibrant ncategories admits
direct and inverse limits. At the end we speculate (without proofs) on
some applications of the notion of limit, including homotopy fiber
product and homotopy coproduct for ncategories, the notion of
nstack, representable functors, and finally on a somewhat different
note, a notion of relative Malcev completion of the higher homotopy at
a representation of the fundamental group.
9) Sjoerd Crans, Generalized centers of braided and sylleptic
monoidal 2categories, Adv. Math. 136 (1998), 183223.
Recent developments in higherdimensional algebra due to Kapranov and
Voevodsky, Day and Street, and Baez and Neuchl include definitions of
braided, sylleptic and symmetric monoidal 2categories, and a center
construction for monoidal 2categories which gives a braided monoidal
2category. I give generalized center constructions for braided and
sylleptic monoidal 2categories which give sylleptic and symmetric
monoidal 2categories respectively, and I correct some errors in the
original center construction for monoidal 2categories.

Previous issues of "This Week's Finds" and other expository articles
on mathematics and physics, as well as some of my research papers,
can be obtained at
http://math.ucr.edu/home/baez/
For a table of contents of all the issues of This Week's Finds, try
http://math.ucr.edu/home/baez/twf.html
A simple jumpingoff point to the old issues is available at
http://math.ucr.edu/home/baez/twfshort.html
If you just want the latest issue, go to
http://math.ucr.edu/home/baez/this.week.html
twf_ascii/week108000064400020410000157000000411560774011336300141370ustar00baezhttp00004600000001Also available at http://math.ucr.edu/home/baez/week108.html
September 22, 1997
This Week's Finds in Mathematical Physics  Week 108
John Baez
In the Weeks to come I want to talk about quantum gravity, and
especially the relation between general relativity and spinors, since
Barrett and Crane and I have some new papers out about how you can
describe "quantum 4geometries"  geometries of spacetime which have a
kind of quantum discreteness at the Planck scale  starting from the
mathematics of spinors.
But first I want to say a bit about CTCS '97  a conference on
category theory and computer science organized by Eugenio Moggi and
Giuseppe Rosolini. It was so wellorganized that they handed
us the conference proceedings when we arrived:
1) Eugenio Moggi and Giuseppe Rosolini, eds., Category Theory and
Computer Science, Lecture Notes in Computer Science 1290, Springer
Verlag, Berlin, 1997.
It was held in Santa Margherita Ligure, a picturesque little Italian
beach town near Genoa  the perfect place to spend all day in the
basement of a hotel listening to highly technical lectures. It's near
Portofino, famous for its big yachts full of rich tourists, but I didn't
get that far. The food was great, though, and it was nice to see the
lazy waves of the Mediterranean, so different from the oceans I know and
love. The vegetation was surprisingly similar to that in Riverside:
lots of palm trees and cacti.
I spoke about ncategories, with only the barest mention of their
possible relevance to computer science. But I was just the token
mathematical physicist in the crowd; most of the other participants were
pretty heavily into "theoretical computer science"  a subject that
covers a lot of newfangled aspects of what used to be called "logic".
What's neat is that I almost understood some of these talks, thanks to
the fact that category theory provides a highly general language for
talking about processes.
What's a computer, after all, but a physical process that simulates
fairly arbitrary processes  including other physical processes? As
we simulate more and more physics with better and better computers based
on more and more physics, it seems almost inevitable that physics and
computer science will come to be seen as two ends of a more general
theory of processes. No?
A nice example of an analogy between theoretical computer science and
mathematical physics was provided by Gordon Plotkin (in the plane,
on the way back, when I forced him to explain his talk to me). Computer
scientists like to define functions recursively. For example, we
can define a function from the natural numbers to the natural numbers:
f: N > N
by its value at 0 together with a rule to get its value at n+1 from
its value at n:
f(0) = c
f(n+1) = g(f(n))
Similarly, physicists like to define functions by differential equations.
For example, we can define a function from the real numbers to the real
numbers:
f: R > R
by its value at 0 together with a rule to get its derivative from its
value:
f(0) = c
f'(t) = g(f(t))
In both cases a question arises: how do we know we've really defined a
function? And in both cases, the answer involves a "fixedpoint
theorem". In both cases, the equations above define the function f *in
terms of itself*. We can write this using an equation of the form:
f = F(f)
where F is some operator that takes functions to functions. We say
f is a "fixed point for F" if this holds. A fixedpoint theorem is
something that says there exists a solution, preferably unique, of
this sort of equation.
But how do we describe this operator F more precisely in these examples?
In the case of the definition by recursion, here's how: for any function
f: N > N, we define the function F(f): N > N by
F(f)(0) = c
F(f)(n+1) = g(f(n))
The principle of mathematical induction says that any operator F of this
sort has a unique fixed point.
Similarly, we can formulate the differential equation above as a fixed
point problem by integrating both sides, obtaining:
f(t) = c + integral_0^t g(f(s)) ds
which is an example of an "integral equation". If we call the function
on the right hand side F(f), then this integral equation says
f = F(f)
In this case, "Picard's theorem on the local existence and uniqueness of
solutions of ordinary differential equations" is what comes to our
rescue and asserts the existence of a unique fixed point.
You might wonder how Picard's theorem is proved. The basic idea of the
proof is very beautiful, because it *takes advantage* of the frightening
circularity implicit in the equation f = F(f). I'll sketch this idea,
leaving out all the details.
So, how do we solve this equation? Let's see what we can do with it.
There's not much to do, actually, except substitute the left side into
the right side and get:
f = F(F(f)).
Hmm. Now what? Well, we can do it again:
f = F(F(F(f)))
and again:
f = F(F(F(f)))).
Are we having fun yet? It look like we're getting nowhere fast...
or even worse, getting nowhere *slowly*! Can we repeat this process so
much that the f on the righthand side goes away, leaving us with the
solution we're after:
f = F(F(F(F(F(F(F(F(F(F(F(F(F(F(F(F(F(F(F(F(F(F(F(F(F(F(F(F(F(F(F(F..... ?
Well, actually, yes, if we're smart. What we do is this. We start by
*guessing* the solution to our equation. How do we guess? Well, our
solution f should have f(0) = 0, so just start with any function with
this property. Call it f_1. Then we improve this initial guess
repeatedly by letting
f_2 = F(f_1)
f_3 = F(f_2)
f_4 = F(f_3)
and so on. Now for the fun part: we show that these guesses get closer
and closer to each other... so that they converge to some function f
with f = F(f)! Voila! With a little more work we can show that no
matter what our initial guess was, our subsequent guesses approach the
same function f, so that the solution f is unique.
I'm glossing over some details, of course. To prove Picard's theorem we
need to assume the function g is reasonably nice (continuous isn't nice
enough, we need something like "Lipschitz continuous"), and our initial
guess should be reasonably nice (continuous will do here). Also,
Picard's theorem only shows that there's a solution defined on some
finite time interval, not the whole real line. (This little twist is
distressing to Plotkin since it complicates the analogy with
mathematical induction. But there must be some slick way to save the
analogy; it's too cute not to be important!)
You can read about Picard's theorem and other related fixedpoint
theorems in any decent book on analysis. Personally I'm fond of:
2) Michael Reed and Barry Simon, Methods of Modern Mathematical Physics.
Vol. 1: Functional Analysis. Vol. 2: Fourier Analysis, SelfAdjointness.
Vol. 3: Scattering Theory. Vol. 4: Analysis of Operators. Academic
Press, New York, 1980.
which is sort of the bible of analysis for mathematical physicists.
Now, it may seem a bit overelaborate to reformulate the principle of
mathematical induction as a fixed point theorem. However, this way of
looking at recursion is the basis of a lot of theoretical computer
science. It applies not only to recursive definitions of functions
but also recursive definitions of "types" like those given in "Backus
Naur form"  a staple of computer science.
Let me take a simple example that Jim Dolan told me about. Suppose we
have some set of "letters" and we want to define the set of all nonempty
"words" built from these letters. For example, if our set of letters
was L = {a,b,c} then we would get an infinite set W of words like a, ca,
bb, bca, cbabba, and on.
In BackusNaur form we might express this as follows:
letter ::= a  b  c
word ::= 
In English the first line says "a letter is either a, b, or c", while
the second says "a word is either a letter or a word followed by a
letter". The second one is the interesting part because it's recursive.
In the language of category theory we could say the same thing as
follows. Let L be our set of letters. Given any set S, let
F(S) = L + S x L
where + means disjoint union and x means Cartesian product. Then the
set of "words" built from the letters in L satisfies W = F(W), or in
other words,
W = L + W x L.
This says "a word is either a letter or an ordered pair consisting of a
word followed by a letter." In short, we have a fixed point on our
hands!
How do we solve this equation? Well, now I'm going to show you
something they never showed you when you first learned set theory. We
just use the usual rules of algebra:
W = L + W x L
W  W x L = L
W x (1  L) = L
W = L/(1  L)
and then expand the answer out as a Taylor series, getting
W = L + L x L + L x L x L + ...
This says "a word is either a letter or an ordered pair of letters or an
ordered triple of letters or..." Black magic, but it works!
Now, you may wonder exactly what's going on  when we're allowed to
subtract and divide sets and expand functions of sets in Taylor series
and all that. I'm not an expert on this, but one place to look is in
Joyal's work on "analytic functors" (functors that you can expand in
Taylor series):
3) Andre Joyal, Une th'eorie combinatoire des s'eries formelles,
Advances in Mathematics 42 (1981), 182.
Before I explain a little of the idea behind this black magic, let me do
another example. I already said that the principle of mathematical
induction could be thought of as guaranteeing the existence of certain
fixed points. But underlying this is something still more basic: the
set of natural numbers is also defined by a fixed point property!
Suppose we take our set of letters above to be set {0} which has only
one element. Then our set of words is {0,00,000,0000,0000,...}. We
can think of this as a funny way of writing the set of natural numbers,
so let's call it N. Also, let's follow von Neumann and define
1 = {0},
which is sensible since it's a set with one element. Then our fixed
point equation says:
N = N + 1
This is the basic fixed point property of the natural numbers.
At this point some of you may be squirming... this stuff looks a bit
weird when you first see it. To make it more rigorous I need to bring
in some category theory, so I'll assume you've read "week73" and
"week76" where I explained categories and functors and isomorphisms.
If you've got a function F: S > S from some set to itself, a fixed
point of F is just an element x for which F(x) is *equal* to x. But now
suppose we have a functor F: C > C from some category to itself.
What's a fixed point of this?
Well, we could define it as an object x of C for which F(x) = x.
But if you know a little category theory you'll know that this sort of
"strict" fixed point is very boring compared to a "weak" fixed point:
an object x of C equipped with an *isomorphism*
f: F(x) > x
Equality is dull, isomorphism is interesting. It's also very
interesting to consider a more general notion: a "lax" fixed point,
meaning an object x equipped with just a *morphism*
f: F(x) > x
Let's consider an example. Take our category C to be the category
of sets. And take our functor F to be the functor
F(x) = x + 1
by which we mean "disjoint union of the set x with the oneelement set"
 I leave it to you to check that this is a functor. A lax fixed
point of F is thus a set x equipped with a function
f: x + 1 > x
so the natural numbers N = {0,00,000,...} is a lax fixed point in an
obvious way... in fact a weak fixed point. So when I wrote N = N + 1
above, I was lying: they're not equal, they're just isomorphic.
Similarly with those other equations involving sets.
Now, just as any function from a set to itself has a *set* of fixed
points, any functor F from a category C to itself has a *category* of
lax fixed points. An object in this category is just an object x of C
equipped with a morphism f: F(x) > x, and a morphism from this object
to some other object g: F(y) > y is just a commutative square:
f
F(x) > x
 
F(h)  h
 
V g V
F(y) > y
In our example, the natural numbers is actually the "initial" lax
fixed point, meaning that in the category of lax fixed points there
is exactly one morphism from this object to any other.
So that's the real meaning of these funny recursive definitions in
BackusNaur form: we have a functor F from some category like Set to
itself, and we are defining an object by saying that it's the initial
lax fixed point of this functor. It's a soupedup version of defining
an element of a set as the unique fixed point of a function!
I should warn you that category theorists and theoretical computer
scientists usually say "algebra" of a functor instead of "lax fixed
point" of a functor. Anyway, this gives a bit of a flavor of what those
folks talk about.

Addendum: Here's an interesting email that Doug Merritt sent me
after reading the above stuff:
A little web searching and discussion with Andras Kornai yields the
following info.
The original work on representing grammars as power series is
N. Chomsky and M.P. Schutzenberger.
The algebraic theory of contextfree languages. In Computer
Programming and Formal Systems. NorthHolland Publishing Company, 1963.
...where Schutzenberger supplied the formal power series aspect,
basically just as the usual generating function trick.
The algebraic connection was developed through the 60's and 70's,
culminating in the work of Samuel Eilenberg, founder of category
theory, such as in
Eilenberg, Samuel
"Automata, Languages and Machines", NY, Academic
Press, 1974.
A lot of the work in the area comes under the heading "syntactic
semigroups", which is fairly selfexplanatory (and yields a lot of
hits when web surfing).
The question of expanding a grammar via synthetic division as usual
comes down to the question of whether it is represented as a complete
division algebra or not. Grammars are typically nonabelian, however
in order to use more powerful mathematical machinery, frequently
commutativity is often nonetheless assumed, and the grammar forced
into that Procrustean bed.
I happened across an interesting recent paper (actually a '94 PhD thesis)
that brings all the modern machinery to bear on this sort of thing (e.g.
explaining how to represent grammars by power series via the Langrange
Inversion Formula, and multinonterminal (multivariable) grammars via
the Generalized LIF), and that is even quite readable:
Ole Vilhelm Larsen
Computing OrderIndependent Statistical Characteristics
of Stochastic Contextfree Languages
html: http://cwis.auc.dk/phd/fulltext/larsen/html/index.html
acrobat: http://cwis.auc.dk/phd/fulltext/larsen/pdf/larsen.pdf
The html and acrobat thesis is in English, but the web pages are in
Danish, which is why I explicitly give both URLs. The abstract is too
long to quote here.
You probably know all this better than I, but: As for fixed points, the
original theorem by Banach applies only to contractive mappings, but
beginning in '68 a flood of new theorems applying to various different
noncontractive situations began to appear, and research continues hot
and heavy. One danger of simply assuming fixed points is that there may
be orbits rather than attractive basins, which I alluded to briefly in
my sci.math FAQ entry (which has become somewhat mangled over the
years) concerning the numeric solution of f(x) = x^x via direct fixed
point recurrence (F(F(F(F(F...(guess)..)))). The orbits cause oscillatory
instability in some regions such that it becomes appropriate to switch to
a different technique.
Anyway that's merely to say that there are indeed spaces where one
can't just assume a fixed point theorem and that this can have practical
implications.
Hope that's of some interest.
Doug

Doug Merritt doug@netcom.com
Professional Wildeyed Visionary Member, Crusaders for a Better Tomorrow
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Previous issues of "This Week's Finds" and other expository articles on
mathematics and physics, as well as some of my research papers, can be
obtained at
http://math.ucr.edu/home/baez/
For a table of contents of all the issues of This Week's Finds, try
http://math.ucr.edu/home/baez/twf.html
A simple jumpingoff point to the old issues is available at
http://math.ucr.edu/home/baez/twfshort.html
If you just want the latest issue, go to
http://math.ucr.edu/home/baez/this.week.html
s. Vol. 2: Fourier Analysis, SelfAdjointness.
Vol. 3: Scattering Theory. Vol. 4: Analysis of Operators. Academic
Press, New York, 1980.
which is sort of the bible of analysis for mathematical physicists.
Now, it may seem a bit overelaborate to reformulate the principle of
mathematical induction as a fixed point theorem. However, this way of
looking at recursion is the basis of a lot of theortwf_ascii/week109000064400020410000157000000427251015243177400141430ustar00baezhttp00004600000001Also available at http://math.ucr.edu/home/baez/week109.html
September 27, 1997
This Week's Finds in Mathematical Physics  Week 109
John Baez
In the Weeks to come I want to talk about quantum gravity. A lot of
cool things have been happening in this subject lately. But I want to
start near the beginning....
In the 1960's, John Wheeler came up with the notion of "spacetime foam".
The idea was that at very short distance scales, quantum fluctuations of
the geometry of spacetime are so violent that the usual picture of a
smooth spacetime with a metric on it breaks down. Instead, one should
visualize spacetime as a "foam", something roughly like a superposition
of all possible topologies which only looks smooth and placid on large
enough length scales. His arguments for this were far from rigorous;
they were based on physical intuition. Electromagnetism and all other
fields exhibit quantum fluctuations  so gravity should too. A wee
bit of dimensional analysis suggests that these fluctuations become
significant on a length scale around the Planck length, which is
about 10^{35} meters. This is very small, much smaller than what we
can probe now. Around this length scale, there's no reason to suspect
that "perturbative quantum gravity" should apply, where you treat
gravitational waves as tiny ripples on flat spacetime, quantize these,
and get a theory of "gravitons". Indeed, the nonrenormalizability
of quantum gravity suggests otherwise.
Wheeler didn't know what formalism to use to describe "spacetime foam",
but he was more concerned with building up a rough picture of it. Since
he is so eloquent, especially when he's giving handwaving arguments, let
me quote him here:
"No point is more central than this, that empty space is not empty. It
is the seat of the most violent physics. The electromagnetic field
fluctuates. Virtual pairs of positive and negative electrons, in
effect, are constantly being created and annihilated, and likewise pairs
of mu mesons, pairs of baryons, and pairs of other particles. All these
fluctuations coexist with the quantum fluctuations in the geometry and
topology of space. Are they additional to those geometrodynamic
zeropoint disturbances, or are they, in some sense not now
wellunderstood, mere manifestations of them?"
That's from:
1) Charles Misner, Kip Thorne and John Wheeler, Gravitation
Freeman Press, 1973.
It's in the famous last chapter called "Beyond the end of time". Strong
stuff! This is what got me interested in quantum gravity in college.
Later I came to prefer less florid writing, and realized how hard it was
to turn gripping prose into actual theories... but back then I ate it up
uncritically.
Part of Wheeler's vision was that ultimately physics is all about
geometry, and that particles might be manifestations of this geometry.
For example, electronpositron pairs might be ends of wormholes threaded
by electric field lines:
"In conclusion, the vision of Riemann, Clifford and Einstein, of a
purely geometric basis for physics, today has come to a higher state of
development, and offers richer prospects  and presents deeper
problems, than ever before. The quantum of action adds to this
geometrodynamics new features, of which the most striking is the
presence of fluctuations of the wormhole type throughout all space. If
there is any correspondence between this virtual foamlike structure and
the physical vacuum as it has come to be known through quantum
electrodynamics, then there seems to be no escape from identifying these
wormholes with `undressed electrons'. Completely different from these
`undressed electrons', according to all available evidence, are the
electrons and other particles of experimental physics. For these
particles the geometrodynamic picture suggests the model of collective
disturbances in a virtual foamlike vacuum, analogous to different kinds
of phonons or excitons in a solid."
That quote is from:
2) John Wheeler, Geometrodynamics, Academic Press, New York, 1962.
There were many problems with getting this wormhole picture of particles
to work. First, there was  and is!  no experimental evidence that
wormholes exist, virtual or otherwise. The main reason for believing in
virtual wormholes was the quantummechanical idea that "whatever is not
forbidden is required"... an idea which must be taken with a grain of
salt. Second, there was no mathematical model of "spacetime foam" or
"virtual wormholes". It was just a vague notion.
However, Wheeler was mainly worried about two other problems. First,
how can we relate a space with a wormhole to one without? Since the two
have different topologies, there can't be any continuous way of going
from one to the other. In response to this problem, he suggested that
the description of spacetime in terms of a smooth manifold was not
fundamental, and that we really need some more other description, some
sort of "pregeometry". Second, what about the fact that electrons have
spin 1/2? This means that when you turn one around 360 degrees it
doesn't come back to the same state: it picks up a phase of 1. Only
when you turn it around twice does it come back to its original state!
This is nicely described using the mathematics of "spinors", but *not* so
nicely described in terms of wormholes.
In his freewheeling, intuitive manner, Wheeler fastened on this
second problem as a crucial clue to the nature of "pregeometry":
"It is impossible to accept any description of elementary particles
that does not have a place for spin 1/2. What, then, has any purely
geometric description to offer in explanation of spin 1/2 in general?
More particularly and more importantly, what place is there in quantum
geometrodynamics for the neutrino  the only entity of halfintegral
spin that is a pure field in its own right, in the sense that it has
zero rest mass and moves at the speed of light? No clear or satisfactory
answer is known to this question today. Unless and until an answer
is forthcoming, *pure geometrodynamics must be judged deficient as a
basis of elementary particle physics*."
Physics moves in indirect ways. Though Wheeler's words inspired many
students of relativity, progress on "spacetime foam" was quite slow.
It's not surprising, given the thorny problems and the lack of a precise
mathematical model. Quite a bit later, Hawking and others figured out
how to do calculations involving virtual wormholes, virtual black holes
and such using a technique called "Euclidean quantum gravity". Pushed
to its extremes, this leads to a theory of spacetime foam, though not
yet a rigorous one (see "week67").
But long before that, Newman, Penrose, and others started finding
interesting relationships between general relativity and the mathematics
of spin1/2 particles... relationships that much later would yield a
theory of spacetime foam in which spinors play a crucial part!
The best place to read about spinorial techniques in general relativity
is probably:
3) Roger Penrose and Wolfgang Rindler, Spinors and SpaceTime. Vol. 1:
TwoSpinor Calculus and Relativistic Fields. Vol. 2: Spinor and Twistor
Methods in SpaceTime Geometry. Cambridge University Press, Cambridge,
19851986.
There are roughly 3 main aspects to Penrose's work on spinors and
general relativity. The first is the "spinor calculus", described in
volume 1 of these books. By now this is a standard tool in relativity,
and you can find introductions to it in many textbooks, like
"Gravitation" or Wald's more recent text:
4) Robert M. Wald, General Relativity, University of Chicago Press,
Chicago, 1984.
The second is "twistor theory", described in volume 2. This is
mathematically more elaborate, and it includes an ambitious program to
reformulate the laws of physics in such a way that massless spin1/2
particles, rather than points of spacetime, play the basic role.
The third is the theory of "spin networks", which was a very radical,
purely combinatorial approach to describing the geometry of space.
Penrose's inability to extend it to *spacetime* is what made him turn
later to twistor theory. Probably the best explanation of Penrose's
original spin network ideas can be found in the thesis of one of his
students:
5) John Moussouris, Quantum models of spacetime based on recoupling
theory, Ph.D. thesis, Department of Mathematics, Oxford University,
1983.
Here I want to talk about the spinor calculus, which is the most widely
used of these ideas. It's all about the rotation group in 3 dimensions
and the Lorentz group in 3+1 dimensions (by which we mean 3 space
dimensions and 1 time dimension). A lot of physics is based on these
groups. For general stuff about rotation groups and spinors in *any*
dimension, see "week61" and "week93". Here I'll be concentrating on
stuff that only works when we start with *3* space dimensions.
Now I will turn up the math level a notch....
In the quantum mechanics of angular momentum, what matters is not the
representations of the rotation group SO(3), but of its double cover
SU(2). This group has one irreducible unitary representation of each
dimension d = 1, 2, 3, etc.. Physicists prefer to call these the
"spinj" representations, where j = 0, 1/2, 1, etc.. The relation is
of course that 2j + 1 = d.
The spin0 representation is the trivial representation. Physicists
call vectors in this representation "scalars", since they are just
complex numbers. Particles transforming in the spin0 representation of
SU(2) are also called scalars. Examples include pions and other mesons.
The only *fundamental* scalar particle in the Standard Model is the
Higgs boson  hypothesized but still not seen.
The spin1/2 representation is the fundamental representation, in which
SU(2) acts on C^2 in the obvious way. Physicists call vectors in this
representation "spinors". Examples of spin1/2 particles include
electrons, protons, neutrons, and neutrinos. The fundamental spin1/2
particles in the Standard Model are the leptons (electron, muon, tau
and their corresponding neutrinos) and quarks.
The spin1 representation comes from turning elements of SU(2) into 3x3
matrices using the double cover SU(2) > SO(3). This is therefore also
called the "vector" representation. The spin1 particles in the
Standard Model are the gauge fields: the photon, the W and Z, and the
gluons.
Though you can certainly make composite particles of higher spin, like
hadrons and atomic nuclei, there are no fundamental particles of spin
greater than 1 in the Standard Model. But the Standard Model doesn't
cover gravity. In gravity, the spin2 representation is very important.
This comes from letting SO(3), and thus SU(2), act on symmetric
traceless 3x3 matrices in the obvious way (by conjugation). In
perturbative quantum gravity, gravitons are expected to be spin2
particles. Why is this? Well, a cheap answer is that the metric on
space is given by a symmetric 3x3 matrix. But this is not very
satisfying... I'll give a better answer later.
Now, the systematic way to get all these representations is to build
them out of the spin1/2 representation. SU(2) acts on C^2 in an
obvious way, and thus acts on the space of polynomials on C^2. The
space of homogeneous polynomials of degree 2j is thus a representation
of SU(2) in its own right, called the spinj representation. Since
multiplication of polynomials is commutative, in math lingo we say the
spinj representation is the "symmetrized tensor product" of 2j copies
of the spin1/2 representation. This is the mathematical sense in which
spin 1/2 is fundamental!
(In some sense, this means we can think of a spinj particle as built
from 2j indistinguishable spin1/2 bosons. But there is something odd
about this, since in physics we usually treat spin1/2 particles as
fermions and form *antisymmetrized* tensor products of them!)
Now let's go from space to spacetime, and consider the Lorentz
group, SO(3,1). Again it's not really this group but its double
cover that matters in physics; its double cover is SL(2,C). Note
that SL(2,C) has SU(2) as a subgroup just as SO(3,1) has SO(3) as
a subgroup; everything fits together here, in a very pretty way.
Now, while SU(2) has only one 2dimensional irreducible representation,
SL(2,C) has two, called the lefthanded and righthanded spinor
representations. The "lefthanded" one is the fundamental
representation, in which SL(2,C) acts on C^2 in the obvious way. The
"righthanded" one is the conjugate of this, in which we take the
complex conjugate of the entries of our matrix before letting it act on
C^2 in the obvious way. These two representations become equivalent
when we restrict to SU(2)... but for SL(2,C) they're not! For example,
when we study particles as representations of SL(2,C), it turns out that
neutrinos are lefthanded, while antineutrinos are righthanded.
All the irreducible representations of SL(2,C) on complex vector spaces
can be built up from the lefthanded and righthanded spinor
representations. Here's how: take the symmetrized tensor product
of 2j copies of the lefthanded spin representation and tensor it with
the symmetrized tensor product of 2k copies of the righthanded one.
We call this the (j,k) representation.
People in general relativity have a notation for all this stuff. They
write lefthanded spinors as gadgets with one "unprimed subscript", like
this:
v_A
where A = 1,2. Righthanded spinors are gadgets with one "primed
subscript", like:
w_{A'}
where A' = 1,2. As usual, fancier tensors have more subscripts.
For example, guys in the (j,k) representation have j unprimed subscripts
and k primed ones, and don't change when we permute the unprimed
subscripts among themselves, or the primed ones among themselves.
Now SO(3,1) has an obvious representation on R^4, called the "vector"
representation for obvious reasons. If we think of this as a
representation of SL(2,C), it's the (1,1) representation. So when
Penrose writes a vector in 4 dimensions, he can do it either the old
way:
v_a
where a = 0,1,2,3, or the new spinorial way:
v_{AA'}
where A,A' = 1,2.
Similarly, we can write *any* tensor using spinors with twice as many
indices. This may not seem like a great step forward, but it actually
was... because it lets us slice and dice concepts from general relativity
in interesting new ways.
For example, the Riemann curvature tensor describing the curvature of
spacetime is really important in relativity. It has 20 independent
components but it can split up into two parts, the Ricci tensor and Weyl
tensor, each of which have 10 independent components. Thanks to
Einstein's equation, the Ricci tensor at any point of spacetime is
determined by the matter there (or more precisely, by the flow of energy
and momentum through that point). In particular, the Ricci tensor is
zero in the vacuum. The Weyl tensor
C_{abcd}
describes aspects of curvature like gravitational waves or tidal forces
which can be nonzero even in the vacuum. In spinorial notation this is
C_{AA'BB'CC'DD'}
but we can also write it as
C_{AA'BB'CC'DD'} = Phi_{ABCD} epsilon_{A'B'} epsilon_{C'D'}
+ complex conjugate thereof
where
0 1
epsilon =
1 0
and Phi is the "Weyl spinor". The Weyl spinor is symmetric in all its 4
indices so it lives in the (2,0) representation of SL(2,C). Note that
this is a 5dimensional complex representation, so the Weyl spinor has
10 real degrees of freedom, just like the Weyl tensor  but these
degrees of freedom have been encoded in a very efficient way! Even
better, we see here why, in perturbative quantum gravity, the graviton
is a spin2 particle!
I'm only scratching the surface here, but the point is that spinorial
techniques are really handy all over general relativity. A great
example is Witten's spinorial proof of the positive energy theorem:
6) Edward Witten, A new proof of the positive energy theorem,
Commun. Math. Phys. 80 (1981), 381402.
This says that for any spacetime that looks like flat Minkowski space
off at spatial infinity, but possibly has gravitational radiation and
matter in the middle, the "ADM mass" is greater than or equal to zero as
long as the matter satisfies the "dominant energy condition", which says
that the speed of energy flow is less than the speed of light. What's
the ADM mass? Well, basically the idea is this: if we go off towards
spatial infinity, where spacetime is almost flat and general relativity
effects aren't too big, we can imagine measuring the mass of the stuff
in the middle by seeing how fast a satellite would orbit it. That's the
ADM mass. If the satellite is *attracted* by the stuff in the middle,
the ADM mass is positive. The proof of the positive energy theorem was
really complicated before Witten used spinors, which let you write the
ADM mass as an integral of an obviously nonnegative quantity.
Next time I'll talk about spin networks and how they show up in recent
work on quantum gravity. We'll see that the idea of building up everything
from the spin1/2 representation of SU(2) assumes grandiose proportions:
in this setup, *space itself* is built from spinors!

Previous issues of "This Week's Finds" and other expository articles on
mathematics and physics, as well as some of my research papers, can be
obtained at
http://math.ucr.edu/home/baez/
For a table of contents of all the issues of This Week's Finds, try
http://math.ucr.edu/home/baez/twf.html
A simple jumpingoff point to the old issues is available at
http://math.ucr.edu/home/baez/twfshort.html
If you just want the latest issue, go to
http://math.ucr.edu/home/baez/this.week.html
ress,
Chicago, 1984.
The second is "twistwf_ascii/week11000064400020410000157000000164500774011336300140470ustar00baezhttp00004600000001Newsgroups: sci.physics.research,sci.math
Subject: This Week's Finds in Mathematical Physics [Week 11]
Date:
From: baez@ucrmath.ucr.edu
Approved: jbaez@math.mit.edu
Week 11
I'm hitting the road again tomorrow and will be going to the Quantum
Topology conference in Kansas until Sunday, so I thought I'd post this
week's finds early. As a result they'll be pretty brief. Let me start
with one that I mentioned in week9 but is now easier to get:
1) Unique determination of an inner product by adjointness relations
in the algebra of quantum observables, by Alan D. Rendall, 10 pages, now
available as grqc/9303026.
and then mention another thing I've gotten as a spinoff from the
gravity conference at UCSB:
2) An algebraic approach to the quantization of constrained systems:
finite dimensional examples, by Ranjeet S. Tate, Syracuse University
physics department PhD dissertation, August 1992, SUGP92/81. (Tate
is now at rstate@cosmic.physics.ucsb.edu, but please don't ask him for
copies unless you're pretty serious, because it's big.)
Both the technical problems of "canonical" quantum gravity and one of the
main conceptual problems  the problem of time  stem from the fact that
general relativity is a system in which the initial data have
constraints. So improving our understanding of quantizing constrained
classical systems is important in understanding quantum gravity.
Let me say a few words about these constraints and what I mean by
"canonical" quantum gravity.
First consider the wave equation in 2 dimensions. This is
an equation for a function from R^2 to R, say phi(t,x), where t is a
timelike and x is a spacelike coordinate. The equation is simply
d^2 phi/dt^2  d^2phi/dx^2 = 0.
Now this equation can be rewritten as an evolutionary equation for
initial data as follows. We consider pairs of functions (Q,P) on R 
which we think of phi and d phi/dt on "space", that is, on a surface t =
constant. And we rewrite the secondorder equation above as a
firstorder equation:
d/dt (Q,P) = (P, d^2Q/dx^2). 1)
This is a standard trick. We call the space of pairs (Q,P) the "phase
space" of the theory. In canonical quantization, we treat this a lot
like the space R^2 of pairs (q,p) describing the initial position and
momentum of a particle. Note that for a harmonic oscillator we have an
equation a whole lot like 1):
d/dt (q,p) = (p, q).
This is why when we quantize the wave equation it's a whole lot like the
harmonic oscillator.
Now in general relativity things are similar but more complicated.
The analog of the pairs (phi, d phi/dt) are pairs (Q,P) where Q is the
metric on spacetime restricted to a spacelike hypersurface  that is,
the "metric on space at a given time"  and P is concocted from the
extrinsic curvature of that hypersurface as it sits in spacetime.
Now the name of the game is to turn Einstein's equation for the metric
into a firstorder equation sort of like 1). The problem is, in general
relativity there is no godgiven notion of time. So we need to *pick* a
"lapse function" on our hypersurface, and a "shift vector field" on our
hypersurface, which say how we want to push our hypersurface forwards in
time. The lapse function says at each point how much we push it in the
normal direction, while the shift vector field says at each point how
much we push it in some tangential direction. These are utterly
arbitrary and give us complete flexibility in how we want to push the
hypersurface forwards. Even if spacetime was flat, we could push the
hypersurface forwards in a dull way like:
 new
____________________ old
or in a screwy way like

/ \ /\
/  \
 new
____________________ old
Of course, in general relativity spacetime is usually not flat, which
makes it ultimately impossible to decide what counts as a "dull way" and
what counts as a "screwy way," which is why we simply allow all possible
ways.
Anyway, having *chosen* a lapse function and shift vector field, we can
rewrite Einstein's equations as an evolutionary equation. This is a bit
of a mess, and it's called the ADM (ArnowittDeserMisner) formalism.
Schematically, it goes like
d/dt (Q,P) = (stuff, stuff'). 2)
where both "stuff" and "stuff'" depend on both Q and P in a pretty
complex way.
But there is a catch. While the evolutionary equations are equivalent
to 6 of Einstein's equations (Einstein's equation for general relativity
is really 10 scalar equations packed into one tensor equation), there
are 4 more of Einstein's equations which turn into *constraints* on Q
and P. 1 of these constraints is called the Hamiltonian constraint and
is closely related to the lapse function; the other 3 are called the
momentum or diffeomorphism constraints and are closely related to the
shift vector field.
For those of you who know Hamiltonian mechanics, the reason why the
Hamiltonian constraint is called what it is is that we can write it as
H(Q,P) = 0
for some combination of Q and P, and this H(Q,P) acts a lot like a
Hamiltonian for general relativity in that we can rewrite 2) using the
Poisson brackets on the "phase space" of all (Q,P) pairs as
d/dt Q = {P,H(Q,P)}
d/dt P = {Q,H(Q,P)}.
The funny thing is that H is not zero on the space of all (Q,P) pairs,
so the equations above are nontrivial, but it does vanish on the submanifold
of pairs satisfying the constraints, so that, in a sense, "the
Hamiltonian of general relativity is zero". But one must be careful in
saying this because it can be confusing! It has confused lots of people
worrying about the problem of time in quantum gravity, where they
naively think "What  the Hamiltonian is zero? That means there's no
dynamics at all!"
The problem in quantizing general relativity in the "canonical" approach
is largely figuring out what to do with the constraints. It was Dirac
who first seriously tackled such problems, but the constraints in
general relativity always seemed intractible (when quantizing) until
Ashtekar invented his "new variables" for quantum gravity, that all of a
sudden make the constraints look a lot simpler. Ashtekar also has
certain generalizations of Dirac's general approach to quantizing
systems with constraints, and part of what Tate (who was a student of
Ashtekar) is doing is to study a number of toy models to see how
Ashtekar's ideas work.
I should note that there are lots of other ways to handle problems with
constraints, like BRST quantization, that aren't mentioned here at all.
Well, I'm off to Kansas and I hope to return with a bunch of goodies and
some gossip about 4manifold invariants, topological quantum field
theories and the like. Lee Smolin will be talking there too so I will
try to extract some information about quantum gravity from him.

Previous editions of "This Week's Finds," and other expository posts
on mathematical physics, are available by anonymous ftp from
math.princeton.edu, thanks to Francis Fung. They are in the directory
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papers discussed in each week of "This Week's Finds."
Please don't ask me about hepth and grqc; instead, read the
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twf_ascii/week110000064400020410000157000000402721062321375500141260ustar00baezhttp00004600000001Also available at http://math.ucr.edu/home/baez/week110.html
October 4, 1997
This Week's Finds in Mathematical Physics  Week 110
John Baez
Last time I sketched Wheeler's vision of "spacetime foam", and his
intuition that a good theory of this would require taking spin1/2
particles very seriously. Now I want to talk about Penrose's "spin
networks". These were an attempt to build a purely combinatorial
description of spacetime starting from the mathematics of spin1/2
particles. He didn't get too far with this, which is why he moved on to
invent twistor theory. The problem was that spin networks gave an
interesting theory of *space*, but not of spacetime. But recent work on
quantum gravity shows that you can get pretty far with spin network
technology. For example, you can compute the entropy of quantum black
holes. So spin networks are quite a flourishing business.
Okay. Building space from spin! How does it work?
Penrose's original spin networks were purely combinatorial gadgets:
graphs with edges labelled by numbers j = 0, 1/2, 1, 3/2,... These
numbers stand for total angular momentum or "spin". He required that
three edges meet at each vertex, with the corresponding spins j1, j2, j3
adding up to an integer and satisfying the triangle inequalities
j1  j2 <= j3 <= j1 + j2
These rules are motivated by the quantum mechanics of angular momentum:
if we combine a system with spin j1 and a system with spin j2, the spin
j3 of the combined system satisfies exactly these constraints.
In Penrose's setup, a spin network represents a quantum state of the
geometry of space. To justify this interpretation he did a lot of
computations using a special rule for computing a number from any spin
network, which is now called the "Penrose evaluation" or "chromatic
evaluation". In "week22" I said how this works when all the edges have
spin 1, and described how this case is related to the fourcolor
theorem. The general case isn't much harder, but it's a real pain to
describe without lots of pictures, so I'll just refer you to the
original papers:
1) Roger Penrose, Angular momentum: an approach to combinatorial spacetime,
in Quantum Theory and Beyond, ed. T. Bastin, Cambridge U. Press, Cambridge,
1971, pp. 151180. Also available at http://math.ucr.edu/home/baez/penrose/
Roger Penrose, Applications of negative dimensional tensors, in Combinatorial
Mathematics and its Applications, ed. D. Welsh, Academic Press, New York,
1971, pp. 221244.
Roger Penrose, On the nature of quantum geometry, in Magic Without Magic,
ed. J. Klauder, Freeman, San Francisco, 1972, pp. 333354.
R. Penrose, Combinatorial quantum theory and quantized directions, in
Advances in Twistor Theory, eds. L. Hughston and R. Ward, Pitman Advanced
Publishing Program, San Francisco, 1979, pp. 301317.
It's easier to explain the *physical meaning* of the Penrose evaluation.
Basically, the idea is this. In classical general relativity, space is
described by a 3dimensional manifold with a Riemannian metric: a recipe
for measuring distances and angles. In the spin network approach to
quantum gravity, the geometry of space is instead described as a
superposition of "spin network states". In other words, spin networks
form a basis of the Hilbert space of states of quantum gravity, so we
can write any state Psi as
Psi = Sum c_i psi_i
where psi_i ranges over all spin networks and the coefficients c_i are
complex numbers. The simplest state is the one corresponding to good
old flat Euclidean space. In this state, each coefficient c_i is just
the Penrose evaluation of the corresponding spin network psi_i.
Actually, this interpretation wasn't fully understood until later, when
Rovelli and Smolin showed how spin networks arise naturally in the
socalled "loop representation" of quantum gravity. They also came up
with a clearer picture of the way a spin network state corresponds to a
possible geometry of space. The basic picture is that spin network
edges represent flux tubes of area: an edge labelled with spin j
contributes an area proportional to sqrt(j(j+1)) to any surface it
pierces.
The cool thing is that Rovelli and Smolin didn't postulate this, they
*derived* it. Remember, in quantum theory, observables are given by
operators on the Hilbert space of states of the physical system in
question. You typically get these by "quantizing" the formulas for the
corresponding classical observables. So Rovelli and Smolin took the
usual formula for the area of a surface in a 3dimensional manifold with
a Riemannian metric and quantized it. Applying this operator to a spin
network state, they found the picture I just described: the area of a
surface is a sum of terms proportional to sqrt(j(j+1)), one for each
spin network edge poking through it.
Of course, I'm oversimplifying both the physics and the history here.
The tale of spin networks and loop quantum gravity is rather long. I've
discussed it already in "week55" and "week99", but only sketchily. If
you want more details, try:
2) Carlo Rovelli, Loop quantum gravity, preprint available as
grqc/9710008, also available as a webpage on Living Reviews in
Relativity at http://www.livingreviews.org/Articles/Volume1/19981rovelli/
The abstract gives a taste of what it's all about:
"The problem of finding the quantum theory of the gravitational field,
and thus understanding what is quantum spacetime, is still open. One of
the most active of the current approaches is loop quantum gravity. Loop
quantum gravity is a mathematically welldefined, nonperturbative and
background independent quantization of general relativity, with its
conventional matter couplings. The research in loop quantum gravity
forms today a vast area, ranging from mathematical foundations to
physical applications. Among the most significant results obtained are:
(i) The computation of the physical spectra of geometrical quantities
such as area and volume; which yields quantitative predictions on
Planckscale physics. (ii) A derivation of the BekensteinHawking black
hole entropy formula. (iii) An intriguing physical picture of the
microstructure of quantum physical space, characterized by a
polymerlike Planck scale discreteness. This discreteness emerges
naturally from the quantum theory and provides a mathematically
welldefined realization of Wheeler's intuition of a spacetime "foam".
Longstanding open problems within the approach (lack of a scalar
product, overcompleteness of the loop basis, implementation of reality
conditions) have been fully solved. The weak part of the approach is the
treatment of the dynamics: at present there exist several proposals,
which are intensely debated. Here, I provide a general overview of
ideas, techniques, results and open problems of this candidate theory of
quantum gravity, and a guide to the relevant literature."
For a nice picture of Rovelli standing in front of some spin networks,
check out:
3) Carlo Rovelli's homepage, http://www.phyast.pitt.edu/~rovelli/
which also has links to many of his papers.
You'll note from this abstract that the biggest problem in loop quantum
gravity is finding an adequate description of *dynamics*. This is
partially because spin networks are better suited for describing space
than spacetime. For this reason, Rovelli, Reisenberger and I have been
trying to describe spacetime using "spin foams"  sort of like soap
suds with all the bubbles having faces labelled by spins. Every slice
of a spin foam is a spin network.
But I'm getting ahead of myself! I should note that the spin networks
appearing in the loop representation are different from those Penrose
considered, in two important ways.
First, they can have more than 3 edges meeting at a vertex, and the
vertices must be labelled by "intertwining operators", or "intertwiners"
for short. This is a concept coming from group representation theory;
as described in "week109", what we've been calling "spins" are really
irreducible representations of SU(2). If we orient the edges of a spin
network, we should label each vertex with an intertwiner from the tensor
product of representations on the "incoming" edges to the tensor product
of representations labelling the "outgoing" edges. When 3 edges
labelled by spins j1, j2, j3 meet at a vertex, there is at most one
intertwiner
f: j1 tensor j2 > j3,
at least up to a scalar multiple. The conditions I wrote down  the
triangle inequality and so on  are just the conditions for a nonzero
intertwiner of this sort to exist. That's why Penrose didn't label his
vertices with intertwiners: he considered the case where there's
essentially just one way to do it! When more edges meet at a vertex,
there are more intertwiners, and this extra information is physically
very important. One sees this when one works out the "volume operators"
in quantum gravity. Just as the spins on edges contribute *area* to
surfaces they pierce, the intertwiners at vertices contribute *volume*
to regions containing them!
Second, in loop quantum gravity the spin networks are *embedded* in some
3dimensional manifold representing space. Penrose was being very
radical and considering "abstract" spin networks as a purely
combinatorial replacement for space, but in loop quantum gravity, one
traditionally starts with general relativity on some fixed spacetime and
quantizes that. Penrose's more radical approach may ultimately be the
right one in this respect. The approach where we take classical physics
and quantize it is very important, because we understand classical
physics better, and we have to start somewhere. Ultimately, however,
the world is quantummechanical, so it would be nice to have an approach
to space based purely on quantummechanical concepts. Also, treating
spin networks as fundamental seems like a better way to understand the
"quantum fluctuations in topology" which I mentioned in "week109".
However, right now it's probably best to hedge ones bets and work hard
on both approaches.
Lately I've been very excited by a third, hybrid approach:
4) Andrea Barbieri, Quantum tetrahedra and simplicial spin networks,
preprint available as grqc/9707010.
Barbieri considers "simplicial spin networks": spin networks living in a
fixed 3dimensional manifold chopped up into tetrahedra. He only
considers spin networks dual to the triangulation, that is, spin
networks having one vertex in the middle of each tetrahedron and one
edge intersecting each triangular face.
In such a spin network there are 4 edges meeting at each vertex,
and the vertex is labelled with an intertwiner of the form
f: j1 tensor j2 > j3 tensor j4
where j1,...,j4 are the spins on these edges. If you know about the
representation theory of SU(2), you know that j1 tensor j2 is a direct
sum of representations of spin j5, where j5 goes from j1  j2 up to
j1 + j2 in integer steps. So we get a basis of intertwining operators:
f: j1 tensor j2 > j3 tensor j4
by picking one factoring through each representation j5:
j1 tensor j2 > j5 > j3 tensor j4
where:
a) j1 + j2 + j5 is an integer and j1  j2 <= j5 <= j1 + j2
b) j3 + j4 + j5 is an integer and j3  j4 <= j5 <= j3 + j4.
Using this, we get a basis of simplicial spin networks by labelling all
the edges *and vertices* by spins satisfying the above conditions.
Dually, this amounts to labelling each tetrahedron and each triangle
in our manifold with a spin! Let's think of it this way.
Now focus on a particular simplicial spin network and a particular
tetrahedron. What do the spins j1,...,j5 say about the geometry of the
tetrahedron? By what I said earlier, the spins j1,...,j4 describe the
areas of the triangular faces: face number 1 has area proportional to
sqrt(j1(j1+1)), and so on. What about j5? It also describes an area.
Take the tetrahedron and hold it so that faces 1 and 2 are in front,
while faces 3 and 4 are in back. Viewed this way, the outline of the
tetrahedron is a figure with four edges. The midpoints of these four
edges are the corners of a parallelogram, and the area of this
parallelogram is proportional to sqrt(j5(j5+1)). In other words, there
is an area operator corresponding to this parallelogram, and our spin
network state is an eigenvector with eigenvalue proportional to
sqrt(j5(j5+1)). Finally, there is also a *volume operator*
corresponding to the tetrahedron, whose action on our spin network state
is given by a more complicated formula involving the spins j1,...,j5.
Well, that either made sense or it didn't... and I don't think either of
us want to stick around to find out which! What's the bottom line, you
ask? First, we're seeing how an ordinary tetrahedron is the classical
limit of a "quantum tetrahedron" whose faces have quantized areas and
whose volume is also quantized. Second, we're seeing how to put
together a bunch of these quantum tetrahedra to form a 3dimensional
manifold equipped with a "quantum geometry"  which can dually be seen
as a spin network. Third, all this stuff fits together in a truly
elegant way, which suggests there is something good about it. The
relationship between spin networks and tetrahedra connects the theory of
spin networks with approaches to quantum gravity where one chops up
space into tetrahedra  like the "Regge calculus" and "dynamical
triangulations" approaches.
Next week I'll say a bit about using spin networks to study quantum
black holes. Later I'll talk about *dynamics* and spin foams.
Meanwhile, I've been really lagging behind in describing new papers as
they show up... so here are a few interesting ones:
5) C. Nash, Topology and physics  a historical essay, to appear in
A History of Topology, edited by Ioan James, ElsevierNorth Holland,
preprint available as hepth/9709135.
6) Luis AlvarezGaume and Frederic Zamora, Duality in quantum field
theory (and string theory), available as hepth/9709180.
Quoting the abstract:
"These lectures give an introduction to duality in Quantum Field Theory. We
discuss the phases of gauge theories and the implications of the
electricmagnetic duality transformation to describe the mechanism of
confinement. We review the exact results of N=1 supersymmetric QCD and the
SeibergWitten solution of N=2 super YangMills. Some of its extensions to
String Theory are also briefly discussed."
7) Richard E. Borcherds, What is a vertex algebra?, available as
qalg/9709033.
"These are the notes of an informal talk in Bonn describing how to
define an analogue of vertex algebras in higher dimensions."
8) J. M. F. Labastida and Carlos Lozano, Lectures in topological quantum
field theory, 62 pages in LaTeX with 5 figures in encapsulated
Postscript, available as hepth/9709192.
"In these lectures we present a general introduction to topological
quantum field theories. These theories are discussed in the framework of
the MathaiQuillen formalism and in the context of twisted N=2
supersymmetric theories. We discuss in detail the recent developments in
DonaldsonWitten theory obtained from the application of results based
on duality for N=2 supersymmetric YangMills theories. This involves a
description of the computation of Donaldson invariants in terms of
SeibergWitten invariants. Generalizations of DonaldsonWitten theory
are reviewed, and the structure of the vacuum expectation values of
their observables is analyzed in the context of duality for the simplest
case."
9) Martin Markl, Simplex, associahedron, and cyclohedron, preprint
available as alggeom/9707009.
"The aim of the paper is to give an `elementary' introduction to the
theory of modules over operads and discuss three prominent examples of
these objects  simplex, associahedron (= the Stasheff polyhedron) and
cyclohedron (= the compactification of the space of configurations of
points on the circle)."

Previous issues of "This Week's Finds" and other expository articles on
mathematics and physics, as well as some of my research papers, can be
obtained at
http://math.ucr.edu/home/baez/
For a table of contents of all the issues of This Week's Finds, try
http://math.ucr.edu/home/baez/twfcontents.html
A simple jumpingoff point to the old issues is available at
http://math.ucr.edu/home/baez/twfshort.html
If you just want the latest issue, go to
http://math.ucr.edu/home/baez/this.week.html
oblems of this candidate theory of
quantum gravity, and a guide to the relevant literature."
For a nice picture of Rovelli standing in front of some spin networks,
check out:
3) Carlo Rovelli's homepage, http://www.phyast.pitt.edu/~rovelli/
which also has links to many of his papers.
You'll note from this abstract that twf_ascii/week111000064400020410000157000000304540774011336300141300ustar00baezhttp00004600000001Also available at http://math.ucr.edu/home/baez/week111.html
October 24, 1997
This Week's Finds in Mathematical Physics  Week 111
John Baez
This week I'll say a bit about black hole entropy, and next week I'll
say a bit about attempts to compute it using spin networks, as promised.
Be forewarned: all of this stuff should be taken with a grain of salt,
since there is no experimental evidence backing it up. Also, my little
"history" of the subject is very amateur. (In particular, when I say
someone did something in suchandsuch year, all I mean is that it was
published in that year.)
Why is the entropy of black holes so interesting? Mainly because it
serves as a testing ground for our understanding of quantum gravity. In
classical general relativity, any object that falls into a black hole
is, in some sense, lost and gone forever. Once it passes the "event
horizon", it can never get out again. This leads to a potential paradox
regarding the second law of thermodynamics, which claims that the total
entropy of the universe can never decrease. My office desk constantly
increases in entropy as it becomes more cluttered and dusty. I could
reduce its entropy with some work, dusting it and neatly stacking up the
papers and books, but in the process I would sweat and metabolize,
increasing my *own* entropy even more  so I don't bother. If,
however, I could simply dump my desk into a black hole, perhaps I could
weasel around the second law. True, the black hole would be more
massive, but nobody could tell if I'd dumped a clean desk or a dirty
desk into it, so in a sense, the entropy would be *gone*!
Of course there are lots of potential objections to this method of
violating the second law. *Anything* involving the second law of
thermodynamics is controversial, and the idea of violating it by
throwing entropy down black holes is especially so. The whole subject
might have remained a mere curiosity if it hadn't been for the work of
Hawking and Penrose.
In 1969, Penrose showed that, in principle, one could extract energy
from a rotating black hole:
1) Roger Penrose, Gravitational collapse: the role of general
relativity, Rev. del Nuovo Cimento 1, (1969) 272276.
Basically, one can use the rotation of the black hole to speed
up a rock one throws past it, as long as the rock splits and one
piece falls in while the rock is in the "ergosphere"  the region
of spacetime where the "frame dragging" is so strong that any object
inside is *forced* to rotate along with it. This result led to a
wave of thought experiments and theorems involving black holes and
thermodynamics.
In 1971, Hawking proved the "black hole area theorem":
1) Stephen Hawking, Gravitational radiation from colliding black holes,
Phys. Rev. Lett. 26 (1971), 13441346.
The precise statement of this theorem is a bit technical, but loosely,
it says that under reasonable conditions  e.g., no "exotic matter"
with negative energy density or the like  the total area of the event
horizons of any collection of black holes must always increase. This
result sets an upper limit on how much energy one can extract from a
rotating black hole, how much energy can be released in a black hole
collision, etc.
Now, this sounds curiously similar to the second law of thermodynamics,
with the area of the black hole playing the role of entropy! It turned
out to be just the beginning of an extensive analogy between black hole
physics and thermodynamics. I have a long way to go, so I will just
summarize this analogy in one cryptic chart:
BLACK HOLES THERMODYNAMICS
black hole mass M energy E
event horizon area A entropy S
surface gravity K temperature T
FIRST LAW: dM = K dA / 8 pi + work dE = T dS + work
SECOND LAW: A increases S increases
THIRD LAW: can't get K = 0 can't get T = 0
For a more thorough review by someone who really knows this stuff, try:
3) Robert M. Wald, Black holes and thermodynamics, in Symposium on
Black Holes and Relativistic Stars (in honor of S. Chandrasekhar),
December 1415, 1996, preprint available as grqc/9702022.
In 1973, Jacob Bekenstein suggested that we take this analogy really
seriously. In particular, he argued that black holes really do have
entropy proportional to their surface area. In other words, the total
entropy of the world is the entropy of all the matter *plus* some
constant times the area of all the black holes:
4) Jacob Bekenstein, Black holes and entropy, Phys. Rev. D7 (1973),
23332346.
This raises an obvious question  what's the constant?? Also,
in the context of classical general relativity, there are serious
problems with this idea: you can cook up thought experiments where the
total entropy defined this way goes down, no matter what you say the
constant is.
However, in 1975, Hawking showed that black holes aren't quite black!
5) Stephen Hawking, Particle creation by black holes, Commun.
Math. Phys. 43 (1975), 199220.
More precisely, using quantum field theory on curved spacetime, he
showed that a black hole should radiate photons thermally, with a
temperature T proportional to the surface gravity K at the event
horizon. It's important to note that this isn't a "quantum gravity"
calculation; it's a semiclassical approximation. Gravity is treated
classically! One simply assumes spacetime has the "Schwarzschild
metric" corresponding to a black hole. Quantum mechanics enters only in
treating the electromagnetic field. The goal of everyone ever since has
been to reproduce Hawking's result using a fullfledged quantum gravity
calculation. The problem, of course, is to get a theory of quantum
gravity.
Anyway, here is Hawking's magic formula:
T = K / 2 pi
Here I'm working in units where hbar = c = k = G = 1, but it's important
to note that there is secretly an hbar (Planck's constant) hiding in
this formula, so that it *only makes sense quantummechanically*. This
is why Bekenstein's proposal was problematic at the purely classical
level.
This formula means we can take really seriously the analogy between T
and K. We even know how to convert between the two! Of course, we also
know how to convert between the black hole mass M and energy E:
E = M
Thus, using the first law (shown in the chart above), we can convert
between entropy and area as well:
S = A/4
How could we hope to get such a formula using a fullfledged quantum
gravity calculation? Well, in statistical mechanics, the entropy of any
macrostate of a system is the logarithm of the microstates corresponding
to that macrostate. A microstate is a complete precise description of
the system's state, while a macrostate is a rough description. For
example, if I tell you "my desk is dusty and covered with papers", I'm
specifying a macrostate. If there are N ways my desk could meet this
description (i.e., N microstates corresponding to that macrostate), the
entropy of my desk is ln(N).
We expect, or at least hope, that a working quantum theory of gravity
will provide a statisticalmechanical way to calculate the entropy of a
black hole. In other words, we hope that specifying the area A of
the black hole horizon specifies the macrostate, and that there are
about N = exp(A/4) microstates corresponding to this macrostate.
What are these microstates? Much ink has been spilt over this thorny
question, but one reasonable possibility is that they are *states of the
geometry of the event horizon*. If we know its area, there are still
lots of geometries that the event horizon could have... and perhaps, for
some reason, there are about exp(A/4) of them! For this to be true, we
presumably need some theory of "quantum geometry", so that the number of
geometries is finite.
I presume you see what I'm leading up to: the idea of computing black
hole entropy using spin networks, which are designed precisely to
describe "quantum geometries", as sketched in "week55", "week99", and
"week110". I'll talk about this next week.
To be fair to other approaches, I should emphasize that string theorists
have their own rather different ideas about computing black hole entropy
using *their* approach to quantum gravity. A nice review article about
this approach is:
6) Gary Horowitz, Quantum states of black holes, preprint available as
grqc/9704072.
Next time, however, I will only talk about the spin network (also known
as "loop representation") approach, because that's the one I understand.
Okay, let me wrap up by listing a few interesting papers here and there
which are contributing to the entropy of my desk. It's 1:30 am and I'm
getting tired, so I'll just cop out and quote the abstracts. The first
one is a short readable essay explaining the limitations of quantum
field theory. The others are more technical.
7) Roman Jackiw, What is quantum field theory and why have some
physicists abandoned it?, 4 pages, preprint available as hepth/9709212.
"The presentday crisis in quantum field theory is described."
8) Adel Bilal, M(atrix) theory: a pedagogical introduction,
preprint available as hepth/9710136.
"I attempt to give a pedagogical introduction to the matrix model of
Mtheory as developed by Banks, Fischler, Shenker and Susskind
(BFSS). In the first lecture, I introduce and review the relevant
aspects of Dbranes with the emergence of the matrix model action. The
second lecture deals with the appearance of elevendimensional
supergravity and Mtheory in strongly coupled type IIA superstring
theory. The third lecture combines the material of the two previous ones
to arrive at the BFSS conjecture and explains the evidence presented by
these authors. The emphasis is not on most recent developments but on a
hopefully pedagogical presentation."
9) Gregory Gabadadze, Modeling the glueball spectrum by a closed bosonic
membrane, 43 pages, preprint available as hepph/9710402.
"We use an analogy between the YangMills theory Hamiltonian and the
matrix model description of the closed bosonic membrane theory to
calculate the spectrum of glueballs in the large N_c limit. Some
features of the YangMills theory vacuum, such as the screening of the
topological charge and vacuum topological susceptibility are
discussed. We show that the topological susceptibility has different
properties depending on whether it is calculated in the weak coupling or
strong coupling regimes of the theory. A mechanism of the formation of
the pseudoscalar glueball state within pure YangMills theory is
proposed and studied."
Fans of quaternions and octonions might like the following paper:
10) Jose M. FigueroaO'Farrill, Gauge theory and the division algebras,
preprint available as hepth/9710168.
"We present a novel formulation of the instanton equations in
8dimensional YangMills theory. This formulation reveals these
equations as the last member of a series of gaugetheoretical equations
associated with the real division algebras, including flatness in
dimension 2 and (anti)selfduality in 4. Using this formulation we
prove that (in flat space) these equations can be understood in terms of
moment maps on the space of connections and the moduli space of
solutions is obtained via a generalised symplectic quotient: a Kaehler
quotient in dimension 2, a hyperkaehler quotient in dimension 4 and an
octonionic Kaehler quotient in dimension 8. One can extend these
equations to curved space: whereas the 2dimensional equations make
sense on any surface, and the 4dimensional equations make sense on an
arbitrary oriented manifold, the 8dimensional equations only make sense
for manifolds whose holonomy is contained in Spin(7). The interpretation
of the equations in terms of moment maps further constraints the
manifolds: the surface must be orientable, the 4manifold must be
hyperkaehler and the 8manifold must be flat."

Previous issues of "This Week's Finds" and other expository articles on
mathematics and physics, as well as some of my research papers, can be
obtained at
http://math.ucr.edu/home/baez/
For a table of contents of all the issues of This Week's Finds, try
http://math.ucr.edu/home/baez/twf.html
A simple jumpingoff point to the old issues is available at
http://math.ucr.edu/home/baez/twfshort.html
If you just want the latest issue, go to
http://math.ucr.edu/home/baez/this.week.html
twf_ascii/week112000064400020410000157000000554761015243221400141320ustar00baezhttp00004600000001Also available at http://math.ucr.edu/home/baez/week112.html
November 3, 1997
This Week's Finds in Mathematical Physics  Week 112
John Baez
This week I will talk about attempts to compute the entropy of a
black hole by counting its quantum states, using the spin network
approach to quantum gravity.
But first, before the going gets tough and readers start dropping like
flies, I should mention the following science fiction novel:
1) Greg Egan, Distress, HarperCollins, 1995.
I haven't been keeping up with science fiction too carefully lately, so
I'm not really the best judge. But as far as I can tell, Egan is one of
the few practitioners these days who bites off serious chunks of reality
 who really tries to face up to the universe and its possibilies in
their full strangeness. Reality is outpacing our imagination so fast
that most attempts to imagine the future come across as miserably
unambitious. Many have a deliberately "retro" feel to them  space
operas set in Galactic empires suspiciously similar to ancient Rome,
cyberpunk stories set in dark urban environments borrowed straight from
film noire, complete with cynical voiceovers... is science fiction
doomed to be an essentially *nostalgic* form of literature?
Perhaps we are becoming too wise, having seen how our wildest
imaginations of the future always fall short of the reality, blindly
extrapolating the current trends while missing out on the really
interesting twists. But still, science fiction writers have to try to
imagine the unimaginable, right? If they don't, who will?
But how do we *dare* imagine what things will be like in, say, a
century, or a millenium? Vernor Vinge gave apt expression to this
problem in his novel featuring the marooned survivors of a "singularity"
at which the rate of technological advance became, momentarily,
*infinite*, and most of civilization inexplicably... disappeared. Those
who failed to catch the bus were left wondering just where it went.
Somewhere unimaginable, that's all they know.
"Distress" doesn't look too far ahead, just to 2053. Asexuality is
catching on bigtime... as are the "ultramale" and "ultrafemale" options,
for those who don't like this gender ambiguity business. Voluntary
Autists are playing around with eliminating empathy. And some scary
radical secessionists are redoing their genetic code entirely, replacing
good old ATCG by base pairs of their own devising. Fundamental physics,
thank god, has little new to offer in the way of new technology. For
decades, it's drifted off introspectively into more and more abstract
and mathematical theories, with few new experiments to guide it. But
this is the year of the Einstein Centenary Conference! Nobel laureate
Violet Masala will unveil her new work on a Theory of Everything. And
rumors have it that she may have finally cracked the problem, and found
 yes, that's right  the final, correct and true theory of physics.
As science reporter Andrew Worth tries to bone up for his interviews
with Masala, he finds it's not so easy to follow the details of the
various "AllTopology Models" that have been proposed to explain the
10dimensionality of spacetime in the Standard Unified Field Theory. In
one of the most realistic passages of imagined mathematical prose I've
ever seen in science fiction, he reads "At least two conflicting
generalized measures can be applied to T, the space of all topological
spaces with countable basis. Perrini's measure [Perrini, 2012] and
Saupe's measure [Saupe, 2017] are both defined for all bounded subsets
of T, and are equivalent when restricted to M  the space of
ndimensional paracompact Hausdorff manifolds  but they yield
contradictory results for sets of more exotic spaces. However, the
physical significance (if any) of this discrepancy remains obscure...."
But, being a hardy soul and a good reporter, Worth is eventually able to
explain to us readers what's at stake here, and *why* Masala's new work
has everyone abuzz. But that's really just the beginning. For in
addition to this respectable work on AllTopology Models, there is a lot
of somewhat cranky stuff going on in "anthrocosmology", involving
sophisticated and twisted offshoots of the anthropic principle. Some
argue that when the correct Theory of Everything is found, a kind of
cosmic selfreferential feedback loop will be closed. And then there's
no telling *what* will happen!
Well, I won't give away any more. It's fun: it made me want to run out
and do a lot more mathematical physics. And it raises a lot of deep
issues. At the end it gets a bit too "actionpacked" for my taste, but
then, my idea of excitement is lying in bed thinking about ncategories.
Now for the black holes.
In "week111", I left off with a puzzle. In a quantum theory of gravity,
the entropy of a black hole should be the logarithm of the number of its
microstates. This should be proportional to the area of the event
horizon. But what *are* the microstates? String theory has one answer
to this, but I'll focus on the loop representation of quantum gravity.
This approach to quantum gravity is very geometrical, which suggests
thinking of the black hole microstates as "quantum geometries" of the
black hole event horizon. But how are these related to the description
of the geometry of the surrounding space in terms of spin networks?
Starting in 1995, Smolin, Krasnov, and Rovelli proposed some answers to
these puzzles, which I have already mentioned in "week56", "week57", and
"week87". The ideas I'm going to talk about now are a further
development of this earlier work, but instead of presenting everything
historically, I'll just present the picture as I see it now. For more
details, try the following paper:
2) Abhay Ashtekar, John Baez, Alejandro Corichi and Kirill Krasnov,
Quantum geometry and black hole entropy, preprint available as
grqc/9710007.
This is a summary of what will eventually be a longer paper with two
parts, one on the "black hole sector" of classical general relativity,
and one on the quantization of this sector. Let me first say a bit
about the classical aspects, and then the quantum aspects.
One way to get a quantum theory of a black hole is to figure out what a
black hole is classically, get some phase space of classical states, and
then quantize *that*. For this, we need some way of saying which
solutions of general relativity correspond to black holes. This is
actually not so easy. The characteristic property of a black hole is
the presence of an event horizon  a surface such that once you pass
it you can never get back out without going faster than light. This
makes it tempting to find "boundary conditions" which say "this surface
is an event horizon", and use those to pick out solutions corresponding
to black holes.
But the event horizon is not a local concept. That is, you can't tell
just by looking at a small patch of spacetime if it has an event horizon
in it, since your ability to "eventually get back out" after crossing a
surface depends on what happens to the geometry of spacetime in the
future. This is bad, technically speaking. It's a royal pain to deal
with nonlocal boundary conditions, especially boundary conditions that
depend on *solving the equations of motion to see what's going to happen
in the future just to see if the boundary conditions hold*.
Luckily, there is a purely local concept which is a reasonable
substitute for the concept of event horizon, namely the concept of
"outer marginally trapped surface". This is a bit technical  and my
speciality is not this classical general relativity stuff, just the
quantum side of things, so I'm no expert on it!  but basically it
works like this.
First consider an ordinary sphere in ordinary flat space. Imagine light
being emitted outwards, the rays coming out normal to the surface of the
sphere. Clearly the crosssection of each little imagined circular ray
will *expand* as it emanates outwards. This is measured quantitatively
in general relativity by a quantity called... the expansion parameter!
Now suppose your sphere surrounds a spherically symmetric black hole.
If the sphere is huge compared to the size of the black hole, the above
picture is still pretty accurate, since the light leaving the sphere is
very far from the black hole, and gravitational effects are small. But
now imagine shrinking the sphere, making its radius closer and closer to
the Schwarzschild radius (the radius of the event horizon). When the
sphere is just a little bigger than the Schwarzschild radius, the
expansion of light rays going out from the sphere is very small. This
might seem paradoxical  how can the outgoing light rays not expand?
But remember, spacetime is seriously warped near the event horizon, so
your usual flat spacetime intuitions no longer apply. As we approach
the event horizon itself, the expansion parameter goes to zero!
That's roughly the definition of an "outer marginally trapped surface".
A more mathematical but still rough definition is: "an outer marginally
trapped surface is the boundary S of some region of space such that the
expansion of the outgoing family of null geodesics normal to S is
everywhere less than or equal to zero."
We require that our space have some sphere S in it which is an outer
marginally trapped surface. We also require other boundary conditions
to hold on this surface. I won't explain them in detail. Instead, I'll
say two important extra features they have: they say the black hole is
nonrotating, and they disallow gravitational waves falling into S. The
first condition here is a simplifying assumption: we are only studying
black holes of zero angular momentum in this paper! The second
condition is only meant to hold for the time during which we are
studying the black hole. It does not rule out gravitational waves far
from the black hole, waves that might *eventually* hit the black hole.
These should not affect the entropy calculation.
Now, in addition to their physical significance, the boundary conditions
we use also have an interesting *mathematical* meaning. Like most other
field theories, general relativity is defined by an action principle,
meaning roughly that one integrates some quantity called the Lagrangian
over spacetime to get an "action", and finds solutions of the field
equations by looking for minima of this action. But when one studies
field theories on a region with boundary, and imposes boundary
conditions, one often needs to "add an extra boundary term to the
action"  some sort of integral over the boundary  to get things to
work out right. There is a whole yoga of finding the right boundary
term to go along with the boundary conditions... an arcane little art...
just one of those things theoretical physicists do, that for some reason
never find their way into the popular press.
But in this case the boundary term is allimportant, because it's...
THE CHERNSIMONS ACTION!
(Yes, I can see people worldwide, peering into their screens, thinking
"Eh? Am I supposed to remember what that is? What's he getting so
excited about now?" And a few cognoscenti thinking "Oh, *now* I get it.
All this fussing about boundary conditions was just an elaborate ruse to
get a topological quantum field theory on the event horizon!")
So far we've been studying general relativity in honest 4dimensional
spacetime. ChernSimons theory is a closely related field theory one
dimension down, in 3dimensional spacetime. As time passes, the surface
of the black hole traces out a 3dimensional submanifold of our
4dimensional spacetime. When we quantize our classical theory of
gravity with our chosen boundary conditions, the ChernSimons term will
give rise to a "ChernSimons field theory" living on the surface of the
black hole. This field theory will describe the geometry of the surface
of the black hole, and how it changes as time passes.
Well, let's not just talk about it, let's do it! We quantize our theory
using standard spin network techniques *outside* the black hole, and
ChernSimons theory *on the event horizon*, and here is what we get.
States look like this. Outside the black hole, they are described by
spin networks (see "week110"). The spin network edges are labelled by
spins j = 0, 1/2, 1, and so on. Spin network edges can puncture the
black hole surface, giving it area. Each spinj edge contributes
an area proportional to sqrt(j(j+1)). The total area is the sum of
these contributions.
Any choice of punctures labelled by spins determines a Hilbert space of
states for ChernSimons theory. States in this space describe the
intrinsic curvature of the black hole surface. The curvature is zero
except at the punctures, so that *classically*, near any puncture, you
can visualize the surface as a cone with its tip at the puncture. The
curvature is concentrated at the tip. At the *quantum* level, where the
puncture is labelled with a spin j, the curvature at the puncture is
described by a number j_z ranging j to j in integer steps.
Now we ask the following question: "given a black hole whose area is
within epsilon of A, what is the logarithm of the number of microstates
compatible with this area?" This should be the entropy of the black
hole. To figure it out, first we work out all the ways to label
punctures by spins j so that the total area comes within epsilon of A.
For any way to do this, we then count the allowed ways to pick numbers
j_z describing the intrinsic curvature of the black hole surface. Then
we sum these up and take the logarithm.
That's roughly what we do, anyway, and for black holes much bigger than
the Planck scale we find that the entropy is proportional to the area.
How does this compare with the result of Bekenstein and Hawking,
described in "week111"? Remember, they computed that
S = A/4
where S is the entropy and A is the area, measured in units where c =
hbar = G = k = 1. What we get is
S = (ln 2 / 4 pi gamma sqrt(3)) A
To compare these results, you need to know what is that mysterious
"gamma" factor in the second equation! It's often called the Immirzi
parameter, because many people learned of it from the following paper:
3) Giorgio Immirzi, Quantum gravity and Regge calculus, in
Nucl. Phys. Proc. Suppl. 57 (1997) 6572, preprint available as
grqc/9701052.
However, it was first discovered by Barbero:
4) Fernando Barbero, Real Ashtekar variables for Lorentzian signature
spacetimes, Phys. Rev. D51 (1995), 55075510, preprint available as
grqc/9410014.
It's an annoying unavoidable arbitrary dimensionless parameter that
appears in the loop representation, which nobody had noticed before
Barbero. It's still rather mysterious. But it works a bit like this.
In ordinary quantum mechanics we turn the position q into an operator,
namely multiplication by x, and also turn the momentum p into an
operator, namely i d/dx. The important thing is the canonical
commutation relations: pq  qp = i. But we could also get the
canonical commutation relations to hold by defining
p = i gamma d/dx
q = x/gamma
since the gammas cancel out! In this case, putting in a gamma factor
doesn't affect the physics. One gets "equivalent representations of the
canonical commutation relations". In the loop representation, however,
the analogous trick *does* affect the physics  different choices of
the Immirzi parameter give different physics! For more details try:
5) Carlo Rovelli and Thomas Thiemann, The Immirzi parameter in quantum
general relativity, preprint available as grqc/9705059.
How does the Immirzi parameter affect the physics? It *determines the
quantization of area*. You may notice how I keep saying "each spinj
edge of a spin network contributes an area proportional to sqrt(j(j+1))
to any surface it punctures"... without ever saying what the constant
of proportionality is! Well, the constant is
8 pi gamma
Until recently, everyone went around saying the constant was 1. (As for
the factor of 8pi, I'm no good at these things, but apparently at least
some people were getting that wrong, too!) Now Krasnov claims to have
gotten these damned factors straightened out once and for all:
5) Kirill Krasnov, On the constant that fixes the area spectrum in
canonical quantum gravity, preprint available as grqc/9709058.
So: it seems we can't determine the constant of proportionality in the
entropyarea relation, because of this arbitrariness in the Immirzi
parameter. But we can, of course, use the BekensteinHawking formula
together with our formula for black hole entropy to determine gamma,
obtaining
gamma = ln(2) / sqrt(3) pi
This may seem like cheating, but right now it's the best we can do. All
we can say is this: we have a theory of the microstates of a black hole,
which predicts that entropy is proportional to area for largish black
holes, and which taken together with the BekensteinHawking calculation
allows us to determine the Immirzi parameter.
What do the funny constants in the formula
S = (ln 2 / 4 pi gamma sqrt(3)) A
mean? It's actually simple. The states that contribute most to the
entropy of a black hole are those where nearly all spin network edges
puncturing its surface are labelled by spin 1/2. Each spin1/2 puncture
can have either j_z = 1/2 or j_z = 1/2, so it contributes ln(2) to the
entropy. On the other hand, each spin1/2 edge contributes 4 pi gamma
sqrt(3) to the area of the black hole. Just to be dramatic, we can call
ln 2 the "quantum of entropy" since it's the entropy (or information)
contained in a single bit. Similarly, we can call 4 pi gamma sqrt(3)
the "quantum of area" since it's the area contributed by a spin1/2
edge. These terms are a bit misleading since neither entropy nor area
need come in *integral* multiples of this minimal amount. But anyway,
we have
S = (quantum of entropy / quantum of area) A
What next? Well, one thing is to try to use these ideas to study
Hawking radiation. That's hard, because we don't understand
*Hamiltonians* very well in quantum gravity, but Krasnov has made some
progress....
6) Kirill Krasnov, Quantum geometry and thermal radiation from black holes,
preprint available as grqc/9710006.
Let me just quote the abstract:
"A quantum mechanical description of black hole states proposed recently
within the approach known as loop quantum gravity is used to study the
radiation spectrum of a Schwarzschild black hole. We assume the
existence of a Hamiltonian operator causing transitions between
different quantum states of the black hole and use Fermi's golden rule
to find the emission line intensities. Under certain assumptions on the
Hamiltonian we find that, although the emission spectrum consists of
distinct lines, the curve enveloping the spectrum is close to the Planck
thermal distribution with temperature given by the thermodynamical
temperature of the black hole as defined by the derivative of the
entropy with respect to the black hole mass. We discuss possible
implications of this result for the issue of the Immirzi gammaambiguity
in loop quantum gravity."
This is interesting, because Bekenstein and Mukhanov have recently
noted that if the area of a quantum black hole is quantized
in *evenly spaced steps*, there will be large deviations from the Planck
distribution of thermal radiation:
7) Jacob D. Bekenstein and V. F. Mukhanov, Spectroscopy of the quantum
black hole, preprint available as grqc/9505012.
However, in the loop representation the area is not quantized in evenly
spaced steps: the area A can be any sum of quantities like 8 pi gamma
sqrt(j(j+1)), and such sums become very densely packed for large A.
Let me conclude with a few technical comments about how ChernSimons
theory shows up here. For a long time I've been studying the "ladder of
dimensions" relating field theories in dimensions 2, 3, and 4, in part
because this gives some clues as to how ncategories are related to
topological quantum field theory, and in part because it relates quantum
gravity in spacetime dimension 4, which is mysterious, to ChernSimons
theory in spacetime dimension 3, which is wellunderstood. It's neat
that one can now use this ladder to study black hole entropy. It's
worth comparing Carlip's calculation of black hole entropy in spacetime
dimension 3 using a 2dimensional field theory (the WessZuminoWitten
model) on the surface traced out by the black hole event horizon  see
"week41". Both the theories we use and those Carlip uses, are all part
of the same big ladder of theories! Something interesting is going on
here.
But there's a twist in our calculation which really took me by surprise.
We do not use SU(2) ChernSimons theory on the black hole surface, we
use U(1) ChernSimons theory! The reason is simple. The boundary
conditions we use, which say the black hole surface is "marginally outer
trapped", also say that its extrinsic curvature is zero. Thus the
curvature tensor reduces, at the black hole surface, to the intrinsic
curvature. Curvature on a 3dimensional space is so(3)valued, but the
intrinsic curvature on the surface S is so(2)valued. Since so(3) =
su(2), general relativity has a lot to do with SU(2) gauge theory. But
since so(2) = u(1), the field theory on the black hole surface can be
thought of as a U(1) gauge theory.
(Experts will know that U(1) is a subgroup of SU(2) and this is why we
look at all values of j_z going from j to j: we are decomposing
representations of SU(2) into representations of this U(1) subgroup.)
Now U(1) ChernSimons theory is a lot less exciting than SU(2)
ChernSimons theory so mathematically this is a bit of a disappointment.
But U(1) ChernSimons theory is not utterly boring. When we are
studying U(1) ChernSimons theory on a punctured surface, we are
studying flat U(1) connections modulo gauge transformations. The space
of these is called a "Jacobian variety". When we quantize U(1)
ChernSimons theory using geometric quantization, we are looking for
holomorphic sections of a certain line bundle on this Jacobian variety.
These are called "theta functions". Theta functions have been
intensively studied by string theorists and number theorists, who use
them do all sorts of wonderful things beyond my ken. All I know about
theta functions can be found in the beginning of the following two
books:
8) Junichi Igusa, Theta Functions, SpringerVerlag, Berlin, 1972.
9) David Mumford, Tata Lectures on Theta, 3 volumes, Birkhauser, Boston,
19831991.
Theta functions are nice, so it's fun to see them describing states of a
quantum black hole!

Previous issues of "This Week's Finds" and other expository articles on
mathematics and physics, as well as some of my research papers, can be
obtained at
http://math.ucr.edu/home/baez/
For a table of contents of all the issues of This Week's Finds, try
http://math.ucr.edu/home/baez/twf.html
A simple jumpingoff point to the old issues is available at
http://math.ucr.edu/home/baez/twfshort.html
If you just want the latest issue, go to
http://math.ucr.edu/home/baez/this.week.html
in integer steps.
Now we ask the following question: "given a black hole whose area is
within epsilon of A, what is the logarithm of the number of microstates
compatible with this area?" This twf_ascii/week113000064400020410000157000000423051015243230600141200ustar00baezhttp00004600000001Also available at http://math.ucr.edu/home/baez/week113.html
November 26, 1997
This Week's Finds in Mathematical Physics  Week 113
John Baez
This week I'd like to talk about "spin foams". People have already
thought a lot about using spin networks to describe the quantum geometry
of 3dimensional space at a given time. This is great for kinematical
aspects of quantum gravity, but not so good for dynamics. For dynamics,
it would be nice to have a description of the quantum geometry of
4dimensional *spacetime*. That's where spin foams come in.
If we use spin networks to describe the geometry of space at the Planck
scale, how might we describe spacetime? Well, space is supposed to be a
kind of slice of spacetime, so let's recall what a spin network is, and
see what it could be a slice of.
First of all, spin network is a graph: a bunch of vertices connected by
edges. What gives a graph when you slice it? Foam! Consider the soap
suds you get while washing the dishes. If we idealize it as a bunch
of 2dimensional surfaces meeting along edges, and imagine intersecting
it with a plane, we see that the result is typically a graph.
Topologists call this sort of space a "2dimensional complex". It's a
generalization of a graph because we can form it by starting with a
bunch of "vertices", then connecting these with a bunch of "edges" to
obtain a graph, and then taking a bunch of 2dimensional discs or
"faces" and attaching them along their boundaries to this graph.
Mathematically, there's no reason to stop in dimension 2. Topologists
are interested in complexes of all dimensions. However, 2dimensional
complexes have been given special attention, thanks to a number of
famous unsolved problems involving them. A great place to learn about
them is:
1) C. HogAngeloni, W. Metzler, and A. Sieradski, Twodimensional
Homotopy and Combinatorial Group Theory, London Mathematical Society
Lecture Note Series 197, Cambridge U. Press, Cambridge, 1993.
However, a spin network is not *merely* a graph: it's a graph with edges
labelled by irreducible representations of some symmetry group and
vertices labelled by intertwiners. If you don't know what this means,
don't panic! If we take our symmetry group to be SU(2), things simplify
tremendously. If we take our graph to have 4 edges meeting at every
vertex, things simplify even more. In this case, all we need to do is
label each vertex and each edge with a number j = 0, 1/2, 1, 3/2,...
called a "spin".
In this special case, we can get our spin network as a slice of a
2dimensional complex with faces and edges labelled by spins. Such a
thing looks a bit like a foam of soap bubbles with edges and faces labelled
by spins  hence the term "spin foam"! More generally, a spin foam is a
2dimensional complex with faces labelled by irreducible representations
of some group and edges labelled by intertwining operators. When we
slice a spin foam, each of its faces gives a spin network edge, and each
of its edges gives a spin network vertex.
Actually, if you've ever looked carefully at soap suds, you'll know that
generically 3 faces meet along each edge. Spin foams like this are
important for quantum gravity in 3 spacetime dimensions. In 4 spacetime
dimensions it seems especially interesting to use spin foams of a
different sort, with 4 faces meeting along each edge. When we slice one
of these, we get a spin network with 4 edges meeting at each vertex.
What's so interesting about spin foams with 4 faces meeting along each
edge? Well, suppose we take a 4dimensional manifold representing
spacetime and triangulate it  that is, chop it up into 4simplices,
the 4dimensional analogs of tetrahedra. We get a mess of 4simplices,
which have tetrahedra as faces, which in turn have triangles as faces.
Now we can form a spin foam with one vertex in the middle of each
4simplex, one edge intersecting each tetrahedron, and one face
intersecting each triangle. This trick is called "Poincare duality":
each ddimensional piece of our spin foam intersects a (4d)dimensional
piece of our triangulation. Since each tetrahedron in our triangulated
manifold has 4 triangular faces, our spin foam will dually have 4 faces
meeting at each edge. Since each 4simplex has 5 tetrahedra and 10
triangles, each spin foam vertex will have 5 edges and 10 faces meeting
at it.
This seems to be a particularly interesting sort of spin foam for
quantum gravity in 4 dimensions: a spin foam dual to a triangulation of
spacetime. If we slice such a spin foam, we generically get a spin
network dual to a triangulation of space!
I discussed Barbieri's work on such spin networks in "week110". A spin
network like this has a nice interpretation as a "3dimensional quantum
geometry", that is, a quantum state of the geometry of space. Each spin
network edge labelled by spin j gives an area proportional to
sqrt(j(j+1)) to the triangle it intersects. There's also a formula for
the volume of each tetrahedron, involving the spin on the corresponding
spin network vertex, together with the spins on the 4 spin network edges
that meet there.
It would be nice to have a similar geometrical interpretation of spin
foams dual to triangulations of spacetime. Some recent steps towards
this can be found in the following papers:
2) John Barrett and Louis Crane, Relativistic spin networks
and quantum gravity, 9 pages, preprint available as grqc/9709028.
3) John Baez, Spin foam models, 39 pages, preprint available as
grqc/9709052 or in Postscript form as http://math.ucr.edu/home/baez/foam.ps.
Perhaps I can be forgiven some personal history here. Michael Reisenberger
has been pushing the idea of spin foams (though not the terminology) for
quite a while... see for example his paper:
4) Michael Reisenberger, Worldsheet formulations of gauge theories
and gravity, preprint available as grqc/9412035.
More recently, Carlo Rovelli and he gave a heuristic derivation of a spin
foam approach to quantum gravity starting with the socalled canonical
quantization approach:
5) Michael Reisenberger and Carlo Rovelli, ``Sum over surfaces''
form of loop quantum gravity, Phys. Rev. D56 (1997), 34903508,
preprint available as grqc/9612035.
I started giving talks about spin foams in the spring of this year.
Following the ideas of Reisenberger and Rovelli, I was trying to
persuade everyone to think of spin foams as higherdimensional analogs
of Feynman diagrams.
Mathematically, a Feynman diagram is just a graph with edges labelled by
representations of some group. But physically, a Feynman diagram
describes a *process* in which a bunch of particles interact. Its edges
correspond to the worldlines traced out by some particles as time
passes, while its vertices represent interactions. Different quantum
field theories use Feynman diagrams with different kinds of vertices.
For any Feynman diagram in our theory, we want to compute a number
called an "amplitude". The absolute value squared of this amplitude
gives the probability that the process in question will occur.
We calculate this amplitude by computing a number for each for each edge
and each vertex and multiplying all these numbers together. The numbers
for edges are called "propagators"  they describe the amplitude for a
particle to go from here to there. The numbers for vertices are called
"vertex amplitudes"  they describe the amplitude for various
interactions to occur.
Similarly, a spin foam is a 2dimensional complex with faces labelled by
representations and edges labelled by intertwiners. Each spin foam face
corresponds to the "worldsheet" traced out by a spin network edge as
time passes. So, in addition to thinking of a spin foam as a
"4dimensional quantum geometry", we can think of it as a kind of
*process*. The goal of the spin foam approach to quantum gravity is to
compute an amplitude for each spin foam. Following what we know about
Feynman diagrams, it seems reasonable to do it by computing a number for
each spin foam face, edge, and vertex, and then multiplying them all
together.
Quantum gravity is related to a simpler theory called BF theory, which
has a spin foam formulation known as the CraneYetter model  see
"week36", "week58", and "week98". There are various clues suggesting
that that the numbers for faces and edges  the "propagators" 
should be computed in quantum gravity just as in the CraneYetter model.
The problem is the vertex amplitudes! The vertices are crucial because
these represent the interactions: the places where something really
"happens". The number we compute for a vertex represents the amplitude
for the corresponding interaction to occur. Until we know this, we
don't know the dynamics of our theory!
The "spin foam vertex amplitudes for quantum gravity" became my holy
grail. Whenever I gave a talk on this stuff I would go around
afterwards asking everyone if they could help me figure them out. I
would lay out all the clues I had and beg for assistance... or at least
a spark of inspiration. In March I gave a talk a talk at Penn State
proposing a candidate for these vertex amplitudes  a candidate I no
longer believe in. Afterwards Carlo Rovelli told me about his attempts
to work out something similar with Louis Crane and Lee Smolin...
attempts that never quite got anywhere. We had a crack at it together
but it didn't quite gel. In May I asked John Barrett about the vertex
amplitudes at a conference in Warsaw. He said he couldn't guess them.
I couldn't get *anyone* to guess an answer. In June, at a quantum
gravity workshop in Vienna, I asked Roger Penrose a bunch of questions
about spinors, hoping that this might be the key  see "week109".
I learned a lot of interesting stuff, but I didn't find the holy grail.
I kept on thinking. I started getting some promising ideas, and by the
summer I was hard at work on the problem, calculating furiously. I was
also writing a big fat paper about spin foams: the general formalism,
the relation to triangulations, the relationships to category theory,
and so on. I was very happy with it  but I didn't want to finish it
until I knew the spin foam vertex amplitudes. That would be the icing
on the cake, I thought.
Then one weekend Louis Crane sent me email saying he and John Barrett
had written a paper proposing a model of quantum gravity. Aaargh! Had
they beat me to the holy grail? I frantically wrote up everything I had
while waiting for Monday, when their paper would appear on the preprint
server grqc. On Monday I downloaded it and yes, they had beaten me.
It was a skinny little paper and I absorbed it more or less instantly.
They didn't say a word about spin foams  they were working dually
with triangulations  but from my viewpoint, what they had done was to
give a formula for the spin foam vertex amplitudes.
Oh well. When you can't beat 'em, join 'em! I finished up my paper,
explaining how their formula fit in with what I'd written already, and
put it on the the preprint server the following weekend.
What did they do to get their formula? Well, the key trick was not to
use SU(2) as the symmetry group, but instead use SU(2) x SU(2). This is
the double cover of SO(4), the rotation group in 4 dimensions. Following
the idea behind Ashtekar's new variables for general relativity, I was
only using the "lefthanded half" of this group, that is, one of the
SU(2) factors. But the geometry of the 4simplex, and its relation to
quantum theory, is in some ways more easily understood using the full
SU(2) x SU(2) symmetry group.
Not surprisingly, an irreducible representation of SU(2) x SU(2) is
described by a pair of spins (j,k). The reason is that we can take the
spinj representation of the "lefthanded" SU(2) and the spink
representation of the "righthanded" SU(2) and tensor them to get an
irreducible representation of SU(2) x SU(2). If we use SU(2) x SU(2) as
our group, our spin foams dual to triangulations will thus have every
face and every edge labelled by a *pair* of spins. However, Barrett and
Crane's work suggests that the only spin foams with nonzero amplitudes
are those for which both spins labelling a face or edge are equal! Thus
in a way we are back down to SU(2)  but we think of it all a bit
differently.
I'm tempted to go into detail and explain exactly how the model works,
because it involves a lot of beautiful geometry. But it takes a while,
so I won't. First you need to really grok the phase space of all
possible geometries of the 4simplex. Then you need to quantize this
phase space, obtaining the "Hilbert space of a quantum 4simplex". Then
you need to note that there's a special linear functional on this
Hilbert space, called the "Penrose evaluation"  see "week110".
Putting all this together gives the vertex amplitudes for quantum
gravity... we hope.
Anyway, back to my little personal story....
Though I'd been working on my paper before Barrett and Crane started,
and they finished before me, Michael Reisenberger started one even
earlier and finished even later! Indeed, he has been working on a
spin foam model of quantum gravity for several years now  see
"week86". He took a purely lefthanded SU(2) approach, a bit different
what I'd been trying, but closely related. He told lots of people
about it, but unfortunately he's very slow to publish.
When I heard Barrett and Crane were about to come out with their paper,
I emailed Reisenberger warning him of this. He doesn't like being
scooped any more than I do. Unfortunately I only had his old email
address in Canada, and now he's down in Uruguay, so he never got that
email. Thus Barrett and Crane's paper, followed by mine, came as a
a big shock to him! Luckily, this motivated him to hurry and come out
with a version of his paper:
6) Michael Reisenberger, A lattice worldsheet sum for 4d Euclidean
general relativity, 50 pages, preprint available as grqc/9711052.
Let me quote the abstract:
A lattice model for four dimensional Euclidean quantum general
relativity is proposed for a simplicial spacetime. It is shown how this
model can be expressed in terms of a sum over worldsheets of spin
networks, and an interpretation of these worldsheets as spacetime
geometries is given, based on the geometry defined by spin networks in
canonical loop quantized GR. The spacetime geometry has a Planck scale
discreteness which arises "naturally" from the discrete spectrum of
spins of SU(2) representations (and not from the use of a spacetime
lattice). The lattice model of the dynamics is a formal quantization of
the classical lattice model of [Reisenberger's paper "A lefthanded
simplicial action for euclidean general relativity], which reproduces,
in a continuum limit, Euclidean general relativity.'
To wrap up my little history, let me say what's been happening lately.
There is still a lot of puzzlement and mystery concerning these spin
foam models... it's far from clear that they really work as hoped for.
We may be doing things a little bit wrong, or we may be on a completely
wrong track. The phase space of the 4simplex involves some tricky
constraint equations which could kill us if we're not dealing with them
right. Barbieri has suggested a modified version of Barrett and Crane's
approach which may overcome some problems with the constraints:
7) Andrea Barbieri, Space of the vertices of relativistic spin networks,
2 pages, preprint available as grqc/9709076.
John Barrett visited me last week and we made some progress on this
issue, but it's still very touchy.
Also, all the work cited above deals with socalled "Euclidean" quantum
gravity  that's why it uses the double cover of the rotation group
SO(4). For "Lorentzian" quantum gravity we'd need instead to use the
double cover of the Lorentz group SO(3,1). This group is isomorphic to
SL(2,C). As explained in "week109", the finitedimensional irreducible
representations of SL(2,C) are also described by pairs of spins, so the
Lorentzian theory should be similar to the Euclidean theory. However,
most work so far has dealt with the Euclidean case; this needs to be
addressed.
Finally, Crane has written a bit more about the geometrical significance of
his work with Barrett:
8) Louis Crane, On the interpretation of relativistic spin networks and
the balanced state sum, 4 pages, preprint available as grqc/9710108.
Next week I'll talk about other developments in the loop representation
of quantum gravity, some arising from Thiemann's work on the Hamiltonian
constraint. After that, I think I want to talk about something
completely different, like homotopy theory. Lately I've been trying to
make "This Week's Finds" very elementary and readable  relatively
speaking, I mean  but I'm getting in the mood for writing in a more
technical and incomprehensible manner, and homotopy theory is an ideal
subject for that sort of writing.

Previous issues of "This Week's Finds" and other expository articles on
mathematics and physics, as well as some of my research papers, can be
obtained at
http://math.ucr.edu/home/baez/
For a table of contents of all the issues of This Week's Finds, try
http://math.ucr.edu/home/baez/twf.html
A simple jumpingoff point to the old issues is available at
http://math.ucr.edu/home/baez/twfshort.html
If you just want the latest issue, go to
http://math.ucr.edu/home/baez/this.week.html
The numbers for vertices are called
"vertex amplitudes"  they describe the amplitude for various
interactions to occur.
Similarly, a spin foam is a 2dimensional complex with faces labelled by
representations and edges labelled by intertwiners. Each spin foam face
corresponds to the "worldsheet" traced out twf_ascii/week114000064400020410000157000000323771015243242400141320ustar00baezhttp00004600000001Also available at http://math.ucr.edu/home/baez/week114.html
January 12, 1998
This Week's Finds in Mathematical Physics  Week 114
John Baez
Classes have started! But I just flew back yesterday from the Joint
Mathematics Meetings in Baltimore  the big annual conference organized
by the AMS, the MAA, SIAM, and other societies. Over 4000
mathematicians could be seen wandering in clumps about the glitzy harbor
area and surrounding crimeridden slums, arguing about abstractions,
largely oblivious to the world around them. Everyone ate the obligatory
crab cakes for which Baltimore is justly famous. Some of us drank a bit
too much beer, too.
Witten gave a plenary talk on "Mtheory", which was great fun even
though he didn't actually say what Mtheory is. Steve Sawin and I ran a
session on quantum gravity and lowdimensional topology, so I'll say a
bit about what went on there. There was also a nice session on homotopy
theory in honor of J. Michael Boardman. I'll talk about that and
various other things next week.
A lot of the buzz in our session concerned the new "spin foam" approach
to quantum gravity which I discussed in "week113". The big questions
are: how do you test this approach without impractical computer
simulations? Lee Smolin's paper below suggests one way. Should you
only sum over spin foams that are dual to a particular triangulation of
spacetime, or should you sum over all spin foams that fit in a
particular 4dimensional spacetime manifold, or should you sum over
*all* spin foams? There was a lot of argument about this. In addition
to the question of what is physically appropriate, there's the
mathematical problem of avoiding divergent infinite sums. Perhaps the
sum required to answer any truly physical question only involves
finitely many spin foams  that's what I hope. Finally, should the
time evolution operators constructed using spin foams be thought of as
describing true time evolution, or merely the projection onto the kernel
of the Hamiltonian constraint? While it sounds a bit technical, this
question is crucial for the interpretation of the theory; it's part of
what they call "the problem of time".
Carlo Rovelli spoke about how spin foams arise in canonical quantum
gravity, while John Barrett and Louis Crane discussed them in the
context of discretized path integrals for quantum gravity, also known as
state sum models. As in the more traditional "Regge calculus" approach,
these models start by chopping spacetime into simplices. The biggest
difference is that now *areas of triangles* play a more important role
than lengths of edges. But Barrett, Crane and others are starting to
explore the relationships:
1) John W. Barrett, Martin Rocek, Ruth M. Williams, A note on area
variables in Regge calculus, preprint available as grqc/9710056.
2) Jarmo Makela, Variation of area variables in Regge calculus
preprint available as grqc/9801022.
Also, there's been some progress on extracting Einstein's equation for
general relativity as a classical limit of the BarrettCrane state
sum model. Let me quote the abstract of this paper:
3) Louis Crane and David N. Yetter, On the classical limit of the
balanced state sum, preprint available as grqc/9712087.
"The purpose of this note is to make several advances in the
interpretation of the balanced state sum model by Barrett and Crane in
grqc/9709028 as a quantum theory of gravity. First, we outline a
shortcoming of the definition of the model pointed out to us by Barrett
and Baez in private communication, and explain how to correct
it. Second, we show that the classical limit of our state sum reproduces
the EinsteinHilbert lagrangian whenever the term in the state sum to
which it is applied has a geometrical interpretation. Next we outline a
program to demonstrate that the classical limit of the state sum is in
fact dominated by terms with geometrical meaning. This uses in an
essential way the alteration we have made to the model in order to fix
the shortcoming discussed in the first section. Finally, we make a brief
discussion of the Minkowski signature version of the model."
Lee Smolin talked about his ideas for relating spin foam models
to string theory. He has a new paper on this, so I'll just
quote the abstract:
4) Lee Smolin, Strings as perturbations of evolving spinnetworks,
preprint available as hepth/9801022.
"A connection between nonperturbative formulations of quantum gravity
and perturbative string theory is exhibited, based on a formulation of
the nonperturbative dynamics due to Markopoulou. In this formulation
the dynamics of spin network states and their generalizations is
described in terms of histories which have discrete analogues of the
causal structure and many fingered time of Lorentzian spacetimes.
Perturbations of these histories turn out to be described in terms of
spin systems defined on 2dimensional timelike surfaces embedded in the
discrete spacetime. When the history has a classical limit which is
Minkowski spacetime, the action of the perturbation theory is given to
leading order by the spacetime area of the surface, as in bosonic string
theory. This map between a nonperturbative formulation of quantum
gravity and a 1+1 dimensional theory generalizes to a large class of
theories in which the group SU(2) is extended to any quantum group or
supergroup. It is argued that a necessary condition for the
nonperturbative theory to have a good classical limit is that the
resulting 1+1 dimensional theory defines a consistent and stable
perturbative string theory."
Fotini Markopolou spoke about her recent work with Smolin on
formulating spin foam models in a manifestly local, causal
way.
5) Fotini Markopoulou and Lee Smolin, Quantum geometry with intrinsic
local causality, preprint available as grqc/9712067.
"The space of states and operators for a large class of background
independent theories of quantum spacetime dynamics is defined. The SU(2)
spin networks of quantum general relativity are replaced by labelled
compact twodimensional surfaces. The space of states of the theory is
the direct sum of the spaces of invariant tensors of a quantum group G_q
over all compact (finite genus) oriented 2surfaces. The dynamics is
background independent and locally causal. The dynamics constructs
histories with discrete features of spacetime geometry such as causal
structure and multifingered time. For SU(2) the theory satisfies the
Bekenstein bound and the holographic hypothesis is recast in this
formalism."
The main technical idea in this paper is to work with "thickened" or
"framed" spin networks, which amounts to replacing graphs by
solid handlebodies. One expects this "framing" business to be
important for quantum gravity with nonzero cosmological constant.
This framing business also appears in the qdeformed version of
Barrett and Crane's model and in my "abstract" version of
their model, which assumes no background spacetime manifold.
Markopoulou and Smolin don't specify a choice of dynamics; instead,
they describe a *class* of theories which has my model as a
special case, though their approach to causality is better suited
to Lorentzian theories, while mine is Euclidean.
As I've often noted, spin foams are about spacetime geometry,
or dynamics, while spin networks are a way of describing the
geometry of space, or kinematics. Kinematics is always easier
than dynamics, so the spin network approach to the quantum geometry
of space has been much better worked out than the new spin foam stuff.
Abhay Ashtekar gave an overview of these kinematical issues in his
talk on "quantum Riemannian geometry", and Kirill Krasnov described
how our understanding of these already allows us to compute the entropy
of black holes (see "week112"). Here it's worth mentioning that the
second part of Ashtekar's paper with Jerzy Lewandowski is finally out:
6) Abhay Ashtekar and Jerzy Lewandowski, Quantum theory of geometry II:
volume operators, preprint available as grqc/9711031.
"A functional calculus on the space of (generalized) connections was
recently introduced without any reference to a background metric. It is
used to continue the exploration of the quantum Riemannian geometry.
Operators corresponding to volume of threedimensional regions are
regularized rigorously. It is shown that there are two natural
regularization schemes, each of which leads to a welldefined operator.
Both operators can be completely specified by giving their action on
states labelled by graphs. The two final results are closely related
but differ from one another in that one of the operators is sensitive to
the differential structure of graphs at their vertices while the second
is sensitive only to the topological characteristics. (The second
operator was first introduced by Rovelli and Smolin and De Pietri and
Rovelli using a somewhat different framework.) The difference between
the two operators can be attributed directly to the standard
quantization ambiguity. Underlying assumptions and subtleties of
regularization procedures are discussed in detail in both cases because
volume operators play an important role in the current discussions of
quantum dynamics."
Before spin foam ideas came along, the basic strategy in the loop
representation of quantum gravity was to start with general relativity
on a smooth manifold and try to quantize it using the "canonical
quantization" approach. Here the most important and difficult thing
is to implement the "Hamiltonian constraint" as an operator on the
Hilbert space of kinematical states, so you can write down the
WheelerdeWitt equation, which is, quite roughly speaking, the
quantum gravity analog of Schrodinger's equation. (For a summary of
this approach, try "week43".)
The most careful attempt to do this so far is the work of Thiemann:
7) Thomas Thiemann, Quantum spin dynamics (QSD), preprint
available as grqc/9606089.
Quantum spin dynamics (QSD) II, preprint available as
grqc/9606090.
QSD III: Quantum constraint algebra and physical scalar
product in quantum general relativity, preprint available as
grqc/9705017.
QSD IV: 2+1 Euclidean quantum gravity as a model to test
3+1 Lorentzian quantum gravity, preprint available as grqc/9705018.
QSD V: Quantum gravity as the natural regulator of matter
quantum field theories, preprint available as grqc/9705019.
QSD VI: Quantum Poincare algebra and a quantum positivity of energy
theorem for canonical quantum gravity, preprint available as
grqc/9705020
Kinematical Hilbert spaces for fermionic and Higgs quantum
field theories, grqc/9705021
If everything worked as smoothly as possible, the Hamiltonian constraint
would satisfy nice commutation relations with the other constraints of
the theory, giving a representation of something called the "Dirac
algebra". However, as Don Marolf explained in his talk, this doesn't
really happen, at least in a large class of approaches including
Thiemann's:
8) Jerzy Lewandowski and Donald Marolf, Loop constraints: A habitat and
their algebra, preprint available as grqc/9710016.
9) Rodolfo Gambini, Jerzy Lewandowski, Donald Marolf, and Jorge Pullin,
On the consistency of the constraint algebra in spin network quantum
gravity, preprint available as grqc/9710018.
This is very worrisome... as everything concerning quantum gravity
always is. Personally these results make me want to spend less time on
the Hamiltonian constraint, especially to the extent that it assumes a
the old picture of spacetime as a smooth manifold, and more time on
approaches that start with a discrete picture of spacetime. However,
the only way to make serious progress is for different people to push on
different fronts simultaneously.
There were a lot of other interesting talks, but since I'm concentrating
on quantum gravity here I won't describe the ones that were mainly about
topology. I'll wrap up by mentioning Steve Carlip's talk on spacetime
foam. He gave a nice illustration to how hard it is to "sum over
topologies" by arguing that this sum diverges for negative values of the
cosmological constant. He has a paper out on this:
10) Steven Carlip, Spacetime foam and the cosmological constant,
Phys. Rev. Lett. 79 (1997) 40714074, preprint available as
grqc/9708026.
Again, I'll quote the abstract:
"In the saddle point approximation, the Euclidean path integral for
quantum gravity closely resembles a thermodynamic partition function,
with the cosmological constant Lambda playing the role of temperature
and the ``density of topologies'' acting as an effective density of
states. For Lambda < 0, the density of topologies grows superexponentially,
and the sum over topologies diverges. In thermodynamics, such a
divergence can signal the existence of a maximum temperature. The same
may be true in quantum gravity: the effective cosmological constant may
be driven to zero by a rapid rise in the density of topologies."

Previous issues of "This Week's Finds" and other expository articles on
mathematics and physics, as well as some of my research papers, can be
obtained at
http://math.ucr.edu/home/baez/
For a table of contents of all the issues of This Week's Finds, try
http://math.ucr.edu/home/baez/twf.html
A simple jumpingoff point to the old issues is available at
http://math.ucr.edu/home/baez/twfshort.html
If you just want the latest issue, go to
http://math.ucr.edu/home/baez/this.week.html
ergroup. It is argued that a necessary condition for the
nonperturbative theory to have a good classical limit is that the
resulting 1+1 dimensional theory defines a consistent and stable
perturbative string theory."
Fotini Markopolou spoke about her rectwf_ascii/week115000064400020410000157000000430211015243252600141220ustar00baezhttp00004600000001Also available at http://math.ucr.edu/home/baez/week115.html
February 1, 1998
This Week's Finds in Mathematical Physics  Week 115
John Baez
These days I've been trying to learn more homotopy theory. James Dolan
got me interested in it by explaining how it offers many important clues
to ncategory theory. Ever since, we've been trying to understand what
the homotopy theorists have been up to for the last few decades. Since
trying to explain something is often the best way to learn it, I'll talk
about this stuff for several Weeks to come.
Before plunging in, though, I'd like mention yet another novel by Greg
Egan:
1) Greg Egan, Diaspora, Orion Books, 1997.
The main character of this book, Yatima, is a piece of software... and a
mathematician. The tale begins in 2975 with ver birth as an "orphan", a
citizen of the polis born of no parents, its mind seed chosen randomly
by the conceptory. Yatima learns mathematics in a virtual landscape
called the Truth Mines. To quote (with a few small modifications):
The luminous object buried in the cavern floor broadcast the
definition of a topological space: a set of points, grouped into
`open subsets' which specified how the points were connected to
one another  without appealing to notions like `distance' or
`dimension'. Short of a raw set with no structure at all, this was
about as basic as you could get: the common ancestor of virtually every
entity worth of the name `space', however exotic. A single tunnel led
into the cavern, providing the link to the necessary prior concepts, and
half a dozen tunnels led out, slanting gently `down' into the bedrock,
pursuing various implications of the definition. Suppose T is a
topological space... then what follows? These routes were paved with
small gemstones, each one broadcasting an intermediate result on the way
to a theorem.
Every tunnel in the Mines was built from the steps of a watertight
proof; every theorem, however deeply buried, could be traced back to
every one of its assumptions. And to pin down exactly what was meant by
a `proof', every field of mathematics used its own collection of formal
systems: sets of axioms, definitions, and rules of deduction, along with
the specialised vocabulary needed to state theorems and conjectures
precisely.
[....]
The library was full of the ways past miners had fleshed out the
theorems, and Yatima could have had those details grafted in alongside
the raw data, granting ver the archived understanding of thousands of
Konishi citizens who'd travelled this route before. The right
mindgrafts would have enabled ver effortlessly to catch up with all
the living miners who were pushing the coal face ever deeper in their
own inspired directions... at the cost of making ver, mathematically
speaking, little more than a patchwork clone of them, capable only of
following in their shadow.
If ve ever wanted to be a miner in vis own right  making and testing
vis own conjectures at the coal face, like Gauss and Euler, Riemann and
LeviCivita, deRham and Cartan, Radiya and Blanca  then Yatima knew
there were no shortcuts, no alternatives to exploring the Mines first
hand. Ve couldn't hope to strike out in a fresh direction, a route no
one had ever chosen before, without a new take on the old results. Only
once ve'd constructed vis own map of the Mines  idiosyncratically
crumpled and stained, adorned and annotated like no one else's  could
ve begin to guess where the next rich vein of undiscovered truths lay
buried.
The tale ends in a universe 267,904,176,383,054 duality transformations
away from ours, at the end of a long quest. What does Yatima do then?
Keep studying math! "It would be a long, hard journey to the coal face,
but this time there would be no distractions."
I won't give away any more of the plot. Suffice it to say that this is
hard science fiction  readers in search of carefully drawn characters
may be disappointed, but those who enjoy virtual reality, wormholes, and
philosophy should have a rollicking good ride. I must admit to being
biased in its favor, since it refers to a textbook I wrote. A science
fiction writer who actually knows the GaussBonnet theorem! We should
be very grateful.
Okay, enough fun  it's time for homotopy theory. Actually homotopy
theory is *tremendously* fun, but it takes quite a bit of persistence to
come anywhere close to the coal face. The original problems motivating
the subject are easy to state. Let's call a topological space simply
a "space", and call a continuous function between these simply a "map".
Two maps f,g: X > Y are "homotopic" if one can be continuously deformed
to the other, or in other words, if there is a "homotopy" between them:
a continuous function F: [0,1] x X > Y with
F(0,x) = f(x)
and
F(1,x) = g(x).
Also, two spaces X and Y are "homotopy equivalent" if there are
functions f: X > Y and g: Y > X for which fg and gf are homotopic to
the identity. Thus, for example, a circle, an annulus, and a solid
torus are all homotopy equivalent.
Homotopy theorists want to classify spaces up to homotopy equivalence.
And given two spaces X and Y, they want to understand the set [X,Y] of
homotopy classes of maps from X to Y. However, these are very hard
problems! To solve them, one needs highpowered machinery.
There are two roughly two sides to homotopy theory: building machines,
and using them to do computations. Of course these are fundamentally
inseparable, but people usually tend to prefer either one or the other
activity. Since I am a mathematical physicist, always on the lookout
for more tools for my own work, I'm more interested in the nice shiny
machines homotopy theorists have built than in the terrifying uses to
which they are put.
What follows will strongly reflect this bias: I'll concentrate on a
bunch of elegant concepts lying on the interface between homotopy theory
and category theory. This realm could be called "homotopical algebra".
Ideas from this realm can be applied, not only to topology, but to many
other realms. Indeed, two of its most famous practitioners, James
Stasheff and Graeme Segal, have spent the last decade or so using it in
string theory! I'll eventually try to say a bit about how that works,
too.
Okay.... now I'll start listing concepts and tools, starting with the
more fundamental ones and then working my way up. This will probably
only make sense if you've got plenty of that commodity known as
"mathematical sophistication". So put on some Coltrane, make yourself a
cafe macchiato, kick back, and read on. If at any point you feel a
certain lack of sophistication, you might want to reread "The Tale of
nCategories", starting with "week73", where a bunch of the basic terms
are defined.
A. Presheaf Categories. Given a category C, a "presheaf" on C is a
contravariant functor F: C > Sets. The original example of this is
where C is the category whose objects are open subsets of a topological
space X, with a single morphism f: U > V whenever the open set U is
contained in the open set V. For example, there is the presheaf of
continuous realvalued functions, for which F(U) is the set of all
continuous real functions on U, and for any inclusion f: U > V,
F(f): F(V) > F(U) is the "restriction" map which assigns to any
continuous function on V its restriction to U. This is a great way of
studying functions in all neighborhoods of X at once.
However, I'm bringing up this subject for a different reason, related to
a different kind of example. Suppose that C is a category whose objects
are "shapes" of some kind, with morphisms f: x > y corresponding to
ways the shape x can be included as a "piece" of the shape y. Then a
presheaf on C can be thought of as a geometrical structure built by
gluing together these shapes along their common pieces.
For example, suppose we want to describe directed graphs as presheaves.
A directed graph is a bunch of vertices and edges, where the edges have
a direction specified. Since they are made of two "shapes", the vertex
and the edge, we'll cook up a little category C with two object, V and
E. There are two ways a vertex can be included as a piece of an edge,
either as its "source" or its "target". Our category C, therefore, has
two morphisms, S: V > E and T: V > E. These are the only morphisms
except for identity morphisms  which correspond to how the edge is
part of itself, and the vertex is part of itself! Omitting identity
morphisms, our little category C looks like this:
S
>
V E
>
T
Now let's work out what a presheaf on C is. It's a contravariant
functor F: C > Set. What does this amount to? Well, it amounts
to a set F(V) called the "set of vertices", a set F(E) called the
"set of edges", a function F(S): F(E) > F(V) assigning to each edge
its source, and a function F(T): F(E) > F(V) assigning to each edge
its target. That's just a directed graph!
Note the role played by contravariance here: if a little shape V is
included as a piece of a big shape E, our category gets a morphism
S: V > E, and then in our presheaf we get a function F(S): F(E) > F(V)
going the *other way*, which describes how each big shape has a bunch of
little shapes as pieces.
Given any category C there is actually a *category* of presheaves on C.
Given presheaves F,G: C > Sets, a morphism M from F to G is just a
natural transformation M: F => G. This is beautifully efficient way of
saying quite a lot. For example, if C is the little category described
above, so that F and G are directed graphs, a natural transformation
M: F => G is the same as:
a map M(V) sending each vertex of the graph F to a vertex of the
graph G,
and
a map M(E) sending each edge of the graph F to a edge of the
graph G,
such that
M(V) of the source of any edge e of F equals the source of
M(E) of e,
and
M(V) of the target of any edge e of F equals the target of
M(E) of e.
Whew! Easier just to say M is a natural transformation between
functors!
For more on presheaves, try:
2) Saunders Mac Lane and Ieke Moerdijk, Sheaves in Geometry and Logic:
a First Introduction to Topos Theory, SpringerVerlag, New York, 1992.
B. The Category of Simplices. This is a very important example
of a category whose objects are shapes  namely, simplices  and
whose morphisms correspond to the ways one shape is a piece of another.
The objects of Delta are called 1, 2, 3, etc., corresponding to the
simplex with 1 vertex (the point), the simplex with 2 vertices (the
interval), the simplex with 3 vertices (the triangle), and so on. There
are a bunch of ways for an lowerdimensional simplex to be a face of a
higher dimensional simplex, which give morphisms in Delta. More
subtly, there are also a bunch of ways to map a higherdimensional
simplex down into a lowerdimensional simplex, called "degeneracies".
For example, we can map a tetrahedron down into a triangle in a way that
carries the vertices {0,1,2,3} of the tetrahedron into the vertices
{0,1,2} of the triangle as follows:
0 > 0
1 > 0
2 > 1
3 > 2
These degeneracies also give morphisms in Delta.
We could list all the morphisms and the rules for composing them
explicitly, but there is a much slicker way to describe them. Let's use
the old trick of thinking of the natural number n as being the totally
ordered nelement set {0,1,2,...,n1} of all natural numbers less than
n. Thus for example we think of the object 4 in Delta, the tetrahedron,
as the totally ordered set {0,1,2,3}. These correspond to the 4
vertices of the tetrahedron. Then the morphisms in Delta are just all
orderpreserving maps between these totally ordered sets. So for
example there is a morphism f: {0,1,2,3} > {0,1,2} given by the
orderpreserving map with
f(0) = 0
f(1) = 0
f(2) = 1
f(3) = 2
The rule for composing morphisms is obvious: just compose the maps!
Slick, eh?
We can be slicker if we are willing to work with a category *equivalent*
to Delta (in the technical sense described in "week76"), namely, the
category of *all* nonempty totally ordered sets, with orderpreserving
maps as morphisms. This has a lot more objects than just {0}, {0,1},
{0,1,2}, etc., but all of its objects are isomorphic to one of these.
In category theory, equivalent categories are the same for all practical
purposes  so we brazenly call this category Delta, too. If we do
so, we have following *incredibly* slick description of the category
of simplices: it's just the category of finite nonempty totally ordered
sets!
If you are a true mathematician, you will wonder "why not use the empty
set, too?" Generally it's bad to leave out the empty set. It may seem
like "nothing", but "nothing" is usually very important. Here it
corresponds to the "empty simplex", with no vertices! Topologists often
leave this one out, but sometimes regret it later and put it back in
(the buzzword is "augmentation"). True category theorists, like Mac
Lane, never leave it out. They define Delta to be the category of *all*
totally ordered finite sets. For a beautiful introduction to this approach,
try:
3) Saunders Mac Lane, Categories for the Working Mathematician,
Springer, Berlin, 1988.
C. Simplicial sets. Now we put together the previous two ideas: a
"simplicial set" is a presheaf on the category of simplices! In other
words, it's a contravariant functor F: Delta > Set. Geometrically,
it's basically just a bunch of simplices stuck together along their
faces in an arbitrary way. We can think of it as a kind of purely
combinatorial version of a "space". That's one reason simplicial sets
are so popular in topology: they let us study spaces in a truly elegant
algebraic context. We can define all the things topologists love 
homology groups, homotopy groups (see "week102"), and so on  while
never soiling our hands with open sets, continuous functions and the
like. To see how it's done, try:
4) J. Peter May, Simplicial Objects in Algebraic Topology, Van Nostrand,
Princeton, 1968.
Of course, not everyone prefers the austere joys of algebra to the
earthy pleasures of geometry. Algebraic topologists thrill to
categories, functors and natural transformations, while geometric
topologists like drawing pictures of hideously deformed multiholed
doughnuts in 4 dimensional space. It's all a matter of taste.
Personally, I like both!
D. Simplicial objects. We can generalize the heck out of the notion of
"simplicial set" by replacing the category Set with any other category
C. A "simplical object in C" is defined to be a contravariant functor
F: Delta > C. There's a category whose objects are such functors and
whose morphisms are natural transformations between them.
So, for example, a "simplicial abelian group" is a simplicial object
in the category of abelian groups. Just as we may associate to any
set X the free abelian group on X, we may associate to any simplicial
set X the free simplicial abelian group on X. In fact, it's more than
analogy: the latter construction is a spinoff of the former! There
is a functor
L: Set > Ab
assigning to any set the free abelian group on that set (see "week77").
Given a simplicial set
X: Delta > Set
we may compose with L to obtain a simplicial abelian group
XL: Delta > Ab
(where I'm writing composition in the funny order that I like to use).
This is the free simplicial abelian group on the simplicial set X!
Later I'll talk about how to compute the homology groups of a simplicial
abelian group. Combined with the above trick, this will give a very
elegant way to define the homology groups of a simplicial set. Homology
groups are a very popular sort of invariant in algebraic topology; we
will get them with an absolute minimum of sweat.
Just as a good firework show ends with lots of explosions going off
simultaneously, leaving the audience stunned, deafened, and content, I
should end with a blast of abstraction, just for the hell of it. Those
of you who remember my discussion of "theories" in "week53" can easily
check that there is a category called the "theory of abelian groups".
This allows us to define an "abelian group object" in any category with
finite limits. In particular, since the category of simplicial sets has
finite limits (any presheaf category has all limits), we can define an
abelian group object in the category of simplicial sets. And now for a
very pretty result: abelian group objects in the category of simplicial
sets are the same as simplicial abelian groups! In other words, an
abstract "abelian group" living in the world of simplicial sets is the
same as an abstract "simplicial set" living in the world of abelian
groups. I'm very fond of this kind of "commutativity of abstraction".
Finally, I should emphasize that all of this stuff was first explained
to me by James Dolan. I just want to make these explanations available
to everyone.

Previous issues of "This Week's Finds" and other expository articles on
mathematics and physics, as well as some of my research papers, can be
obtained at
http://math.ucr.edu/home/baez/
For a table of contents of all the issues of This Week's Finds, try
http://math.ucr.edu/home/baez/twf.html
A simple jumpingoff point to the old issues is available at
http://math.ucr.edu/home/baez/twfshort.html
If you just want the latest issue, go to
http://math.ucr.edu/home/baez/this.week.html
ome kind, with morphisms f: x > y corresponding to
ways the shape x can be included as a "piece" of the shape y. Then a
presheaf on C can be thought of as a geometrical structure built by
gluing together these shapes along their common pieces.
For example, suppose we want to describe directed graphs as presheaves.
A directed graph is a bunch of vertices and edges, where the edges have
a direction specified. Since they are made of two "shapes", the vertex
and the edge, we'll cook up a twf_ascii/week116000064400020410000157000000425511103537250000141260ustar00baezhttp00004600000001Also available at http://math.ucr.edu/home/baez/week116.html
February 7, 1998
This Week's Finds in Mathematical Physics  Week 116
John Baez
While general relativity and the Standard Model of particle physics are
very different in many ways, they have one important thing in common:
both are gauge theories. I will not attempt to explain what a gauge
theory is here. I just want to recommend the following nice book on the
early history of this subject:
1) Lochlainn O'Raifeartaigh, The Dawning of Gauge Theory, Princeton
U. Press, Princeton, 1997.
This contains the most important early papers on the subject, translated
into English, together with detailed and extremely intelligent
commentary. It starts with Hermann Weyl's 1918 paper "Gravitation and
Electricity", in which he proposed a unification of gravity and
electromagnetism. This theory was proven wrong by Einstein in a
oneparagraph remark which appears at the end of Weyl's paper 
Einstein noticed it would predict atoms of variable size!  but it
highlighted the common features of general relativity and Maxwell's
equations, which were later generalized to obtain the modern concept of
gauge theory.
It also contains Theodor Kaluza's 1921 paper "On the Unification Problem
of Physics" and Oskar Klein's 1926 paper "Quantum Theory and
FiveDimensional Relativity". These began the trend, currently very
popular in string theory, of trying to unify forces by postulating
additional dimensions of spacetime. It's interesting how gauge theory
has historical roots in this seemingly more exotic notion. The original
KaluzaKlein theory assumed a 5dimensional spacetime, with the extra
dimension curled into a small circle. Starting with 5dimensional
general relativity, and using the U(1) symmetry of the circle, they
recovered 4dimensional general relativity coupled to a U(1) gauge
theory  namely, Maxwell's equations. Unfortunately, their theory also
predicted an unobserved spin0 particle, which was especially
problematic back in the days before mesons were discovered.
I wasn't familiar with another item in this book, Wolfgang Pauli's
letter to Abraham Pais entitled "MesonNucleon Interactions and
Differential Geometry". This theory, "written down July 2225 1953 in
order to see how it looks", postulated 2 extra dimensions in the shape
of a small sphere. The letter begins, "Split a 6dimensional space into
a (4+2)dimensional one." At the time, mesonnucleon interactions were
believe to have an SU(2) symmetry corresponding to conservation of
"isospin". Pauli obtained a theory with this symmetry group using the
SU(2) symmetry of the sphere.
Apparently Pauli got a lot of his inspiration from Weyl's 1929 paper
"Electron and Gravitation", also reprinted in this volume. This
masterpiece did all the following things: it introduced the concept of
2component spinors (see "week109"), considered the possibility that the
laws of physics violate parity and time reversal symmetry, introduced
the tetrad formulation of general relativity, introduced the notion of a
spinor connection, and explicitly derived electromagnetism from the
gauge principle! A famously critical fellow, Pauli lambasted Weyl's
ideas on parity and time reversal violation  which are now known to
be correct. But even he conceded the importance of deriving Maxwell's
equations from the gauge principle, saying "Here I must admit your
ability in Physics". And he incorporated many of the ideas into his
1953 letter.
An allaround good read for anyone seriously interested in the history
of physics! It's best if you already know some gauge theory.
Now let me continue the tour of homotopy theory I began last week. I
was talking about simplices. Simplices are amphibious creatures, easily
capable of living in two different worlds. On the one hand, we can
think of them as topological spaces, and on the other hand, as purely
algebraic gadgets: objects in the category of finite totally ordered
sets, which we call Delta. This gives simplices a special role as a
bridge between topology and algebra.
This week I'll begin describing how this works. Next time we'll
get into some of the cool spinoffs. I'll keep up the format of
listing tools one by one:
E. Geometric realization. In "week115" I talked about simplicial sets.
A simplicial set is a presheaf on the category Delta. Intuitively, it's
a purely combinatorial way of describing a bunch of abstract simplices
glued together along their faces. We want a process that turns such
things into actual topological spaces, and also a process that turns
topological spaces back into simplicial sets.
Let's start with the first one. Given a simplicial set X, we can
form a space X called the "geometric realization" of X by gluing
spaces shaped like simplices together in the pattern given by X. Given
a morphism between simplicial sets there's an obvious continuous map
between their geometric realizations, so geometric realization is
actually a functor
 : SimpSet > Top
from the category of simplicial sets, SimpSet, to the category of
topological space, Top.
It's straightforward to fill in the details. But if we want to be
slick, we can define geometric realization using the magic of
adjoint functors  see below.
F. Singular Simplicial Set. The basic idea here is that given a
topological space X, its "singular simplicial set" Sing(X) consists of
all possible ways of mapping simplices into X. This gives a functor
Sing: Top > SimpSet.
We make this precise as follows.
By thinking of simplices as spaces in the obvious way, we can
associate a space to any object of Delta, and also a continuous map
to any morphism in Delta. Thus there's a functor
i: Delta > Top.
For any space X we define
Sing(X): Delta > Set
by
Sing(X)() = hom(i(),X)
where the blank slot indicates how Sing(X) is waiting to eat a simplex
and spit out the set of all ways of mapping it  thought of as a
space!  into the space X. The blank slot also indicates how Sing(X)
is waiting to eat a *morphism* between simplices and spit out a
*function* between sets.
Having said what Sing does to *spaces*, what does it do to *maps*? The
same formula works: for any map f: X > Y between topological spaces,
we define
Sing(f)() = hom(i(),f).
It may take some headscratching to understand this, but if you work it
out, you'll see it works out fine. If you feel like you are drowning
under a tidal wave of objects, morphisms, categories, and functors,
don't worry! Medical research has determined that people actually grown
new neurons when learning category theory.
In fact, even though it might not seem like it, I'm being incredibly
pedagogical and nurturing. If I were really trying to show off, I would
have compressed the last couple of paragraphs into the following one
line:
Sing()() = hom(i(),).
where Sing becomes a functor using the fact that for any category C
there's a functor hom: C^{op} x C > Set, where C^{op} denotes the
"opposite" of C, the category with all its arrows reversed. (See
"week78" for an explanation of this.)
Or I could have said this: form the composite
i x 1 hom
Delta^{op} x Top > Top^{op} x Top > Set
and dualize this to obtain
Sing: Top > SimpSet.
These are all different ways of saying the same thing.
Forming the singular simplical set of a space is not really an "inverse"
to geometric realization, since if we take a simplicial set X, form its
geometric realization, and then form the singular simplicial set of
that, we get something much bigger than X. However, if you think about
it, there's an obvious map from X into Sing(X). Similarly, if we
start with a topological space X, there's an obvious map from Sing(X)
down to X.
What this means is that Sing is the right adjoint of  , or in other
words,   is the left adjoint of Sing. Thus if we want to be slick,
we can just *define* geometric realization to be the left adjoint of
Sing. (See "week77""week79" for an exposition of adjoint functors.)
G. Chain Complexes. Now gird yourself for some utterly unmotivated
definitions! If you've taken a basic course in algebraic topology, you
have probably learned about chain complexes already, and if you haven't,
you probably aren't reading this anymore  so I'll just plunge in.
A "chain complex" C is a sequence of abelian groups and "boundary"
homomorphisms like this:
d_1 d_2 d_3
C_0 < C_1 < C_2 < C_3 < ...
satisfying the magic equation
d_i d_{i+1} x = 0
This equation says that the image of d_{i+1} is contained in the kernel
of d_i, so we may define the "homology groups" to be the quotients
H_i(C) = ker(d_i) / im(d_{i+1})
The study of this stuff is called "homological algebra". You can read
about it in such magisterial tomes as:
2) Henri Cartan and Samuel Eilenberg, Homological Algebra, Princeton
University Press, 1956.
or
3) Saunders Mac Lane, Homology, SpringerVerlag, Berlin, 1995.
But it you want something a bit more userfriendly, try:
4) Joseph J. Rotman, An Introduction to Homological Algebra,
Academic Press, New York, 1979.
The main reason chain complexes are interesting is that they are similar
to topological spaces, but simpler. In "singular homology theory", we
use a certain functor to convert topological spaces into chain
complexes, thus reducing topology problems to simpler algebra problems.
This is usually one of the first things people study when they study
algebraic topology. In sections G. and H. below, I'll remind you how
this goes.
Though singular homology is very useful, not everybody gets around to
learning the deep reason why! In fact, chain complexes are really just
another way of talking about a certain especially simple class of
topological spaces, called "products of EilenbergMacLane spaces of
abelian groups". In such spaces, topological phenomena in different
dimensions interact in a particularly trivial way. Singular homology
thus amounts to neglecting the subtler interactions between topology
in different dimensions. This is what makes it so easy to work with 
yet ultimately so limited.
Before I keep rambling on I should describe the category of chain
complexes, which I'll call Chain. The objects are just chain complexes,
and given two of them, say C and C', a morphism f: C > C' is a sequence
of group homomorphisms f_i : C_i > C'_i making the following big
diagram commute:
d_1 d_2 d_3
C_0 < C_1 < C_2 < C_3 < ...
   
f_0 f_1 f_2 f_3
   
V V V V
C'_0 < C'_1 < C'_2 < C'_3 < ...
d'_1 d'_2 d'_3
The reason Chain gets to be so much like the category Top of topological
spaces is that we can define homotopies between morphisms of chain
complexes by copying the definition of homotopies between continuous
maps. First, there is a chain complex called I that's analogous to the
unit interval. It looks like this:
d_1 d_2 d_3 d_4
Z+Z < Z < 0 < 0 < ...
The only nonzero boundary homomorphism is d_1, which is given by
d_1(x) = (x,x).
(Why? We take I_1 = Z and I_0 = Z+Z because the interval is built out
of one 1dimensional thing, namely itself, and two 0dimensional things,
namely its endpoints. We define d_1 the way we do since the boundary of
an oriented interval consists of two points: its initial endpoint, which
is positively oriented, and its final endpoint, which is negatively
oriented. This remark is bound to be obscure to anyone who hasn't
already mastered the mystical analogies between algebra and topology
that underlie homology theory!)
There is a way to define a "tensor product" C x C' of chain complexes C
and C', which is analogous to the product of topological spaces. And
there are morphisms
i,j : C > I x C
analogous to the two maps from a space into its product with the unit
interval:
i,j : X > [0,1] x X
i(x) = (0,x), j(x) = (1,x)
Using these analogies we can define a "chain homotopy" between chain
complex morphisms f,g: C > C' in a way that's completely analogous to a
homotopy between maps. Namely, it's a morphism F: I x C > C' for which
the composite
i F
C > I x C > C'
equals f, and the composite
j F
C > I x C > C'
equals g. Here we are using the basic principle of category theory:
when you've got a good idea, write it out using commutative diagrams
and then generalize the bejeezus out of it!
The nice thing about all this is that a morphism of chain complexes
f: C > C' gives rise to homomorphisms of homology groups,
H_n(f): H_n(C) > H_n(C').
In fact, we've got a functor
H_n: Chain > Ab.
And even better, if f: C > C' and g: C > C' are chain homotopic, then
H_n(f) and H_n(g) are equal. So we say: "homology is homotopyinvariant".
H. The Chain Complex of a Simplicial Abelian Group. Now let me
explain a cool way of getting chain complexes, which goes a long way
towards explaining why they're important. Recall from item D. in
"week115" that a simplicial abelian group is a contravariant functor
C: Delta > Ab. In particular, it gives us an abelian group C_n
for each object n of Delta, and also "face" homomorphisms
partial_0, ...., partial_{n1} : C_n > C_{n1}
coming from all the ways the simplex with (n1) vertices can be a face
of the simplex with n vertices. We can thus can make C into a chain
complex by defining d_n: C_n > C_{n1} as follows:
d_n = sum (1)^i partial_i
The thing to check is that d_n d_{n+1} x = 0 for all x in C_{n+1}.
The alternating signs make everything cancel out! In the immortal words
of the physicist John Wheeler, "the boundary of a boundary is zero".
Unsurprisingly, this gives a functor from simplicial abelian groups
to chain complexes. Let's call it
Ch: SimpAb > Chain
More surprisingly, this is an equivalence of categories! I leave you to
show this  if you give up, look at May's book cited in section C. of
"week115". What this means is that simplicial abelian groups are just
another way of thinking about chain complexes... or vice versa. Thus,
if I were being ultrasophisticated, I could have skipped the chain
complexes and talked only about simplicial abelian groups! This would
have saved time, but more people know about chain complexes, so I wanted
to mention them.
I. Singular Homology. Okay, now that we have lots of nice shiny
machines, let's hook them up and see what happens! Take the "singular
simplicial set" functor:
Sing: Top > SimpSet,
the "free simplicial abelian group on a simplicial set" functor:
L: SimpSet > SimpAb,
and the "chain complex of a simplicial abelian group" functor:
Ch: SimpAb > Chain,
and compose them! We get the "singular chain complex" functor
C: Top > Chain
that takes a topological space and distills a chain complex out of it.
We can then take the homology groups of our chain complex and get the
"singular homology" of our space. Better yet, the functor C: Top >
Chain takes homotopies between maps and sends them to homotopies between
morphisms of chain complexes! It follows that homotopic maps between
spaces give the same homomorphisms between the singular homology groups
of these spaces. Thus homotopyequivalent spaces will have isomorphic
homology groups... so we have gotten our hands on a nice tool for
studying spaces up to homotopy equivalence.
Now that we've got our hands on singular homology, we could easily spend
a long time using it to solve all sorts of interesting problems. I
won't go into that here; you can read about it in all sorts of
textbooks, like:
5) Marvin J. Greenberg, John R. Harper, Algebraic Topology: A First Course,
Benjamin/Cummings, Reading, Massachusetts, 1981.
or
6) William S. Massey, Singular Homology Theory, SpringerVerlag, New York,
1980.
which uses cubes rather than simplices.
What I'm trying to emphasize here is that singular homology is a
composite of functors that are interesting in their own right. I'll
explore their uses a bit more deeply next time.

Quote of the Week:
At a very early age, I made an assumption that a successful physicist
only needs to know elementary mathematics. At a later time, to my great
regret, I realized that this assumption of mine was completely wrong.
 Albert Einstein

Previous issues of "This Week's Finds" and other expository articles on
mathematics and physics, as well as some of my research papers, can be
obtained at
http://math.ucr.edu/home/baez/
For a table of contents of all the issues of This Week's Finds, try
http://math.ucr.edu/home/baez/twf.html
A simple jumpingoff point to the old issues is available at
http://math.ucr.edu/home/baez/twfshort.html
If you just want the latest issue, go to
http://math.ucr.edu/home/baez/this.week.html
ll different ways of saying the same thing.
Forming the singular simplical set of a space is not really an "inverse"
to geometric realization, sinctwf_ascii/week117000064400020410000157000000451631015243272000141310ustar00baezhttp00004600000001Also available at http://math.ucr.edu/home/baez/week117.html
February 14, 1997
This Week's Finds in Mathematical Physics  Week 117
John Baez
A true physicist loves matter in all its states. The phases we all
learned about in school  solid, liquid, and gas  are just the
beginning of the story! There lots of others: liquid crystal, plasma,
superfluid, and neutronium, for example. Today I want to say a little
about two more phases that people are trying to create: quarkgluon
plasma and strange quark matter. The first almost certainly exists; the
second is a matter of much discussion.
1) The E864 Collaboration, Search for charged strange quark matter
produced in 11.5 A GeV/c Au + Pb collisions, Phys. Rev. Lett. 79 (1997)
36123616, preprint available as nuclex/9706004.
Last week I went to a talk on the search for strange quark matter by
one of these collaborators, Kenneth Barish. This talk was based on
Barish's work at the E864 experiment at the "AGS", the alternating
gradient synchrotron at Brookhaven National Laboratory in Long Island,
New York.
What's "strange quark matter"? Well, first remember from "week93" that
in the Standard Model there are bosonic particles that carry forces:
ELECTROMAGNETIC FORCE WEAK FORCE STRONG FORCE
photon W+ 8 gluons
W
Z
and fermionic particles that constitute matter:
LEPTONS QUARKS
electron electron neutrino down quark up quark
muon muon neutrino strange quark charm quark
tauon tauon neutrino bottom quark top quark
(There is also the mysterious Higgs boson, which has not yet been seen.)
The quarks and leptons come in 3 generations each. The only quarks in
ordinary matter are the lightest two, those from the first generation:
the up and down. These are the constituents of protons and neutrons,
which are the only stable particles made of quarks. A proton consists
of two ups and a down held together by the strong force, while a neutron
consists of two downs and a up. The up has electric charge +2/3, while
the down has electric charge 1/3. They also interact via the strong
and weak forces.
The other quarks are more massive and decay via the weak interaction
into up and down quarks. Apart from that, however, they are quite
similar. There are lots of shortlived particles made of various
combinations of quarks. All the combinations we've seen so far are of
two basic sorts. There are "baryons", which consist of 3 quarks, and
"mesons", which consist of a quark and an antiquark. Both of these
should be visualized roughly as a sort of bag with the quarks and a
bunch of gluons confined inside.
Why are they confined? Well, I sketched an explanation in "week94", so
you should read that for more details. For now let's just say the
strong force likes to "stick together", so that energy is minimized if
it stays concentrated in small regions, rather than spreading all over
the place, like the electromagnetic field does. Indeed, the strong
force may even do something like this in the absence of quarks, forming
shortlived "glueballs" consisting solely of gluons and virtual
quarkantiquark pairs. (For more on glueballs, see "week68".)
For reasons I don't really understand, the protons and neutrons in the
nucleus do not coalesce into one big bag of quarks. Even in a neutron
star, the quarks stay confined in their individual little bags. But
calculations suggest that at sufficiently high temperatures or
pressures, "deconfinement" should occur. Basically, under these
conditions the baryons and mesons either smash into each other so hard,
or get so severely squashed, that they burst open. The result should be
a soup of free quarks and gluons: a "quarkgluon plasma".
To get deconfinement to happen is not easy  at low pressures, it's
expected to occur at a temperature of 2 trillion Kelvin! According
to the conventional wisdom in cosmology, the last time deconfinement
was prevalent was about 1 microsecond after the big bang! In the E864
experiment, they are accelerating gold nuclei to energies of 11.5 GeV
per nucleon and colliding them with a fixed target made of lead, which
is apparently *not* enough energy to fully achieve deconfinement 
they believe they are reaching temperatures of about 1 trillion Kelvin.
At CERN they are accelerating lead nuclei to 160 GeV per nucleon
and colliding them with a lead target. They may be getting signs of
deconfinement, but as Jim Carr explained in a recent post to sci.physics,
they're being very cautious about coming out and saying so. By mid1999,
the folks at Brookhaven hope to get higher energies with the Relativistic
Heavy Ion Collider, which will collide two beams of gold nuclei headon
at 100 GeV per nucleon... see "week76" for more on this.
One of the hopedfor signs of deconfinement is "strangeness enhancement".
The lightest quark besides the up and down is the strange quark, and in
the high energies present in a quark gluon plasma, strange quarks should
be formed. Moreover, since Pauli exclusion principle prevents two identical
fermions from being in the same state, it can be energetically favorable
to have strange quarks around, since they can occupy lowerenergy states
which are already packed with ups and downs. They seem to be seeing
strangeness enhancement at CERN:
2) Juergen Eschke, NA35 Collaboration, Strangeness enhancement in sulphur
nucleus collisions at 200 GeV/N, preprint available as hepph/9609242.
As far as I can tell, people are just about as sure that deconfinement
occurs at high temperatures as they would be that tungsten boils at
high temperatures, even if they've never actually seen it happen. A
more speculative possibility is that as quarkgluon plasma cools down
it forms "strange quark matter" in the form of "strangelets": big bags of
up, down, and strange quarks. This is what they're looking for at E864.
Their experiment would only detect strangelets that live long enough
to get to the detector. When their experiment is running they get 10^6
collisions per second. So far they've set an upper bound of 10^{7}
charged strangelets per collision, neutral strangelets being harder to
detect and rule out. For more on strangelets, try this:
3) E. P. Gilson and R. L. Jaffe, Very small strangelets, Phys. Rev. Lett.
71 (1993) 332335, preprint available as hepph/9302270.
Strange quark matter is also of interest in astrophysics. In 1984
Witten wrote a paper proposing that in the limit of large quark number,
strange quark matter could be more stable than ordinary nuclear matter!
4) Edward Witten, Cosmic separation of phases, Phys. Rev. D30 (1984)
272285.
More recently, a calculation of Farhi and Jaffe estimates that in
the limit of large quark number, the energy of strange quark matter is
301 MeV per quark, as compared with 310 Mev/quark for iron56, which
is the most stable nucleus. This raises the possibility that under
suitable conditions, a neutron star could collapse to become a "quark
star" or "strange star". Let me quote the abstract of the following
paper:
5) Dany Page, Strange stars: Which is the ground state of QCD at finite
baryon number?, `High Energy Phenomenology' eds. M. A. Perez & R. Huerta
(World Scientific), 1992, pp. 347  356, preprint available as
astroph/9602043.
Witten's conjecture about strange quark matter (`Strange Matter') being
the ground state of QCD at finite baryon number is presented and stars
made of strange matter (`Strange Stars') are compared to neutron stars.
The only observable way in which a strange star differs from a neutron
star is in its early thermal history and a detailed study of strange star
cooling is reported and compared to neutron star cooling. One concludes
that future detection of thermal radiation from the compact object
produced in the core collapse of SN 1987A could present the first
evidence for strange matter.
Here are a couple of books on the subject, which unfortunately I've
not been able to get ahold of:
6) Strange Quark Matter in Physics and Astrophysics: Proceedings of the
International Workshop on Strange Quark Matter in Physics and Astrophysics,
ed. Jes Madsen, NorthHolland, Amsterdam, 1991.
7) International Symposium on Strangeness and Quark Matter, eds. Georges
Vassiliadis et al, World Scientific, Singapore, 1995.
If anyone out there knows more about the latest theories of strange quark
matter, and can explain them in simple terms, I'd love to hear about it.
Okay, enough of that.
Now, on with my tour of homotopy theory!
So far I've mainly been talking about simplicial sets. I described a
functor called "geometric realization" that turns a simplicial set into
a topological space, and another functor that turns a space into a
simplicial set called its "singular simplicial set". I also showed how
to turn a simplicial set into a simplicial abelian group, and how
to turn one of *those* into a chain complex... or vice versa.
As you can see, the key is to have lots of functors at your disposal, so
you can take a problem in any given context  or more precisely, any
given category!  and move it to other contexts where it may be easier
to solve. Eventually I want to talk about what all these categories
we're using have in common: they are all "model categories". Once we
understand that, we'll be able to see more deeply what's going on in all
the games we've been playing.
But first I want to describe a few more important tricks for turning
this into that. Recall from "week115" that there's a category Delta
whose objects 0,1,2,... are the simplices, with n corresponding to the
simplex with n vertices  the simplex with 0 vertices being the "empty
simplex". We can also define Delta in a purely algebraic way as the
category of finite totally ordered sets, with n corresponding to the
totally ordered set {0,1,....,n1}. The morphisms in Delta are then the
orderpreserving maps. Using this algebraic definition we can do some
cool stuff:
J. The Nerve of a Category. This is a trick to turn a category into a
simplicial set. Given a category C, we cook up the simplicial set
Nerve(C) as follows. The 0dimensional simplices of Nerve(C) are just
the objects of C, which look like this:
x
The 1simplices of Nerve(C) are just the morphisms, which look like
this:
xf>y
The 2simplices of Nerve(C) are just the commutative diagrams that
look like this:
y
/ \
f g
/ \
xh>z
where f: x > y, g: y > z, and h: x > z. And so on! In general,
the nsimplices of Nerve(C) are just the commutative diagrams in
C that look like nsimplices!
When I first heard of this idea I cracked up. It seemed like an insane
sort of joke. Turning a category into a kind of geometrical object
built of simplices? What nerve! What use could this possibly be?
Well, for an application of this idea to computer science, see "week70".
We'll soon see lots of applications within topology. But first, let me
give a slick abstract description of this "nerve" process that turns
categories into simplicial sets. It's really a functor
Nerve: Cat > SimpSet
going from the category of categories to the category of simplicial
sets.
First, a remark on Cat. This has categories as objects and functors
as morphisms. Since the "category of all categories" is a bit
creepy, we really want the objects of Cat to be all the "small"
categories, i.e., those having a mere *set* of objects. This
prevents Russell's paradox from raising its ugly head and disturbing
our fun and games.
Next, note that any partially ordered set can be thought of as a
category whose objects are just the elements of our set, and where we
say there's a single morphism from x to y if x <= y. Composition of
morphisms works out automatically, thanks to the transitivity of "less
than or equal to". We thus obtain a functor
i: Delta > Cat
taking each finite totally ordered set to its corresponding category,
and each orderpreserving map to its corresponding functor.
Now we can copy the trick we played in section F of "week116". For any
category C we define the simplicial set Nerve(C) by
Nerve(C)() = hom(i(),C)
Think about it! If you put the simplex n in the blank slot, we
get hom(i(n),C), which is the set of all functors from that simplex,
*regarded as a category*, to the category C. This is just the
set of all diagrams in C shaped like the simplex n, as desired!
We can say all this even more slickly as follows: take
i x 1 hom
Delta^{op} x Cat > Cat^{op} x Cat > Set
and dualize it to obtain
Nerve: Cat > SimpSet.
I should also point out that topologists usually do this stuff with
the topologist's version of Delta, which does not include the "empty
simplex".
K. The Classifying Space of Category. If compose our new functor
Nerve: Cat > SimpSet
with the "geometric realization" functor
 : SimpSet > Top
defined in section E, we get a way to turn a category into a space,
called its "classifying space". This trick was first used by Graeme
Segal, the homotopy theorist who later became the guru of conformal
field theory. He invented this trick in the following paper:
8) Graeme B. Segal, Classifying spaces and spectral sequences,
Publ. Math. Inst. des Haut. Etudes Scient. 34 (1968), 105112.
As it turns out, every reasonable space is the classifying space of some
category! More precisely, every space that's the geometric realization
of some simplicial set is homeomorphic to the classifying space of some
category. To see this, suppose the space X is the geometric realization
of the simplicial set S. Take the set of all simplices in S and
partially order them by saying x <= y if x is a face of y. Here by
"face" I don't mean just mean a face of one dimension less than that of
y; I'm allowing faces of any dimension less than or equal to that of y.
We obtain a partially ordered set. Now think of this as a category, C.
Then Nerve(C) is the "barycentric subdivision" of S. In other words,
it's a new simplicial set formed by chopping up the simplices of S into
smaller pieces by putting a new vertex in the center of each one. It
follows that the geometric realization of Nerve(C) is homeomorphic to
that of S.
There are lots of interesting special sorts of categories, like
groupoids, or monoids, or groups (see "week74"). These give special
cases of the "classifying space" construction, some of which were
actually discovered before the general case. I'll talk about some of
these more next week, since they are very important in topology.
Also sometimes people take categories that they happen to be interested
in, which may have no obvious relation to topology, and study them by
studying their classifying spaces. This gives surprising ways to apply
topology to all sorts of subjects. A good example is "algebraic
Ktheory", where we start with some sort of category of modules over a
ring.
K. Delta as the free monoidal category on a monoid object. Recall that
a "monoid" is a set with a product and a unit element satisfying
associativity and the left and right unit laws. Categorifying this
notion, we obtain the concept of a "monoidal category": a category C
with a product and a unit object satisfying the same laws. A nice
example of a monoidal category is the category Set with its usual
cartesian product, or the category Vect with its usual tensor product.
We usually call the product in a monoidal category the "tensor product".
Now, the "microcosm principle" says that algebraic gadgets often like to
live inside categorified versions of themselves. It's a bit like the
"homunculus theory", where I have a little copy of myself sitting in my
head who looks out through my eyes and thinks all my thoughts for me.
But unlike that theory, it's true!
For example, we can define a "monoid object" in any monoidal category.
Given a monoidal category A with tensor product x and unit object 1, we
define a monoid object a in A to be an object equipped with a "product"
m: a x a > a
and a "unit"
i: 1 > a
which satisfy associativity and the left and right unit laws (written
out as commutative diagrams). A monoid object in Set is just a monoid,
but a monoid object in Vect is an algebra, and I gave some very
different examples of monoid objects in "week89".
Now let's consider the "free monoidal category on a monoid object". In
other words, consider a monoidal category A with a monoid object a in
it, and assume that A has no objects and no morphisms, and satisfies no
equations, other than those required by the definitions of "monoidal
category" and "monoid object".
Thus the only objects of A are the unit object together with a and its
tensor powers. Similarly, all the morphism of A are built up by
composing and tensoring the morphisms m and i. So A looks like this:
1x1xi
>
1xi 1xix1
> >
i ix1 ix1x1
1 > a > a x a > a x a x a ...
m mx1
< <
1xm
<
Here I haven't drawn all the morphisms, just enough so that every
morphism in A is a composite of morphisms of this sort.
What is this category? It's just Delta! The nth tensor power of a
corresponds to the simplex with n vertices. The morphisms going to the
right describe the ways the simplex with n vertices can be a face of the
simplex with n+1 vertices. The morphisms going to the left correspond
to "degeneracies"  ways of squashing a simplex with n+1 vertices down
into one with n vertices.
So: in addition to its other descriptions, we can define Delta as the
free monoidal category on a monoid object! Next time we'll see how
this is fundamental to homological algebra.

Previous issues of "This Week's Finds" and other expository articles on
mathematics and physics, as well as some of my research papers, can be
obtained at
http://math.ucr.edu/home/baez/
For a table of contents of all the issues of This Week's Finds, try
http://math.ucr.edu/home/baez/twf.html
A simple jumpingoff point to the old issues is available at
http://math.ucr.edu/home/baez/twfshort.html
If you just want the latest issue, go to
http://math.ucr.edu/home/baez/this.week.html
py theory!
So far I've mainly been talking about simplicial sets. I described a
functor called "geometric realization" that turns a simplicial set into
a topological space, and another functor that turns a space into a
simplicial set called its "singular simplicial set". I also showed how
to turn a simplicial set into a simplicial abelian group, and how
to turn one of *those* into a chain cotwf_ascii/week118000064400020410000157000000531751015243332200141330ustar00baezhttp00004600000001Also available at http://math.ucr.edu/home/baez/week118.html
March 14, 1998
This Week's Finds in Mathematical Physics  Week 118
John Baez
Like many people of a certain age, as a youth my interest in mathematics
and physics was fed by the Scientific American, especially Martin
Gardner's wonderful column. Since then the magazine seems to have gone
downhill. For me, the last straw was a silly article on the "death of
proof" in mathematics, written by someone wholly unfamiliar with the
subject. The author of that article later wrote a book proclaiming the
"end of science", and went on to manage a successful chain of funeral
homes.
Recently, however, I was pleased to find a terse rebuttal of this
findesiecle pessimism in an article appearing in  none other than
Scientific American!
1) Michael J. Duff, The theory formerly known as strings, Scientific
American 278 (February 1998), 6469.
The article begins:
At a time when certain pundits are predicting the End of Science
on the grounds that all the important discoveries have already
been made, it is worth emphasizing that the two main pillars of
20thcentury physics, quantum mechanics and Einstein's general
theory of relativity, are mutually incompatible.
To declare the end of science at this point, or even of particle physics
(the two are not the same!) would thus be ridiculously premature. It's
true that the quest for a unified theory of all the forces and particles
in nature is experiencing difficulties. On the one hand, particle
accelerators have become very expensive. On the other hand, it's
truly difficult to envision a consistent and elegant formalism
subsuming both general relativity and the Standard Model of particle
physics  much less one that makes new testable predictions. But hey,
the course of true love never did run smooth.
Duff's own vision certainly has its charms. He has long been advocating
the generalization of string theory to a theory of higherdimensional
"membranes". Nowadays people call these "pbranes" to keep track of the
dimension of the membrane: a 0brane is a point particle, a 1brane is a
string, a 2brane is a 2dimensional surface, and so on.
For a long time, higherdimensional membrane theories were unpopular,
even among string theorists, because the special tricks that eliminate
infinities in string theory don't seem to work for higherdimensional
membranes. But lately membranes are all the rage: it seems they show up
in string theory whether or not you put them in from the start! In
fact, they seem to be the key to showing that the 5 different
supersymmetric string theories are really aspects of a single deeper
theory  sometimes called "Mtheory".
Now, I don't really understand this stuff at all, but I've been trying
to learn about it lately, so I'll say a bit anyway, in hopes that some
real experts will correct my mistakes. Much of what I'll say comes from
the following nice review article:
2) M. J. Duff, Supermembranes, preprint available as hepth/9611203
and also the bible of string theory:
3) Michael B. Green, John H. Schwarz, and Edward Witten, Superstring
Theory, two volumes, Cambridge U. Press, Cambridge, 1987.
Let's start with superstring theory. Here the "super" refers to the
fact that instead of just strings whose vibrational modes correspond to
bosonic particles, we have strings with extra degrees of freedom
corresponding to fermionic particles. We can actually think of the
superstring as a string wiggling around in a "superspace": a kind of
space with extra "fermionic" dimensions in addition to the usual
"bosonic" ones. These extra dimensions are described by coordinates
that anticommute with each other, and commute with the usual bosonic
coordinates (which of course commute with each other). This amounts to
taking the boson/fermion distinction so seriously that we build it into
our description of spacetime from the very start! For more details on
the mathematics of superspace, try:
4) Bryce DeWitt, Supermanifolds, Cambridge U. Press, Cambridge,
2nd edition, 1992.
More deeply, "super" refers to "supersymmetry", a special kind of
symmetry transformation that mixes the bosonic and fermionic
coordinates. We speak of "N = 1 supersymmetry" if there is one
fermionic coordinate for each bosonic coordinate, "N = 2 supersymmetry"
if there are two, and so on.
Like all nice physical theories, we can in principle derive everything
about our theory of superstrings once we know the formula for the
*action*. For bosonic strings, the action is very simple. As time
passes, a string traces out a 2dimensional surface in spacetime called
the "string worldsheet". The action is just the *area* of this worldsheet.
For superstring theory, we thus want a formula for the "superarea" of a
surface in superspace. And we need this to be invariant under
supersymmetry transformations. Suprisingly, this is only possible if
spacetime has dimension 3, 4, 6, or 10. More precisely, these are the
dimensions where N = 1 supersymmetric string theory makes sense as a
*classical* theory.
Note: these dimensions are just 2 more than the dimensions of the four
normed division algebras: the reals, complexes, quaternions and
octonions! This is no coincidence. Robert Helling recently posted a
nice article about this on sci.physics.resarch, which I have appended to
"week104". The basic idea is that we can describe the vibrations of a
string in ndimensional spacetime by a field on the string worldsheet
with n2 components corresponding to the n2 directions transverse to
the worldsheet. To get an action that's invariant under supersymmetry,
we need some magical cancellations to occur. It only works when we can
think of this field as taking values in one of the normed division
algebras!
This is one of the curious things about superstring theory: the basic
idea is simple, but when you try to get it to work, you run into lots of
obstacles which only disappear in certain special circumstances  thanks
to a mysterious conspiracy of beautiful mathematical facts. These
"conspiracies" are probably just indications that we don't understand
the theory as deeply as we should. Right now I'm most interested in the
algebraic aspects of superstring theory  and especially its
relationships to "exceptional algebraic structures" like the octonions,
the Lie group E8, and so on. As I learn superstring theory, I like
keeping track of the various ways these structures show up, like
remembering the clues in a mystery novel.
Interestingly, the *quantum* version of superstring theory is more
delicate than the classical version. When I last checked, it only makes
sense in dimension 10. Thus there's something inherently octonionic
about it! For more on this angle, see:
5) E. Corrigan and T. J. Hollowood, The exceptional Jordan algebra and
the superstring, Commun. Math. Phys., 122 (1989), 393.
6) E. Corrigan and T. J. Hollowood, A string construction of a
commutative nonassociative algebra related to the exceptional Jordan
algebra, Phys. Lett. B203 (1988), 47.
and some more references I'll give later.
There are actually 5 variants of superstring theory in dimension 10, as
I explained in "week72":
1. type I superstrings  these are open strings, not closed loops.
2. type IIA superstrings  closed strings where the left and
rightmoving fermionic modes have opposite chiralities.
3. type IIB superstrings  closed strings where the left and
rightmoving fermionic modes have the same chirality.
4. E8 heterotic superstrings  closed strings where the leftmoving
modes are purely bosonic, with symmetry group E8 x E8.
5. Spin(32)/Z_2 heterotic superstrings  closed strings where the
leftmoving modes are purely bosonic, with symmetry group Spin(32)/Z_2.
To get 4dimensional physics out of any of these, we need to think of our
10dimensional spacetime as a bundle with a little 6dimensional
"CalabiYau manifold" sitting over each point of good old 4dimensional
spacetime. But there's another step that's very useful when trying to
understand the implications of superstring theory for ordinary particle
physics. This is to look at the lowenergy limit. In this limit, only
the lowestenergy vibrational modes of the string contribute, each mode
acting like a different kind of massless particle. Thus in this limit
superstring theory acts like an ordinary quantum field theory.
What field theory do we get? This is a very important question. The
field theory looks simplest in 10dimensional Minkowski spacetime; it
gets more complicated when we curl up 6 of the dimensions and think of
it as a 4dimensional field theory, so let's just talk about the simple
situation.
No matter what superstring theory we start with, the lowenergy limit
looks like some form of "supergravity coupled to superYangMills
fields". What's this? Well, supergravity is basically what we get when
we generalize Einstein's equations for general relativity to superspace.
Similarly, superYangMills theory is the supersymmetric version of the
YangMills equations  which are important in particle physics because
they describe all the forces *except* gravity. So superstring theory
has in it the seeds of general relativity and also the other forces of
nature  or at least their supersymmetric analogues.
Like superstring theory, superYangMills theory only works in spacetime
dimensions 3, 4, 6, and 10. (See "week93" for more on this.) Different
forms of supergravity make sense in different dimensions, as explained
in:
7) Y. Tanii, Introduction to supergravities in diverse dimensions, preprint
available as hepth/9802138.
In particular highest dimension in which supergravity makes sense is 11
dimensions (where one only has N = 1 supergravity). Note that this is
one more than the favored dimension of superstring theory! This puzzled
people for a long time. Now it seems that Mtheory is beginning to
resolve these puzzles. Another interesting discovery is that
11dimensional supergravity is related to the exceptional Lie group E8:
8) Stephan Melosch and Hermann Nicolai, New canonical variables for
d = 11 supergravity, preprint available as hepth/9709277.
But I'm getting ahead of myself here! Right now I'm talking about the
lowenergy limit of 10dimensional superstring theory. I said it
amounts to "supergravity coupled to superYangMills fields", and now
I'm attempting to flesh that out a bit. So: starting from N = 1
supergravity in 11 dimensions we can get a theory of supergravity in 10
dimensions simply by requiring that all the fields be constant in one
direction  a trick called "dimensional reduction". This is called
"type IIA supergravity", because it appears in the lowenergy limit of
type IIA superstrings. There are also two other forms of supergravity
in 10 dimensions: "type IIB supergravity", which appears in the
lowenergy limit of type IIB superstrings, and a third form which
appears in the lowenergy limit of the type I and heterotic
superstrings. These other two forms of supergravity are chiral  that
is, they have a builtin "handedness".
Now let's turn to higherdimensional supersymmetric membranes, or
"supermembranes". Duff summarizes this subject in a chart he calls the
"brane scan". This chart lists the known *classical* theories of
supersymetric pbranes. Of course, a pbrane traces out a
(p+1)dimensional surface as time passes, so from a spacetime point of
view it's p+1 which is more interesting. But anyway, here's Duff's
chart of which supersymmetric pbrane theories are possible in which
dimensions d of spacetime:
d
11 X X ?
10 X X X X X X X X X X
9 X X
8 X
7 X X
6 X X X X X X
5 X X
4 X X X X
3 X X X
2 X
1
0 1 2 3 4 5 6 7 8 9 10 p
We immediately notice some patterns. First, we see horizontal stripes
in dimensions 3, 4, 6, and 10: all the conceivable pbrane theories
exist in these dimensions. I don't know why this is true, but it must be
related to the fact that superstring and superYangMills theories make
sense in these dimensions. Second, there are four special pbrane
theories:
the 2brane in dimension 4
the 3brane in dimension 6
the 5brane in dimension 10
the 2brane in dimension 11
which are related to the real numbers, the complex numbers, the
quaternions and the octonions, respectively. Duff refers us to the
following papers for more information on this:
9) G. Sierra, An application of the theories of Jordan algebras and
Freudenthal triple systems to particles and strings, Class. Quant.
Grav. 4 (1987), 227.
10) J. M. Evans, Supersymmetric YangMills theories and division
algebras, Nucl. Phys. B298 (1988), 92108.
From these four "fundamental" theories of pbranes in d dimensions we
can get theories of (pk)branes in dk dimensions by dimensional
reduction of both the spacetime and the pbrane. Thus we see diagonal
lines slanting down and to the left starting from these "fundamental"
theories. Note that these diagonal lines include the superstring
theories in dimensions 3, 4, 6, and 10!
I'll wrap up by saying a bit about how Mtheory, superstrings and
supergravity fit together. I've already said that: 1) type IIA
supergravity in 10 dimensions is the dimensional reduction of
11dimensional supergravity; and 2) the type IIA superstring has type
IIA supergravity coupled to superYangMills fields as a lowenergy
limit. This suggests the presence of a theory in 11 dimensions that
fills in the question mark below:
lowenergy limit
? > 11dimensional
 supergravity
 
dimensional dimensional
reduction reduction
 
V lowenergy limit V
type IIA > type IIA supergravity in
superstrings 10 dimensions
This conjectured theory is called "Mtheory". The actual details of
this theory are still rather mysterious, but not surprisingly, it's
related to the theory of supersymmetric 2branes in 11 dimensions 
since upon dimensional reduction these give superstrings in 10
dimensions. More surprisingly, it's *also* related to the theory of
*5branes* in 11 dimensions. The reason is that supergravity in 11
dimensions admits "soliton" solutions  solutions that hold their shape
and don't disperse  which are shaped like 5branes. These are now
believed to be yet another shadow of Mtheory.
While the picture I'm sketching may seem baroque, it's really just a
small part of a much more elaborate picture that relates all 5
superstring theories to Mtheory. But I think I'll stop here! Maybe
later when I know more I can fill in some more details. By the way, I
thank Dan Piponi for pointing out that Scientific American article.
For more on this business, check out the following review articles:
11) W. Lerche, Recent developments in string theory, preprint available
as hepth/9710246.
12) John Schwarz, The status of string theory, preprint available as
hepth/9711029.
13) M. J. Duff, Mtheory (the theory formerly known as strings),
preprint available as hepth/9608117.
The first one is especially nice if you're interested in a nontechnical
survey; the other two are more detailed.
Okay. Now, back to my tour of homotopy theory! I had promised to talk
about classifying spaces of groups and monoids, but this post is getting
pretty long, so I'll only talk about something else I promised: the
foundations of homological algebra. So, remember:
As soon as we can squeeze a simplicial set out of something, we have all
sorts of methods for studying it. We can turn the simplicial set into a
space and then use all the methods of topology to study this space. Or
we can turn it into a chain complex and apply homology theory. So it's
very important to have tricks for turning all sorts of gadgets into
simplicial sets: groups, rings, algebras, Lie algebras, you name it!
And here's how....
N. Simplicial objects from adjunctions. Remember from section D of
"week115" that a "simplicial object" in some category is a contravariant
functor from Delta to that category. In what follows, I'll take Delta
to be the version of the category of simplices that contains the empty
simplex. Topologists don't usually do this, so what I'm calling a
"simplicial object", they would call an "augmented simplicial object".
Oh well.
Concretely, a simplicial object in a category amounts to a bunch of
objects x_0, x_1, x_2,... together with morphisms like this:
i_1>
i_0> i_0>
x_0 <d_0 x_1 <d_0 x_2 <d_0 x_3 ...
<d_1 <d_1
<d_2
The morphisms d_j are called "face maps" and the morphisms i_j are
called "degeneracies". They are required to satisfy some equations
which I won't bother writing down here, since you can figure them out
yourself if you read section B of "week114".
Now, suppose we have an adjunction, that is, a pair of adjoint functors:
L>
C D
<R
This means we have natural transformations
e: LR => 1_D
i: 1_C => RL
satisfying a couple of equations, which I again won't write down, since
I explained them in "week79" and "week83".
Then an object d in the category D automatically gives a simplicial
object as follows:
LRL.i.R>
L.i.R> L.i.RLR>
d <e LR(d) <e.LR LRLR(d) <e.LRLR LRLRLR(d) ...
<LR.e <LR.e.LR
<LRLR.e
where . denotes horizontal composition of functors and natural
transformations.
For example, if Gp is the category of abelian groups, we have an
adjunction
L>
Set AbGp
<R
where L assigns to each set the free group on that set, and R assigns to
each group its underlying set. Thus given a group, the above trick
gives us a simplicial object in Gp  or in other words, a simplicial
group. This has an underlying simplicial set, and from this we can cook
up a chain complex as in section H of "week116". This lets us study
groups using homology theory! One can define the homology (and
cohomology) of lots other algebraic gadgets in exactly the same way.
Note: I didn't explain why the equations in the definition of adjoint
functors  which I didn't write down  imply the equations in the
definition of a simplicial object  which I also didn't write down!
The point is, there's a more conceptual approach to understanding why
this stuff works. Remember from section K of last week that Delta is
"the free monoidal category on a monoid object". This implies that
whenever we have a monoid object in a monoidal category M, we get a
monoidal functor
F: Delta > M.
This gives a functor
G: Delta^{op} > M^{op}
So: a monoid object in M gives a simplicial object in M^{op}.
Actually, if M is a monoidal category, M^{op} becomes one too, with the
same tensor product and unit object. So it's also true that a monoid
object in M^{op} gives a simplicial object in M!
Another name for a monoid object in M^{op} is a "comonoid object in M".
Remember, M^{op} is just like M but with all the arrows turned around.
So if we've got a monoid object in M^{op}, it gives us a similar gadget
in M, but with all the arrows turned around. More precisely, a comonoid
object in M is an object, say m, with "coproduct"
c: m > m x m
and "counit"
e: m > 1
morphisms, satisfying "coassociativity" and the left and right "counit
laws". You get these laws by taking associativity and the left/right
unit laws, writing them out as commutative diagrams, and turning all the
arrows around.
So: a comonoid object in a monoidal category M gives a simplicial object
in M. Now let's see how this is related to adjoint functors. Suppose
we have an adjunction, so we have some functors
L>
C D
<R
and natural transformations
e: LR => 1_D
i: 1_C => RL
satisfying the same equations I didn't write before.
Let hom(C,C) be the category whose objects are functors from C to
itself and whose morphisms are natural transformations between such
functors. This is a monoidal category, since we can compose functors
from C to itself. In "week92" I showed that hom(C,C) has a monoid
object in it, namely RL. The product for this monoid object is
R.e.L: RLRL => RL
and the unit is
i: 1_C => RL
Folks often call this sort of thing a "monad".
Similarly, hom(D,D) is a monoidal category containing a comonoid object,
namely LR. The coproduct for this comonoid object is
L.i.R: LR => LRLR
and the counit is
e: LR => 1_D
People call this thing a "comonad". But what matters here is that we've
seen this comonoid object automatically gives us a simplicial object in
hom(D,D)! If we pick any object d of D, we get a functor
hom(D,D) > D
by taking
hom(D,D) x D > D
and plugging in d in the second argument. This functor lets us push
our simplicial object in hom(D,D) forwards to a simplicial object in D.
Voila!

Previous issues of "This Week's Finds" and other expository articles on
mathematics and physics, as well as some of my research papers, can be
obtained at
http://math.ucr.edu/home/baez/
For a table of contents of all the issues of This Week's Finds, try
http://math.ucr.edu/home/baez/twf.html
A simple jumpingoff point to the old issues is available at
http://math.ucr.edu/home/baez/twfshort.html
If you just want the latest issue, go to
http://math.ucr.edu/home/baez/this.week.html
X
5 X X
4 X X X X
3 X X X
2 X
1
0 1 2 3 4 5 6 7 8 9 10 p
We immediately notice some patterns. First, we see horizontal stripes
in dimensions 3, 4, 6, and 10: all the conceivable pbrane theories
exist in these dimensions. I don't know why this is true, but it must be
related to the fact that superstring and sutwf_ascii/week119000064400020410000157000000505741015243333400141370ustar00baezhttp00004600000001Also available at http://math.ucr.edu/home/baez/week119.html
April 13, 1998
This Week's Finds in Mathematical Physics  Week 119
John Baez
I've been slacking off on This Week's Finds lately because I was
busy getting stuff done at Riverside so that I could visit the
Center for Gravitational Physics and Geometry here at Penn State
with a fairly clean slate. Indeed, sometimes my whole life seems
like an endless series of distractions designed to prevent me from
writing This Week's Finds. However, now I'm here and ready to have
some fun....
Recently I've been trying to learn about grand unified theories, or
"GUTs". These were popular in the late 1970s and early 1980s, when
the Standard Model of particle interactions had fully come into its
own and people were looking around for a better theory that would
unify all the forces and particles present in that model  in short,
everything except gravity.
The Standard Model works well but it's fairly baroque, so it's natural
to hope for some more elegant theory underlying it. Remember how it
goes:

GAUGE BOSONS
ELECTROMAGNETIC FORCE WEAK FORCE STRONG FORCE
photon W+ 8 gluons
W
Z

FERMIONS
LEPTONS QUARKS
electron electron neutrino down quark up quark
muon muon neutrino strange quark charm quark
tauon tauon neutrino bottom quark top quark

HIGGS BOSON (not yet seen)

The strong, electromagnetic and weak forces are all described by
YangMills fields, with the gauge group SU(3) x SU(2) x U(1). In what
follows I'll assume you know the rudiments of gauge theory, or at
least that you can fake it.
SU(3) is the gauge group of the strong force, and its 8 generators
correspond to the gluons. SU(2) x U(1) is the gauge group of the
electroweak force, which unifies electromagnetism and the weak force.
It's *not* true that the generators of SU(2) corresponds to the W+, W
and Z while the generator of U(1) corresponds to the photon. Instead,
the photon corresponds to the generator of a sneakier U(1) subgroup
sitting slantwise inside SU(2) x U(1); the basic formula to remember
here is:
Q = I_3 + Y/2
where Q is ordinary electric charge, I_3 is the 3rd component of
"weak isospin", i.e. the generator of SU(2) corresponding to the
matrix
(1/2 0)
(0 1/2)
and Y, "hypercharge", is the generator of the U(1) factor. The role
of the Higgs particle is to spontaneously break the SU(2) x U(1)
symmetry, and also to give all the massive particles their mass.
However, I don't want to talk about that here; I want to focus on the
fermions and how they form representations of the gauge group SU(3) x
SU(2) x U(1), because I want to talk about how grand unified theories
attempt to simplify this picture  at the expense of postulating more
Higgs bosons.
The fermions come in 3 generations, as indicated in the chart above.
I want to explain how the fermions in a given generation are grouped
into irreducible representations of SU(3) x SU(2) x U(1). All the
generations work the same way, so I'll just talk about the first
generation. Also, every fermion has a corresponding antiparticle, but
this just transforms according to the dual representation, so I will
ignore the antiparticles here.
Before I tell you how it works, I should remind you that all the
fermions are, in addition to being representations of SU(3) x SU(2) x
U(1), also spin1/2 particles. The massive fermions  the quarks and
the electron, muon and tauon  are Dirac spinors, meaning that they
can spin either way along any axis. The massless fermions  the
neutrinos  are Weyl spinors, meaning that they always spin
counterclockwise along their axis of motion. This makes sense
because, being massless, they move at the speed of light, so everyone
can agree on their axis of motion! So the massive fermions have two
helicity states, which we'll refer to as "lefthanded" and
"righthanded", while the neutrinos only come in a "lefthanded" form.
(Here I am discussing the Standard Model in its classic form. I'm
ignoring any modifications needed to deal with a possible nonzero
neutrino mass. For more on Standard Model, neutrino mass and
different kinds of spinors, see "week93".)
Okay. The Standard Model lumps the lefthanded neutrino and the
lefthanded electron into a single irreducible representation of
SU(3) x SU(2) x U(1):
(nu_L, e_L) (1,2,1)
This 2dimensional representation is called (1,2,1), meaning
that it's the tensor product of the 1dimensional trivial rep
of SU(3), the 2dimensional fundamental rep of SU(2), and the
1dimensional rep of U(1) with hypercharge 1.
Similarly, the lefthanded up and down quarks fit together as:
(u_L, u_L, u_L, d_L, d_L, d_L) (3,2,1/3)
Here I'm writing both quarks 3 times since they also come in 3 color
states. In other words, this 6dimensional representation is the
tensor product of the 3dimensional fundamental rep of SU(3), the
2dimensional fundamental rep of SU(2), and the 1dimensional rep of
U(1) with hypercharge 1/3. That's why we call this rep (3,2,1/3).
(If you are familiar with the irreducible representations of U(1) you
will know that they are usually parametrized by integers. Here we are
using integers divided by 3. The reason is that people defined the
charge of the electron to be 1 before quarks were discovered, at
which point it turned out that the smallest unit of charge was 1/3 as
big as had been previously believed.)
The righthanded electron stands alone in a 1dimensional rep, since
there is no righthanded neutrino:
e_R (1,1,2)
Similarly, the righthanded up quark stands alone in a 3dimensional
rep, as does the righthanded down quark:
(u_R, u_R, u_R) (3,1,4/3)
(d_R, d_R, d_R) (3,1,2/3)
That's it. If you want to study this stuff, try using the formula
Q = I_3 + Y/2
to figure out the charges of all these particles. For example, since
the righthanded electron transforms in the trivial rep of SU(2), it
has I_3 = 0, and if you look up there you'll see that it has Y = 2.
This means that its electric charge is Q = 1, as we already knew.
Anyway, we obviously have a bit of a mess on our hands! The Standard
Model is full of tantalizing patterns, but annoyingly complicated.
The idea of grand unified theories is to find a pattern lurking in all
this data by fitting the group SU(3) x SU(2) x U(1) into a larger
group. The smallestdimensional "simple" Lie group that works is
SU(5). Here "simple" is a technical term that eliminates, for
example, groups that are products of other groups  these aren't very
"unified". Georgi and Glashow came up with their "minimal" SU(5)
grand unified theory in 1975. The idea is to stick SU(3) x SU(2) into
SU(5) in the obvious diagonal way, leaving just enough room to cram in
the U(1) if you are clever.
Now if you add up the dimensions of all the representations above you
get 2 + 6 + 1 + 3 + 3 = 15. This means we need to find a
15dimensional representation of SU(5) to fit all these particles.
There are various choices, but only one that really works when you
take all the physics into account. For a nice simple account of the
detective work needed to figure this out, see:
1) Edward Witten, Grand unification with and without supersymmetry,
Introduction to supersymmetry in particle and nuclear physics, edited by
O. Castanos, A. Frank, L. Urrutia, Plenum Press, 1984.
I'll just give the answer. First we take the 5dimensional fundamental
representation of SU(5) and pack fermions in as follows:
(d_R, d_R, d_R, e+_R, nubar_R) 5 = (3,1,2/3) + (1,2,1)
Here e+_R is the righthanded positron and nubar_R is the righthanded
antineutrino  curiously, we need to pack some antiparticles in with
particles to get things to work out right. Note that the first 3
particles in the above list, the 3 states of the righthanded down
quark, transform according to the fundamental rep of SU(3) and the
trivial rep of SU(2), while the remaining two transform according to
the trivial rep of SU(3) and the fundamental rep of SU(2). That's how
it has to be, given how we stuffed SU(3) x SU(2) into SU(5).
Note also that the charges of the 5 particles on this list add up to
zero. That's also how it has to be, since the generators of SU(5) are
traceless. Note that the down quark must have charge 1/3 for this to
work! In a sense, the SU(5) model says that quarks *must* have
charges in units of 1/3, because they come in 3 different colors!
This is pretty cool.
Then we take the 10dimensional representation of SU(5) given by
the 2nd exterior power of the fundamental representation  i.e.,
antisymmetric 5x5 matrices  and pack the rest of the fermions in
like this:
( 0 ubar_L ubar_L u_L d_L ) 10 = (3,2,1/3) +
( ubar_L 0 ubar_L u_L d_L ) (1,1,2) +
( ubar_L ubar_L 0 u_L d_L ) (3,1,4/3)
( u_L u_L u_L 0 e+_L )
( d_L u_L d_L e+_L 0 )
Here the ubar is the antiparticle of the up quark  again we've
needed to use some antiparticles. However, you can easily check
that these two representations of SU(5) together with their duals
account for all the fermions and their antiparticles.
The SU(5) theory has lots of nice features. As I already noted, it
explains why the up and down quarks have charges 2/3 and 1/3,
respectively. It also gives a pretty good prediction of something
called the Weinberg angle, which is related to the ratio of the masses
of the W and Z bosons. It also makes testable new predictions! Most
notably, since it allows quarks to turn into leptons, it predicts that
protons can decay  with a halflife of somewhere around 10^{29} or
10^{30} years. So people set off to look for proton decay....
However, even when the SU(5) model was first proposed, it was regarded
as slightly inelegant, because it didn't unify all the fermions of a
given generation in a *single* irreducible representation (together
with its dual, for antiparticles). This is one reason why people
began exploring still larger gauge groups. In 1975 Georgi, and
independently Fritzsch and Minkowski, proposed a model with gauge
group SO(10). You can stuff SU(5) into SO(10) as a subgroup in such a
way that the 5 and 10dimensional representations of SU(5) listed
above both fit into a single 16dimensional rep of SO(10), namely the
chiral spinor rep. Yes, 16, not 15  that wasn't a typo! The SO(10)
theory predicts that in addition to the 15 states listed above there
is a 16th, corresponding to a righthanded neutrino! I'm not sure yet
how the recent experiments indicating a nonzero neutrino mass fit into
this business, but it's interesting.
Somewhere around this time, people noticed something interesting about
these groups we've been playing with. They all fit into the "E series"!
I don't have the energy to explain Dynkin diagrams and the ABCDEFG
classification of simple Lie groups here, but luckily I've already
done that, so you can just look at "week62"  "week65" to learn about
that. The point is, there is an infinite series of simple Lie groups
associated to rotations in real vector spaces  the SO(n) groups, also
called the B and D series. There is an infinite series of them
associated to rotations in complex vector spaces  the SU(n) groups,
also called the A series. And there is infintie series of them
associated to rotations in quaternionic vector spaces  the Sp(n)
groups, also called the C series. And there is a ragged band of 5
exceptions which are related to the octonions, called G2, F4, E6, E7,
and E8. I'm sort of fascinated by these  see "week90", "week91", and
"week106" for more  so I was extremely delighted to find that the E
series plays a special role in grand unified theories.
Now, people usually only talk about E6, E7, and E8, but one can work
backwards using Dynkin diagrams to define E5, E4, E3, E2, and E1.
Let's do it! Thanks go to Allan Adler and Robin Chapman for helping
me understand how this works....
E8 is a big fat Lie group whose Dynkin diagram looks like this:
o

ooooooo
If we remove the rightmost root, we obtain the Dynkin diagram of
a subgroup called E7:
o

oooooo
If we again remove the rightmost root, we obtain the Dynkin diagram
of a subgroup of E7, namely E6:
o

ooooo
This was popular as a gauge group for grand unified models, and
the reason why becomes clear if we again remove the rightmost root,
obtaining the Dynkin diagram of a subgroup we could call E5:
o

oooo
But this is really just good old SO(10), which we were just
discussing! And if we yet again remove the rightmost root, we
get the Dynkin diagram of a subgroup we could call E4:
o

ooo
This is just SU(5)! Let's again remove the rightmost root,
obtaining the Dynkin diagram for E3. Well, it may not be clear
what counts as the rightmost root, but here's what I want to
get when I remove it:
o
oo
This is just SU(3) x SU(2), sitting inside SU(5) in the way we just
discussed! So for some mysterious reason, the Standard Model and
grand unified theories seem to be related to the E series!
We could march on and define E2:
o
o
which is just SU(2) x SU(2), and E1:
o
which is just SU(2)... but I'm not sure what's so great about these
groups.
By the way, you might wonder what's the real reason for removing the
roots in the order I did  apart from getting the answers I wanted to
get  and the answer is, I don't really know! If anyone knows, please
tell me. This could be an important clue.
Now, this stuff about grand unified theories and the E series is
one of the reasons why people like string theory, because heterotic
string theory is closely related to E8 (see "week95"). However, I
must now tell you the *bad* news about grand unified theories. And
it is *very* bad.
The bad news is that those people who went off to detect proton
decay never found it! It became clear in the mid1980s that the
proton lifetime was at least 10^{32} years or so, much larger than
what the SU(5) theory most naturally predicts. Of course, if one is
desperate to save a beautiful theory from an ugly fact, one can resort
to desperate measures. For example, one can get the SU(5) model to
predict very slow proton decay by making the grand unification mass
scale large. Unfortunately, then the coupling constants of the strong
and electroweak forces don't match at the grand unification mass
scale. This became painfully clear as better measurements of the
strong coupling constant came in.
Theoretical particle physics never really recovered from this crushing
blow. In a sense, particle physics gradually retreated from the goal
of making testable predictions, drifting into the wonderland of pure
mathematics... first supersymmetry, then supergravity, and then
superstrings... ever more elegant theories, but never yet a verified
experimental prediction. Perhaps we should be doing something
different, something better? Easy to say, hard to do! If we see a
superpartner at CERN, a lot of this "superthinking" will be vindicated
 so I guess most particle physicists are crossing their fingers and
praying for this to happen.
The following textbook on grand unified theories is very nice,
especially since it begins with a review of the Standard Model:
2) Graham G. Ross, Grand Unified Theories, BenjaminCummings, 1984.
This one is a bit more idiosyncratic, but also good  Mohapatra
is especially interested in theories where CP violation arises via
spontaneous symmetry breaking:
3) Ranindra N. Mohapatra, Unification and Supersymmetry: The Frontiers
of QuarkLepton Physics, SpringerVerlag, 1992.
I also found the following articles interesting:
4) D. V. Nanopoulos, Tales of the GUT age, in Grand Unified Theories
and Related Topics, proceedings of the 4th Kyoto Summer Institute,
World Scientific, Singapore, 1981.
5) P. Ramond, Grand unification, in Grand Unified Theories and Related
Topics, proceedings of the 4th Kyoto Summer Institute, World
Scientific, Singapore, 1981.
Okay, now for some homotopy theory! I don't think I'm ever gonna get
to the really cool stuff... in my attempt to explain everything
systematically, I'm getting worn out doing the preliminaries. Oh well,
on with it... now it's time to start talking about loop spaces! These
are really important, because they tie everything together. However,
it takes a while to deeply understand their importance.
O. The loop space of a topological space. Suppose we have a "pointed
space" X, that is, a topological space with a distinguished point
called the "basepoint". Then we can form the space LX of all "based
loops" in X  loops that start and end at the basepoint.
One reason why LX is so nice is that its homotopy groups are the same
as those of X, but shifted:
pi_i(LX) = pi_{i+1}(X)
Another reason LX is nice is that it's almost a topological group,
since one can compose based loops, and every loop has an "inverse".
However, one must be careful here! Unless one takes special care,
composition will only be associative up to homotopy, and the "inverse"
of a loop will only be the inverse up to homotopy.
Actually we can make composition strictly associative if we work with
"Moore paths". A Moore path in X is a continuous map
f: [0,T] > X
where T is an arbitrary nonnegative real number. Given a Moore path
f as above and another Moore path
g: [0,S] > X
which starts where f ends, we can compose them in an obvious way to
get a Moore path
fg: [0,T+S] > X
Note that this operation is associative "on the nose", not just up to
homotopy. If we define LX using Moore paths that start and end at the
basepoint, we can easily make LX into a topological monoid  that is,
a topological space with a continuous associative product and a unit
element. (If you've read section L, you'll know this is just a monoid
object in Top!) In particular, the unit element of LX is the path
i: [0,0] > X that just sits there at the basepoint of X.
LX is not a topological group, because even Moore paths don't have
strict inverses. But LX is *close* to being a group. We can make
this fact precise in various ways, some more detailed than others.
I'm pretty sure one way to say it is this: the natural map from LX to
its "group completion" is a homotopy equivalence.
P. The group completion of a topological monoid. Let TopMon be the
category of topological monoids and let TopGp be the category of
topological groups. There is a forgetful functor
F: TopGp > TopMon
and this has a left adjoint
G: TopMon > TopGp
which takes a topological monoid and converts it into a topological
group by throwing in formal inverses of all the elements and giving
the resulting group a nice topology. This functor G is called "group
completion" and was first discussed by Quillen (in the simplicial
context, in an unpublished paper), and independently by Barratt and
Priddy:
6) M. G. Barratt and S. Priddy, On the homology of nonconnected
monoids and their associated groups, Comm. Math. Helv. 47 (1972),
114.
For any topological monoid M, there is a natural map from M to
F(G(M)), thanks to the miracle of adjoint functors. This is the
natural map I'm talking about in the previous section!

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posed, it was regarded
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Subject: This Week's Finds in Mathematical Physics (Week 12)
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Week 12
I had a lot of fun at the "Quantum Topology" conference at Kansas State
University, in Manhattan (yes, that's right, Manhattan, Kansas, the
socalled "Little Apple"), and then spent a week recovering. Now I'm
back, ready for the next quarter...
The most novel idea I ran into at the conference was due to Oleg Viro,
who, ironically, is right here at U. C. Riverside. He spoke on work
with Turaev on generalizing the Alexander module (a classical knot
invariant) to get a similar sort of module from any 3dimensional
topological quantum field theory. A "topological quantum field theory,"
or TQFT for short, is (in the language of physics) basically just a
generally covariant quantum field theory, one that thinks all coordinate
systems are equally good, just as general relativity is a generally
covariant classical field theory. For a more precise definition of
TQFTs (which even mathematicians who know nothing of physics can
probably follow), see my article "symmetries," available by anonymous
ftp as described at the end of this article. In any event, I don't
think Viro's work exists in printed form yet; I'll let you all know when
something appears.
The most lively talk was one by Louis Crane and David Yetter, the
organizers of the conference. As I noted a while back, they claimed to
have constructed a FOURdimensional TQFT based on some ideas of Ooguri,
who was working on 4dimensional quantum gravity. This would be
very exciting as long as it isn't "trivial" in some sense, because all
the TQFTs developed so far only work in 3dimensional spacetime.
A rigorous 4dimensional TQFT might bring us within striking distance of
a theory of quantum gravity  this is certainly Crane's goal. Ocneanu,
however, had fired off a note claiming to prove that the CraneYetter
TQFT was trivial, in the sense that the partition function of the field
theory for any compact oriented 4manifold equalled 1! In a TQFT, the
partition function of the field theory on a compact manifold is a
invariant of the manifold, and if it equalled 1 for all manifolds, it
would be an extremely dull invariant, hence a rather trivial TQFT.
So, on popular demand, Crane and Yetter had a special talk at 8 pm in
which they described their TQFT and presented results of
calculations that showed the invariant did NOT equal 1 for all compact
oriented 4manifolds. So far they have only calculated it in some
special cases: S^4, S^3 x S^1, and S^2 x S^2. Amusingly, Yetter ran
through the calculation in the simplest case, S^4, in which the
invariant *does* happen to equal 1. But he persuaded most of us (me at
least) that the invariant really is an invariant and that he can
calculate it. I say "persuade" rather than prove because he didn't
present a proof that it's an invariant; the current proof is grungy and
computational, but Viro and Kauffman (who were there) pointed out some
ways that it could be made more slick, so we should see a comprehensible
proof one of these days. However, it's still up in the air whether this
invariant might be "trivial" in some more sophisticated sense, e.g.,
maybe it's a function of wellknown invariants like the signature and
Euler number. Unfortunately, Ocneanu decided at the last minute not to
attend. Nor did Broda (inventor of another 4manifold invariant that
Ruberman seems to have shown "trivial" in previous This Week's Finds)
show up, though he had been going to.
On a slightly more technical note, Crane and Yetter's TQFT depends on
chopping up the 4manifold into simplices (roughly speaking,
4dimensional versions of tetrahedra). Their calculation involves
drawing projections of these beasts into the plane and applying various
rules; it was quite fun to watch Yetter do it on the blackboard. Turaev
and Viro had constructed such a "simplicial" TQFT in 3 dimensions, and
Ooguri had been working on simplicial quantum gravity. As I note below,
Lee Smolin has a new scheme for doing 4dimensional quantum gravity
using simplices. During the conference he was busy trying to figure out
the relation of his ideas to Crane and Yetter's.
Also while at the conference, I found a terrible error in "week10" in my
description of
Vassiliev invariants contain more information than all knot
polynomials, by Sergey Piunikhin, preprint.
I had said that Piunikhin had discovered a Vassiliev invariant that
could distinguish knots from their orientationreversed versions. No!
The problem was a very hasty reading on my part, together with the
following typo in the paper, that tricked my eyes:
Above constructed Vassiliev knot invariant w of order six does knot
detect orientation of knots.
Ugh! Also, people at the conference said that Piunikhin's claim in this
paper to have found a Vassiliev invariant not coming from quantum group
knot polynomials is misleading. I don't understand that yet.
Here are some papers that have recently shown up...
1) Canonical quantum gravity, by Karel Kuchar, preprint in LaTeX form,
35 pages, available as grqc/9304012.
Kuchar (pronounced Kookahsh, by the way) is one of the grand old men of
quantum gravity, one of the people who stuck with the subject for the
many years when it seemed absolutely hopeless, who now deserves some of
the credit for the field's current resurgence. He has always been very
interested in the problem of time, and for anyone who knows a little
general relativity and quantum field theory, this is a very readable
introduction to some of the key problems in canonical quantum gravity.
I should warn the naive reader, however, that Kuchar's views about the
problem of time expressed in this paper go strongly against those of
many other experts! It is a controversial problem.
Briefly, many people believe that physical observables in quantum
gravity should commute with the Hamiltonian constraint (cf "week11");
this means that they are timeindependent, or constants of motion, and
this makes the dynamics of quantum gravity hard to ferrett out. Kuchar
calls such quantities "perennials." But Rovelli has made a proposal for
how to recover dynamics from perennials, basically by considering 1parameter
families A_t of perennials, ironically called "evolving constants of
motion." Kuchar argues against this proposal on two grounds: first, he
does not think physical observables need to commute with the Hamiltonian
constraint, and second, he argues that there may be very few if any
perennials. The latter point is much more convincing to me than the
former, at least at the *classical* level, where the presence of enough
perennials would be close to the complete integrability of general
relativity, which is most unlikely. But on the quantum level things are
likely to be quite different, and Smolin has recently been at work
attempting to construct perennials in quantum gravity (cf "week"). As
for Kuchar's former point, that observables in quantum gravity need not
be perennials, his arguments seem rather weak. In any event, read and
enjoy, but realize that the subject is a tricky one!
2) 2categories and 2knots, by John Fischer, preprint, last revised Feb.
6 1993. (Fischer is at fischerjohn@math.yale.edu)
This is the easiest way to learn about the 2category of 2knots.
Recall (from "week" and "week4") that a 2knot is a surface embedded in
R^4, which may visualized as a "movie" of knots evolving in time.
Fischer shows that the algebraic structure of 2knots is captured by
a braided monoidal 2category, and he describes this 2category.
3) A new discretization of classical and quantum general relativity, by
Mark Miller and Lee Smolin, 22 pages in LaTeX form, available as
grqc/9304005.
Here Smolin proposes a new simplicial approach to general relativity
(there is an older one known as the Regge calculus) which uses
Ashtekar's "new variables," and works in terms of the
CapovillaDellJacobson version of the Lagrangian.
Let me just quote the abstract, I'm getting tired:
We propose a new discrete approximation to the Einstein equations,
based on the CapovillaDellJacobson form of the action for the
Ashtekar variables. This formulation is analogous to the Regge
calculus in that it results from the application of the exact
equations to a restricted class of geometries. Both a Lagrangian
and Hamiltonian formulation are proposed and we report partial
results about the constraint algebra of the Hamiltonian formulation.
We find that in the limit that the $SO(3)$ gauge symmetry of frame
rotations is reduced to the abelian $U(1)^3$, the discrete versions
of the diffeomorphism constraints commute with each other and with
the Hamiltonian constraint.
4) Higher algebraic structures and quantization, by Dan Freed,
preprint, December 18, 1992, available as hepth/9212115.
This is about TQFTs and the highpowered algebra needed to do justice to
the "ladder of field theories" that one can obtain starting with a
ddimensional TQFT  gerbs, torsors, ncategories and other such scary
things. I am too beat to do this justice.

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twf_ascii/week120000064400020410000157000000364221015243334100141210ustar00baezhttp00004600000001Also available at http://math.ucr.edu/home/baez/week120.html
May 6, 1998
This Week's Finds in Mathematical Physics (Week 120)
John Baez
Now that I'm hanging out with the gravity crowd at Penn State, I might
as well describe what's been going on here lately.
First of all, Ashtekar and Krasnov have written an expository account
of their work on the entropy of quantum black holes:
1) Abhay Ashtekar and Kirill Krasnov, Quantum geometry and black holes,
preprint available as grqc/9804039.
But if you prefer to see a picture of a quantum black hole, without
any words or equations, try this:
2) Kirill Krasnov, picture of a quantum black hole,
http://math.ucr.edu/home/baez/blackhole.eps
You'll see a bunch of spin networks poking the horizon, giving it
area and curvature. Of course, this is just a theory.
Second, there's been a burst of new work studying quantum gravity in
terms of spin foams. A spin foam looks a bit like a bunch of soap
suds  with the faces of the bubbles and the edges where the bubbles
meet labelled by spins j = 0, 1/2, 1, 3/2, etc.. Spin foams are an
attempt at a quantum description of the geometry of spacetime. If
you slice a spin foam with a hyperplane representing "t = 0" you get
a spin network: a graph with its edges and vertices labelled by spins.
Spin networks have been used in quantum gravity for a while now to
describe the geometry of space at a given time, so it's natural to hope
that they're a slice of something that describes the geometry of spacetime.
As usual in quantum gravity, it's too early to tell if this approach
will work. As usual, it has lots of serious problems. But before
going into the problems, let me remind you how spin foams are supposed
to work.
To relate spin foams to more traditional ideas about spacetime, one
can consider spin foams living in a triangulated 4manifold: one spin
foam vertex sitting in each 4simplex, one spin foam edge poking
through each tetrahedron, and one spin foam face intersecting each
triangle. Labelling the spin foam edges and faces with spins is
supposed to endow the triangulated 4manifold with a "quantum
4geometry". In other words, it should let us compute things like the
areas of the triangles, the volumes of the tetrahedra, and the
4volumes of the 4simplices. There are some arguments going on now
about the right way to do this, but it's far from an arbitrary
business: the interplay between group representation theory and
geometry says a lot about how it should go. In the simplified case of
3dimensional spacetime, it's fairly well understood  the hard
part, and the fun part, is getting it to work in 4 dimensions.
Assuming we can do this, the next trick is to compute an amplitude for
each spin foam vertex in a nice way, much as one computes amplitudes
for vertices of Feynman diagrams. A spin foam vertex is supposed to
represent an "event"  if we slice the spin foam by a hyperplane we
get a spin network, and as we slide this slice "forwards in time", the
spin network changes its topology whenever we pass a spin foam vertex.
The amplitude for a vertex tells us how likely it is for this event to
happen. As usual in quantum theory, we need to take the absolute
value of an amplitude and square it to get a probability.
We also need to compute amplitudes for spin foam edges and faces,
called "propagators", in analogy to the amplitudes one computes for
the edges of Feynman diagrams. Multiplying all the vertex amplitudes
and propagators for a given spin foam, one gets the amplitude for the
whole spin foam. This tells us how likely it is for the whole spin
foam to happen.
Barrett and Crane came up with a specific way to do all this stuff,
Reisenberger came up with a different way, I came up with a general
formalism for understanding this stuff, and now people are busy
arguing about the merits of different approaches. Here are some
papers on the subject  I'll pick up where I left off in "week113".
3) Louis Crane, David N. Yetter, On the classical limit of the
balanced state sum, preprint available as grqc/9712087.
The goal here is to show that in the limit of large spins, the
amplitude given by Barrett and Crane's formula approaches
exp(iS)
where S is the action for classical general relativity  suitably
discretized, and in signature ++++. The key trick is to use an idea
invented by Regge in 1961.
Regge came up with a discrete analog of the usual formula for the
action in classical general relativity. His formula applies to a
triangulated 4manifold whose edges have specified lengths. In this
situation, each triangle has an "angle deficit" associated to it.
It's easier to visualize this two dimensions down, where each vertex
in a triangulated 2manifold has an angle deficit given by adding up
angles for all the triangles having it as a corner, and then
subtracting 2 pi. No angle deficit means no curvature: the triangles
sit flat in a plane. The idea works similarly in 4 dimensions.
Here's Regge's formula for the action: take each triangle in your
triangulated 4manifold, take its area, multiply it by its angle
deficit, and then sum over all the triangles.
Simple, huh? In the continuum limit, Regge's action approaches the
integral of the Ricci scalar curvature  the usual action in general
relativity. For more see:
4) T. Regge, General relativity without coordinates, Nuovo Cimento 19
(1961), 558571.
So, Crane and Yetter try to show that in the limit of large spins, the
BarrettCrane spin foam amplitude approaches exp(iS) where S is the
Regge action. There argument is interesting but rather sketchy.
Someone should try to fill in the details!
However, it's not clear to me that the large spin limit is physically
revelant. If spacetime is really made of lots of 4simplices labelled
by spins, the 4simplices have got to be quite small, so the spins
labelling them should be fairly small. It seems to me that the right
limit to study is the limit where you triangulate your 4manifold with
a huge number of 4simplices labelled by fairly small spins. After
all, in the spin network picture of the quantum black hole, it seems
that spin network edges labelled by spin 1/2 contribute most of the
states (see "week112").
When you take a spin foam living in a triangulated 4manifold and
slice it in a way that's compatible with the triangulation, the spin
network you get is a 4valent graph. Thus it's not surprising that
Barrett and Crane's formula for vertex amplitudes is related to an
invariant of 4valent graphs with edges labelled by spins. There's
already a branch of math relating such invariants to representations
of groups and quantum groups, and their formula fits right in. Yetter
has figured out how to generalize this graph invariant to nvalent
graphs with edges labelled by spins, and he's also studied more
carefully what happens when one "qdeforms" the whole business 
replacing the group by the corresponding quantum group. This should
be related to quantum gravity with nonzero cosmological constant, if
all the mathematical clues aren't lying to us. See:
5) David N. Yetter, Generalized BarrettCrane vertices and invariants of
embedded graphs, preprint available as math.QA/9801131.
Barrett has also given a nice formula in terms of integrals for
the invariant of 4valent graphs labelled by spins. This is motivated
by the physics and illuminates it nicely:
6) John W. Barrett, The classical evaluation of relativistic spin
networks, preprint available at math.QA/9803063.
Let me quote the abstract:
The evaluation of a relativistic spin network for the classical
case of the Lie group SU(2) is given by an integral formula over
copies of SU(2). For the graph determined by a 4simplex this
gives the evaluation as an integral over a space of geometries
for a 4simplex.
Okay, so much for the good news. What about the bad news? To explain
this, I need to get a bit more specific about Barrett and Crane's
approach.
Their approach is based on a certain way to describe the geometry of a
4simplex. Instead of specifying lengths of edges as in the old Regge
approach, we specify bivectors for all its faces. Geometrically, a
bivector is just an "oriented area element"; technically, the space of
bivectors is the dual of the space of 2forms. If we have a 4simplex
in R^4 and we choose orientations for its triangular faces, there's an
obvious way to associate a bivector to each face. We get 10 bivectors
this way.
What constraints do these 10 bivectors satisfy? They can't be
arbitrary! First, for any four triangles that are all the faces of
the same tetrahedron, the corresponding bivectors must sum to zero.
Second, every bivector must be "simple"  it must be the wedge
product of two vectors. Third, whenever two triangles are the faces
of the same tetrahedron, the sum of the corresponding bivectors must
be simple.
It turns out that these constraints are almost but *not quite enough*
to imply that 10 bivectors come from a 4simplex. Generically, it
there are four possibilities: our bivectors come from a 4simplex,
the *negatives* of our bivectors come from a 4simplex, their *Hodge
duals* of our bivectors come from a 4simplex, or *the negatives of
their Hodge duals* come from a 4simplex.
If we ignore this and describe the 4simplex using bivectors
satisfying the three constraints above, and then quantize this
description, we get the picture of a "quantum 4simplex" that is the
startingpoint for the BarrettCrane model. But clearly it's
dangerous to ignore this problem.
Actually, I learned about this problem from Robt Bryant over on
sci.math.research, and I discussed it in my paper on spin foam models,
citing Bryant of course. Barrett and Crane overlooked this problem in
the first version of their paper, but now they recognize its
importance. Two papers have recently appeared which investigate it
further:
7) Michael P. Reisenberger, Classical Euclidean general relativity from
``lefthanded area = righthanded area'', preprint available as
grqc/9804061.
8) Roberto De Pietri and Laurent Freidel, so(4) Plebanski Action and
relativistic spin foam model, preprint available as grqc/9804071.
These papers study classical general relativity formulated as a
constrained SO(4) BF theory. The constraints needed here are
mathematically just the same as the constraints needed to ensure that
10 bivectors come from the faces of an actual 4simplex! This is part
of the magic of this approach. But again, if one only imposes the
three constraints I listed above, it's not quite enough: one gets
fields that are either solutions of general relativity *or* solutions
of three other theories! This raises the worry that the BarrettCrane
model is a quantization, not exactly of general relativity, but of
general relativity mixed in with these extra theories.
Here's another recent product of the Center for Classical and
Quantum Gravity here at Penn State:
9) Laurent Freidel and Kirill Krasnov, Discrete spacetime volume for
3dimensional BF theory and quantum gravity, preprint available as
hepth/9804185.
Freidel and Krasnov study the volume of a single 3simplex as an
observable in the context of the TuraevViro model  a topological
quantum field theory which is closely related to quantum gravity in
spacetime dimension 3.
And here are some other recent papers on quantum gravity written by
folks who either work here at the CGPG or at least occasionally
drift through. I'll just quote the abstracts of these:
10) Ted Jacobson, Black hole thermodynamics today, to appear in Proceedings
of the Eighth Marcel Grossmann Meeting, World Scientific, 1998, preprint
available as grqc/9801015.
A brief survey of the major themes and developments of black hole
thermodynamics in the 1990's is given, followed by summaries of the
talks on this subject at MG8 together with a bit of commentary,
and closing with a look towards the future.
11) Rodolfo Gambini, Jorge Pullin, Does loop quantum gravity imply
Lambda = 0?, preprint available as grqc/9803097.
We suggest that in a recently proposed framework for quantum
gravity, where Vassiliev invariants span the the space of states,
the latter is dramatically reduced if one has a nonvanishing
cosmological constant. This naturally suggests that the initial
state of the universe should have been one with Lambda=0.
11) R. Gambini, O. Obregon, and J. Pullin, YangMills analogues of the
Immirzi ambiguity, preprint available as grqc/9801055.
We draw parallels between the recently introduced ``Immirzi
ambiguity'' of the Ashtekarlike formulation of canonical quantum
gravity and other ambiguities that appear in YangMills theories,
like the theta ambiguity. We also discuss ambiguities in the
Maxwell case, and implication for the loop quantization of these
theories.
12) John Baez and Stephen Sawin, Diffeomorphisminvariant spin network
states, to appear in Jour. Funct. Analysis, preprint available as
qalg/9708005 or at http://math.ucr.edu/home/baez/int2.ps
We extend the theory of diffeomorphisminvariant spin network
states from the realanalytic category to the smooth
category. Suppose that G is a compact connected semisimple Lie
group and P > M is a smooth principal Gbundle. A `cylinder
function' on the space of smooth connections on P is a continuous
complex function of the holonomies along finitely many piecewise
smoothly immersed curves in M. We construct diffeomorphisminvariant
functionals on the space of cylinder functions from `spin networks':
graphs in M with edges labeled by representations of G and vertices
labeled by intertwining operators. Using the `group averaging'
technique of Ashtekar, Marolf, Mourao and Thiemann, we equip the
space spanned by these `diffeomorphisminvariant spin network states'
with a natural inner product.
Finally, here are two recent reviews of string theory and
supersymmetry:
13) John H. Schwarz and Nathan Seiberg, String theory, supersymmetry,
unification, and all that, to appear in the American Physical Society
Centenary issue of Reviews of Modern Physics, March 1999, preprint
available as hepth/9803179.
14) Keith R. Dienes and Christopher Kolda, Twenty open questions in
supersymmetric particle physics, 64 pages, preprint available as
hepph/9712322.
I'm afraid I'll slack off on my "tour of homotopy theory" this week.
I want to get to fun stuff like model categories and E_infinity spaces,
but it's turning out to be a fair amount of work to reach that goal!
That's what always happens with This Week's Finds: I start learning
about something and think "oh boy, this stuff is great; I'll write it
up really carefully so that everyone can understand it," but then this
turns out to be so much work that by the time I'm halfway through I'm
off on some other kick.

Previous issues of "This Week's Finds" and other expository articles on
mathematics and physics, as well as some of my research papers, can be
obtained at
http://math.ucr.edu/home/baez/
For a table of contents of all the issues of This Week's Finds, try
http://math.ucr.edu/home/baez/twf.html
A simple jumpingoff point to the old issues is available at
http://math.ucr.edu/home/baez/twfshort.html
If you just want the latest issue, go to
http://math.ucr.edu/home/baez/this.week.html
ates, Nuovo Cimento 19
(1961), 558571.
So, Crane and Yetter try to show that in the limit of large spins, the
BarrettCrane spin foam amplitude approaches exp(iS) where S is the
Regge action. There argument is interesting but rather sktwf_ascii/week121000064400020410000157000000640121035010456000141140ustar00baezhttp00004600000001Also available at http://math.ucr.edu/home/baez/week121.html
May 15, 1998
This Week's Finds in Mathematical Physics (Week 121)
John Baez
This time I want to talk about higherdimensional algebra and its
applications to topology. Marco Mackaay has just come out with a
fascinating paper that gives a construction of 4dimensional TQFTs
from certain "monoidal 2categories".
1) Marco Mackaay, Spherical 2categories and 4manifold invariants,
preprint available as math.QA/9805030.
Beautifully, this construction is just a categorified version of
Barrett and Westbury's construction of 3dimensional topological
quantum field theories from "monoidal categories". Categorification
 the process of replacing equations by isomorphisms  is supposed
to take you up the ladder of dimensions. Here we are seeing it in
action!
To prepare you understand Mackaay's paper, maybe I should explain the
idea of categorification. Since I recently wrote something about
this, I think I'll just paraphrase a bit of that. Some of this is
already familiar to longtime customers, so if you know it all
already, just skip it.
2) John Baez and James Dolan, Categorification, to appear in the
Proceedings of the Workshop on Higher Category Theory and Mathematical
Physics at Northwestern University, Evanston, Illinois, March 1997,
eds. Ezra Getzler and Mikhail Kapranov. Preprint available as
math.QA/9802029 or at http://math.ucr.edu/home/baez/cat.ps
So, what's categorification? This tonguetwisting term, invented by
Louis Crane, refers to the process of finding categorytheoretic
analogs of ideas phrased in the language of set theory, using the
following analogy between set theory and category theory:
elements objects
equations between elements isomorphisms between objects
sets categories
functions functors
equations between functions natural isomorphisms between functors
Just as sets have elements, categories have objects. Just as there
are functions between sets, there are functors between categories.
Interestingly, the proper analog of an equation between elements is
not an equation between objects, but an isomorphism. More generally,
the analog of an equation between functions is a natural isomorphism
between functors.
For example, the category FinSet, whose objects are finite sets and
whose morphisms are functions, is a categorification of the set N of
natural numbers. The disjoint union and Cartesian product of finite
sets correspond to the sum and product in N, respectively. Note that
while addition and multiplication in N satisfy various equational
laws such as commutativity, associativity and distributivity, disjoint
union and Cartesian product satisfy such laws *only up to natural
isomorphism*. This is a good example of how equations between functions
get replaced by natural isomorphisms when we categorify.
If one studies categorification one soon discovers an amazing fact: many
deepsounding results in mathematics are just categorifications of facts
we learned in high school! There is a good reason for this. All along,
we have been unwittingly `decategorifying' mathematics by pretending
that categories are just sets. We `decategorify' a category by
forgetting about the morphisms and pretending that isomorphic objects
are equal. We are left with a mere set: the set of isomorphism classes
of objects.
To understand this, the following parable may be useful. Long ago, when
shepherds wanted to see if two herds of sheep were isomorphic, they
would look for an explicit isomorphism. In other words, they would line
up both herds and try to match each sheep in one herd with a sheep in
the other. But one day, along came a shepherd who invented
decategorification. She realized one could take each herd and `count'
it, setting up an isomorphism between it and some set of `numbers',
which were nonsense words like `one, two, three, ...' specially
designed for this purpose. By comparing the resulting numbers, she
could show that two herds were isomorphic without explicitly
establishing an isomorphism! In short, by decategorifying the category
of finite sets, the set of natural numbers was invented.
According to this parable, decategorification started out as a stroke
of mathematical genius. Only later did it become a matter of dumb
habit, which we are now struggling to overcome by means of
categorification. While the historical reality is far more
complicated, categorification really has led to tremendous progress in
mathematics during the 20th century. For example, Noether
revolutionized algebraic topology by emphasizing the importance of
homology groups. Previous work had focused on Betti numbers, which
are just the dimensions of the rational homology groups. As with
taking the cardinality of a set, taking the dimension of a vector
space is a process of decategorification, since two vector spaces are
isomorphic if and only if they have the same dimension. Noether noted
that if we work with homology groups rather than Betti numbers, we can
solve more problems, because we obtain invariants not only of spaces,
but also of maps.
In modern lingo, the nth rational homology is a *functor* defined
on the *category* of topological spaces, while the nth Betti number is
a mere *function*, defined on the *set* of isomorphism classes of
topological spaces. Of course, this way of stating Noether's insight
is anachronistic, since it came before category theory. Indeed, it
was in Eilenberg and Mac Lane's subsequent work on homology that
category theory was born!
Decategorification is a straightforward process which typically
destroys information about the situation at hand. Categorification,
being an attempt to recover this lost information, is inevitably
fraught with difficulties. One reason is that when categorifying, one
does not merely replace equations by isomorphisms. One also demands
that these isomorphisms satisfy some new equations of their own,
called `coherence laws'. Finding the right coherence laws for a given
situation is perhaps the trickiest aspect of categorification.
For example, a monoid is a set with a product satisfying the associative
law and a unit element satisfying the left and right unit laws. The
categorified version of a monoid is a `monoidal category'. This is a
category C with a product
tensor: C x C > C
and unit object 1. If we naively impose associativity and the left
and right unit laws as equational laws, we obtain the definition of a
`strict' monoidal category. However, the philosophy of
categorification suggests instead that we impose them only up to
natural isomorphism. Thus, as part of the structure of a `weak'
monoidal category, we specify a natural isomorphism
a_{x,y,z}: (x tensor y) tensor z > x tensor (y tensor z)
called the `associator', together with natural isomorphisms
l_x: 1 tensor x > x,
r_x: x tensor 1 > x.
Using the associator one can construct isomorphisms between any two
parenthesized versions of the tensor product of several objects.
However, we really want a *unique* isomorphism. For example, there
are 5 ways to parenthesize the tensor product of 4 objects, which are
related by the associator as follows:
((x tensor y) tensor z) tensor w > (x tensor (y tensor z)) tensor w
 
 
 
V 
(x tensor y) tensor (z tensor w) 
 
 
 
V V
{x tensor (y tensor(z tensor w)) < x tensor ((y tensor z) tensor w)
In the definition of a weak monoidal category we impose a coherence
law, called the `pentagon identity', saying that this diagram
commutes. Similarly, we impose a coherence law saying that the
following diagram built using a, l and r commutes:
(1 tensor x) tensor 1 > 1 tensor (x tensor 1)
 
 
V V
x tensor 1 > x < 1 tensor x
This definition raises an obvious question: how do we know we have
found all the right coherence laws? Indeed, what does `right' even
*mean* in this context? Mac Lane's coherence theorem gives one answer
to this question: the above coherence laws imply that any two
isomorphisms built using a, l and r and having the same source and
target must be equal.
Further work along these lines allow us to make more precise the sense
in which N is a decategorification of FinSet. For example, just as N
forms a monoid under either addition or multiplication, FinSet becomes
a monoidal category under either disjoint union or Cartesian product
if we choose the isomorphisms a, l, and r sensibly. In fact, just as
N is a `rig', satisfying all the ring axioms except those involving
additive inverses, FinSet is what one might call a `rig category'. In
other words, it satisfies the rig axioms up to natural isomorphisms
satisfying the coherence laws discovered by Kelly and Laplaza, who
proved a coherence theorem in this context.
Just as the decategorification of a monoidal category is a monoid, the
decategorification of any rig category is a rig. In particular,
decategorifying the rig category FinSet gives the rig N. This
idea is especially important in combinatorics, where the best proof of
an identity involving natural numbers is often a `bijective proof':
one that actually establishes an isomorphism between finite sets.
While coherence laws can sometimes be justified retrospectively by
coherence theorems, certain puzzles point to the need for a deeper
understanding of the *origin* of coherence laws. For example,
suppose we want to categorify the notion of `commutative monoid'. The
strictest possible approach, where we take a strict monoidal category
and impose an equational law of the form x tensor y = y tensor x, is
almost completely uninteresting. It is much better to start with a weak
monoidal category equipped with a natural isomorphism
B_{x,y}: x tensor y > y tensor x
called the `braiding', and then impose coherence laws called `hexagon
identities' saying that the following two diagrams built from the
braiding and the associator commute:
x tensor (y tensor z) > (y tensor z) tensor x
 ^
 
V 
(x tensor y) tensor z y tensor (z tensor x)
 ^
 
V 
(y tensor x) tensor z > y tensor (x tensor z)
(x tensor y) tensor z > z tensor (x tensor y)
 ^
 
V 
x tensor (y tensor z) (z tensor z) tensor y
 ^
 
V 
x tensor (z tensor y) > (x tensor z) tensor y
This gives the definition of a weak `braided monoidal category'. If
we impose an additional coherence law saying that B_{x,y} is the
inverse of B_{y,x}, we obtain the definition of a `symmetric monoidal
category'. Both of these concepts are very important; which one is
`right' depends on the context. However, neither implies that every
pair of parallel morphisms built using the braiding are equal. A good
theory of coherence laws must naturally account for these facts.
The deepest insights into such puzzles have traditionally come from
topology. In homotopy theory it causes problems to work with spaces
equipped with algebraic structures satisfying equational laws, because
one cannot transport such structures along homotopy equivalences. It
is better to impose laws *only up to homotopy*, with these homotopies
satisfying certain coherence laws, but again only up to homotopy, with
these higher homotopies satisfying their own higher coherence laws,
and so on. Coherence laws thus arise naturally in infinite sequences.
For example, Stasheff discovered the pentagon identity and a sequence
of higher coherence laws for associativity when studying the algebraic
structure possessed by a space that is homotopy equivalent to a loop
space. Similarly, the hexagon identities arise as part of a sequence
of coherence laws for spaces homotopy equivalent to double loop
spaces, while the extra coherence law for symmetric monoidal
categories arises as part of a sequence for spaces homotopy equivalent
to triple loop spaces. The higher coherence laws in these sequences
turn out to be crucial when we try to *iterate* the process of
categorification.
To *iterate* the process of categorification, we need a concept of
`ncategory'  roughly, an algebraic structure consisting of a
collection of objects (or `0morphisms'), morphisms between objects
(or `1morphisms'), 2morphisms between morphisms, and so on up to
nmorphisms. There are various ways of making this precise, and right
now there is a lot of work going on devoted to relating these
different approaches. But the basic thing to keep in mind is that
the concept of `(n+1)category' is a categorification of the concept
of `ncategory'. What were equational laws between nmorphisms in
an ncategory are replaced by natural (n+1)isomorphisms, which need
to satisfy certain coherence laws of their own.
To get a feeling for how these coherence laws are related to homotopy
theory, it's good to think about certain special kinds of ncategory.
If we have an (n+k)category that's trivial up to but not including
the kmorphism level, we can turn it into an ncategory by a simple
reindexing trick: just think of its jmorphisms as (jk)morphisms!
We call the ncategories we get this way `ktuply monoidal
ncategories'. Here is a little chart of what they amount to for
various low values of n and k:
ktuply monoidal ncategories
n = 0 n = 1 n = 2
k = 0 sets categories 2categories
k = 1 monoids monoidal monoidal
categories 2categories
k = 2 commutative braided braided
monoids monoidal monoidal
categories 2categories
k = 3 " " symmetric weakly
monoidal involutory
categories monoidal
2categories
k = 4 " " " " strongly
involutory
monoidal
2categories
k = 5 " " " " " "
One reason James Dolan and I got so interested in this chart is the
`tangle hypothesis'. Roughly speaking, this says that ndimensional
surfaces embedded in (n+k)dimensional space can be described purely
algebraically using the a certain special `ktuply monoidal ncategory
with duals'. If true, this reduces lots of differential topology to
pure algebra! It also helps you understand the parameters n and k:
you should think of n as `dimension' and k as `codimension'.
For example, take n = 1 and k = 2. Knots, links and tangles in
3dimensional space can be described algebraically using a certain
`braided monoidal categories with duals'. This was the first
interesting piece of evidence for the tangle hypothesis. It has
spawned a whole branch of math called `quantum topology', which
people are trying to generalize to higher dimensions.
More recently, Laurel Langford tackled the case n = 2, k = 2. She
proved that 2dimensional knotted surfaces in 4dimensional space can
be described algebraically using a certain `braided monoidal
2category with duals'. These socalled `2tangles' are particularly
interesting to me because of their relation to spin foam models of
quantum gravity, which are also all about surfaces in 4space. For
references, see "week103". But if you want to learn about more about
this, you couldn't do better than to start with:
3) J. S. Carter and M. Saito, Knotted Surfaces and Their Diagrams,
American Mathematical Society, Providence, 1998.
This is a magnificently illustrated book which will really get you
able to *see* 2dimensional surfaces knotted in 4d space. At the end
it sketches the statement of Langford's result.
Another interesting thing about the above chart is that ktuply
monoidal ncategories keep getting `more commutative' as k increases,
until one reaches k = n+2, at which point things stabilize. There is
a lot of evidence suggesting that this `stabilization hypothesis' is
true for all n. Assuming it's true, it makes sense to call a ktuply
monoidal ncategory with k >= n+2 a `stable ncategory'.
Now, where does homotopy theory come in? Well, here you need to look
at ncategories where all the jmorphisms are invertible for all j.
These are called `ngroupoids'. Using these, one can develop a
translation dictionary between ncategory theory and homotopy theory,
which looks like this:
omegagroupoids homotopy types
ngroupoids homotopy ntypes
ktuply groupal omegagroupoids homotopy types of kfold loop spaces
ktuply groupal ngroupoids homotopy ntypes of kfold loop spaces
ktuply monoidal omegagroupoids homotopy types of E_k spaces
ktuply monoidal ngroupoids homotopy ntypes of E_k spaces
stable omegagroupoids homotopy types of infinite loop spaces
stable ngroupoids homotopy ntypes of infinite loop spaces
Zgroupoids homotopy types of spectra
The entries on the lefthand side are very natural from an algebraic
viewpoint; the entries on the righthand side are things topologists
already study. We explain what all these terms mean in the paper, but
maybe I should say something about the first two rows, which are the
most basic in a way. A homotopy type is roughly a topological space
`up to homotopy equivalence', and an omegagroupoid is a kind of
limiting case of an ngroupoid as n goes to infinity. If infinity is
too scary, you can work with homotopy ntypes, which are basically
homotopy types with no interesting topology above dimension n. These
should correspond to ngroupoids.
Using these basic correspondences we can then relate various special
kinds of homotopy types to various special kinds of omegagroupoids,
giving the rest of the rows of the chart. Homotopy theorists know a
lot about the righthand column, so we can use this to get a lot of
information about the lefthand column. In particular, we can work
out the coherence laws for ngroupoids, and  this is the best part,
but the least understood  we can then *guess* a lot of stuff about
the coherence laws for *general* ncategories. In short, we are using
homotopy theory to get our foot in the door of ncategory theory.
I should emphasize, though, that this translation dictionary is
partially conjectural. It gets pretty technical to say what exactly
is and is not known, especially since there's pretty rapid progress
going on. Even in the last few months there have been some
interesting developments. For example, Breen has come out with a
paper relating ktuply monoidal ncategories to Postnikov towers and
various farout kinds of homological algebra:
4) Lawrence Breen, Braided ncategories and Sigmastructures,
Prepublications Matematiques de l'Universite Paris 13, 9806,
January 1998, to appear in the Proceedings of the Workshop on
Higher Category Theory and Mathematical Physics at Northwestern
University, Evanston, Illinois, March 1997, eds. Ezra Getzler
and Mikhail Kapranov.
Also, the following folks have also developed a notion of `iterated
monoidal category' whose nerve gives the homotopy type of a kfold
loop space, just as the nerve of a category gives an arbitrary
homotopy type:
5) C. Balteanu, Z. Fiedorowicz, R. Schwaenzl, and R. Vogt,
Iterated monoidal categories, available at math.AT/9808082.
Anyway, in addition to explaining the relationship between ncategory
theory and homotopy theory, Dolan's and my paper discusses iterated
categorifications of the very simplest algebraic structures: the
natural numbers and the integers. The natural numbers are the free
monoid on one generator; the integers are the free group on one
generator. We believe this is just the tip of the following iceberg:
algebraic strutures and the free such structure on one generator
sets the oneelement set

monoids the natural numbers

groups the integers

ktuply monoidal the braid ngroupoid
ncategories in codimension k

ktuply monoidal the braid omegagroupoid
omegacategories in codimension k

stable ncategories the braid ngroupoid
in infinite codimension

stable omegacategories the braid omegagroupoid
in infinite codimension

ktuply monoidal the ncategory of framed ntangles
ncategories with duals in n+k dimensions

stable ncategories the framed cobordism ncategory
with duals

ktuply groupal the homotopy ntype
ngroupoids of the kth loop space of S^k

ktuply groupal the homotopy type
omegagroupoids of the kth loop space of S^k

stable omegagroupoids the homotopy type
of Loop^{infinity} S^infinity

Zgroupoids the sphere spectrum
You may or may not know the guys on the righthand side, but some of
them are very interesting and complicated, so it's really exciting that
they are all in some sense categorified and/or stabilized versions of
the integers and natural numbers.
Whew! There is more to say, but I'll just mention a few related
papers and then quit.
6) Representation theory of Hopf categories, Martin Neuchl, Ph.D.
dissertation, Department of Mathematics, University of Munich,
1997. Available at http://math.ucr.edu/home/baez/neuchl.ps
Just as the category of representations of a Hopf algebra gives a nice
monoidal category, the 2category of representations of a Hopf category
gives a nice monoidal 2category! Categorification strikes again  and
this is perhaps our best hopes for getting our hands on the data needed
to stick into Mackaay's machine and get concrete examples of a 4d topological
quantum field theories!
7) Jim Stasheff, Grafting Boardman's cherry trees to quantum field theory,
preprint available as math.AT/9803156.
Starting with Boardman and Vogt's work, and shortly thereafter that of
May, operads have become really important in homotopy theory, string
theory, and now ncategory theory; this review article sketches some
of the connections.
8) Masoud Khalkhali, On cyclic homology of A_infinity algebras, preprint
available as math.QA/9805051.
9) Masoud Khalkhali, Homology of L_infinity algebras and cyclic homology,
preprint available as math.QA/9805052.
An A_infinity algebra is an algebra that is associative *up to an
associator* which satisfies the pentagon identity *up to a
pentagonator* which satisfies it's own coherence law up to something,
ad infinitum. The concept goes back to Stasheff's work on A_infinity
spaces  spaces with a homotopy equivalence to a space equipped with
an associative product. (These are the same thing as what I called
E_1 spaces in the translation dictionary between ngroupoid theory and
homotopy theory.) But here it's been transported from Top over
to Vect. Similarly, an L_infinity is a Lie algebra "up to an infinity
of higher coherence laws". LodayQuillen and Tsygan showed that that the
Lie algebra homology of the algebra of stable matrices over an
associative algebra is isomorphic, as a Hopf algebra, to the exterior
algebra of the cyclic homology of the algebra. In the second paper
above, Khalkali gets the tools set up to extend this result to the
category of L_infinity algebras.

Previous issues of "This Week's Finds" and other expository articles on
mathematics and physics, as well as some of my research papers, can be
obtained at
http://math.ucr.edu/home/baez/
For a table of contents of all the issues of This Week's Finds, try
http://math.ucr.edu/home/baez/twf.html
A simple jumpingoff point to the old issues is available at
http://math.ucr.edu/home/baez/twfshort.html
If you just want the latest issue, go to
http://math.ucr.edu/home/baez/this.week.html
to higher dimensions.
More recently, Laurel Langford tackled the case n = 2, k = 2. She
proved that 2dimensional knotted surfaces in 4dimensional space can
be described algebraically using a certain `braided monoidal
2category with duals'. These socalled `2tangles' are particularly
interesting to me because of their relation to spin foam models of
quantum gravity, which are also all about surfaces in 4space. For
references, see "week103". But if you want to learn about more about
this, twf_ascii/week203000064400020410000157000001350541053520271700141320ustar00baezhttp00004600000001Also available at http://math.ucr.edu/home/baez/week203.html
February 24, 2004
This Week's Finds in Mathematical Physics  Week 203
John Baez
Last week I posed this puzzle: to find a "Golden Object".
A couple days ago I got a wonderful solution from Robin Houston, a computer
science grad student at the University of Manchester. So, I want to say a
bit more about the golden number, then describe his solution, and then
describe how he found it.
Supposedly the Greeks thought the most beautiful rectangle was one such
that when you chop a square off one end, you're left with a rectangle
of the same shape. If your original rectangle was 1 unit across and G
units long, after you chop a 1by1 square off the end you're left with
a rectangle that's G1 units across and 1 unit long:
G
.........................
. . .
. . .
. . .
. . .
1 . . . 1
. . .
. . .
. 1 . G1 .
.........................
So, to make the proportions of the little rectangle the same as those of
the big one, you want
"1 is to G as G1 is to 1"
or in other words:
1/G = G  1
or after a little algebra,
G^2 = G + 1
so that
G = (1 + sqrt(5))/2 = 1.618033988749894848204586834365...
while
1/G = 0.618033988749894848204586834365...
and
G^2 = 2.618033988749894848204586834365...
(At this point I usually tell my undergraduates that the pattern
continues like this, with G^3 = 3.618... and so on  just to see if
they'll believe anything I say.)
These days, the number G is called the Golden Number, the Golden Ratio,
or the Golden Section. It's often denoted by the Greek letter Phi,
after the Greek sculptor Phidias. Phidias helped design the Parthenon 
and supposedly packed it full of golden rectangles, to make it as
beautiful as possible.
The golden number is a great favorite among amateur mathematicians, because
it has a flashy sort of charm. You can find it all over the place if you
look hard enough  and if you look too hard, you'll find it even in places
where it's not. It's the ratio of the diagonal to the side of a regular
pentagon! If you like the number 5, you'll be glad to know that
5 + sqrt(5)
G = sqrt[]
5  sqrt(5)
If you don't, maybe you'd prefer this:
G = exp(arcsinh(1/2))
My favorite formulas for the golden number are
G = sqrt(1 + sqrt(1 + sqrt(1 + sqrt(1 + sqrt(1 + sqrt(1 + ...
and the continued fraction:
1
G = 1 + 
1 + 1

1 + 1

1 + 1

1 + 1

1 + 1

.
.
.
These follow from the equations G^2 = G + 1 and G = 1 + 1/G, respectively.
If you chop off the continued fraction for G at any point, you'll see that
G is also the limit of the ratios of successive Fibonacci numbers. See
"week190" for a very different proof of this fact.
However, don't be fooled! The charm of the golden number tends to attract
kooks and the gullible  hence the term "fool's gold". You have to be
careful about anything you read about this number. In particular, if you
think ancient Greeks ran around in togas philosophizing about the "golden
ratio" and calling it "Phi", you're wrong. This number was named Phi
after Phidias only in 1914, in a book called _The Curves of Life_ by the
artist Theodore Cook. And, it was Cook who first started calling 1.618...
the golden ratio. Before him, 0.618... was called the golden ratio! Cook
dubbed this number "phi", the lowercase baby brother of Phi.
In fact, the whole "golden" terminology can only be traced back to 1826,
when it showed up in a footnote to a book by one Martin Ohm, brother of
Georg Ohm, the guy with the law about resistors. Before then, a lot of
people called 1/G the "Divine Proportion". And the guy who started
*that* was Luca Pacioli, a pal of Leonardo da Vinci who translated Euclid's
Elements. In 1509, Pacioli published a 3volume text entitled Divina
Proportione, advertising the virtues of this number. Some people think
da Vinci used the divine proportion in the composition of his paintings.
If so, perhaps he got the idea from Pacioli.
Maybe Pacioli is to blame for the modern fascination with the golden
ratio; it seems hard to trace it back to Greece. These days you can buy
books and magazines about "Elliot Wave Theory", a method for making money
on the stock market using patterns related to the golden number. Or, if
you're more spiritually inclined, you can go to workshops on "Sacred
Geometry" featuring talks about the healing powers of the golden ratio.
But Greek texts seem remarkably quiet about this number.
The first recorded hint of it is Proposition 11 in Book II of Euclid's
"Elements". It also shows up elsewhere in Euclid, especially Proposition
30 of Book VI, where the task is "to cut a given finite straight line in
extreme and mean ratio", meaning a ratio A:B such that
A:B::(A+B):A (i.e., "A is to B as A+B is to A")
This is later used in Proposition 17 of Book XIII to construct
the pentagonal face of a regular dodecahedron.
Of course, Euclid wasn't the first to do all these things; he just wrote
them up in a nice textbook. By now it's impossible to tell who discovered
the golden ratio and just what the Greeks thought about it. For a sane
and detailed look at the history of the golden ratio, try this:
1) J. J. O'Connor and E. F. Robertson, The Golden Ratio,
http://wwwgap.dcs.stand.ac.uk/~history/HistTopics/Golden_ratio.html
While I'm at it, I should point out that Theodore Cook's book introducing
the notation "Phi" is still in print:
2) The Curves of Life: Being an Account of Spiral Formations and Their
Application to Growth in Nature, to Science, and to Art: with Special
Reference to the Manuscripts of Leonardo da Vinci, Dover Publications,
New York, 1979.
If you want to see what Euclid said about the golden ratio, you can
also pick up a cheap copy of the Elements from Dover  but it's probably
quicker to go online. There are a number of good places to find Euclid's
Elements online these days.
Topologists know David Joyce as the inventor of the "quandle"  an
algebraic structure that captures most of the information in a knot.
Now he's writing a hightech edition of Euclid, complete with Java applets:
3) David E. Joyce's edition of Euclid's Elements,
http://aleph0.clarku.edu/~djoyce/java/elements/toc.html
Joyce is carrying on a noble tradition: back in 1847, Oliver Byrne did
a wonderful edition of Euclid complete with lots of beautiful color
pictures and even some popup models. You can see this online at
the Digital Mathematics Archive:
4) Oliver Byrne's edition of Euclid's Elements, online at the Digital
Mathematics Archive, http://www.sunsite.ubc.ca/DigitalMathArchive/
The most famous scholarly English translation of Euclid was done by
Sir Thomas Heath in 1908. You can find it together with an edition
in Greek and a nearly infinite supply of other classical texts at
the Perseus Digital Library:
5) Thomas L. Heath's edition of Euclid's Elements, online at
The Perseus Digital Library, http://www.perseus.tufts.edu/
But I'm digressing! My main point was that while G = (1 + sqrt(5))/2
is a neat number, it's a lot easier to find nuts raving about it on the
net than to find truly interesting mathematics associated with it  or
even interesting references to it in Greek mathematics! The cynic might
conclude that the charm of this number is purely superficial. However,
that would be premature.
First of all, there's a certain sense in which G is "the most irrational
number". To get the best rational approximations to a number you use its
continued fraction expansion. For G, this converges as slowly as possible,
since it's made of all 1's:
1
G = 1 + 
1 + 1

1 + 1

1 + 1

1 + 1

1 + 1

.
.
.
We can make this more precise. For any number x there's a constant
c(x) that says how hard it is to approximate x by rational numbers,
given by
lim inf x  p/q = c(x)/q^2
q > infinity
where q ranges over integers, and p is an integer chosen to minimize
x  p/q. This constant is as big as possible when x is the golden
ratio!
It'd be ironic if the famously "rational" Greeks, who according to legend
even drowned the guy who proved sqrt(2) was irrational, chose the most
irrational number as the proportions of their most beautiful rectangle!
But, it wouldn't be a coincidence. Their obsession with ratios and
proportions led them to ponder the situation where A:B::(A+B):A,
and this proportion instantly implies that A and B are incommensurable,
since if you assume A and B are integers and try to find their greatest
common divisor using Euclid's algorithm, you get stuck in an infinite loop.
Euclid even mentions this idea in Proposition 2 of Book X:
If, when the less of two unequal magnitudes is continually subtracted
in turn from the greater that which is left never measures the one
before it, then the two magnitudes are incommensurable.
He doesn't explicitly come out and apply it to what we now call the golden
ratio  but how could he not have made the connection? For more info on
the Greek use of continued fractions and the Euclidean algorithm, check
out the chapter on "antihyphairetic ratio theory" in this book:
6) D. H. Fowler, The Mathematics of Plato's Academy: A New Reconstruction,
Oxford U. Press, Oxford, 1987.
Anyway, it's actually important in physics that the golden number is so
poorly approximated by rationals. This fact shows up in the Kolmogorov
ArnoldMoser theorem, or "KAM theorem", which deals with small perturbations
of completely integrable Hamiltonian systems. Crudely speaking, these are
classical mechanics problems that have as many conserved quantities as
possible. These are the ones that tend to show up in textbooks, like the
harmonic oscillator and the gravitational 2body problem. The reason is
that you can solve such problems if you can do a bunch of integrals  hence
the term "completely integrable".
The cool thing about a completely integrable system is that time evolution
carries states of the system along paths that wrap around tori. Suppose
it takes n numbers to describe the position of your system. Then it also
takes n numbers to describe its momentum, so the space of states is
2ndimensional. But if the system has n different conserved quantities 
that's basically the maximum allowed  the space of states will be foliated
by ndimensional tori. Any state that starts on one of these tori will
stay on it forever! It will march round and round, tracing out a kind of
spiral path that may or may not ever get back to where it started.
Things are pretty simple when n = 1, since a 1dimensional torus is a
circle, so the state *has* to loop around to where it started. For example,
when you have a pendulum swinging back and forth, its position and momentum
trace out a loop as time passes.
When n is bigger, things get trickier. For example, when you have n
pendulums swinging back and forth, their motion is periodic if the
ratios of their frequencies are rational numbers.
This is how it works for any completely integrable system. For any torus,
there's an ntuple of numbers describing the frequency with which paths on
this torus wind around in each of the n directions. If the ratios of these
frequencies are all rational, paths on this torus trace out periodic orbits.
Otherwise, they don't!
KAM theory says what happens when you perturb such a system a little.
It won't usually be completely integrable anymore. Interestingly, the
tori with rational frequency ratios tend to fall apart due to resonance
effects. Instead of periodic orbits, we get chaotic motions instead.
But the "irrational" tori are more stable. And, the "more irrational" the
frequency ratios for a torus are, the bigger a perturbation it takes to
disrupt it! Thus, the most stable tori tend to have frequency ratios
involving the golden number. As we increase the perturbation, the last
torus to die is called a "golden torus".
You can actually *watch* tori breaking into chaotic dust if you check out
the applet illustrating the "standard map" on this website:
7) Takashi Kanamaru and J. Michael T. Thompson, Introduction to Chaos
and Nonlinear Dynamics, standard map applet,
http://brain.cc.kogakuin.ac.jp/~kanamaru/Chaos/e/Standard/
The "standard map" is a certain dynamical system that's good for
illustrating this effect. You won't actually see 2d tori, just
1d crosssections of them  but it's pretty fun. For more details,
try this:
8) M. Tabor, Chaos and Integrability in Nonlinear Dynamics: An Introduction,
Wiley, New York, 1989.
In short, the golden number is the best frequency ratio for avoiding
resonance!
Some audiophiles even say this means the best shaped room for listening
to music is one with proportions 1:G:G^2. I leave it to you to find the
flaw in this claim. For more dubious claims, check out the ad for expensive
speaker cables at the end of this article.
KAM theory is definitely cool, but we shouldn't rest content with this
when skeptics ask if the golden number is all it's cracked up to be.
I figure it's part of our job as mathematicians to keep on discovering
mindblowing facts about the golden number. A small part, but part:
we shouldn't give up the field to amateurs!
Penrose has done his share. His "Penrose tiles" take crucial advantage
of the selfsimilarity embodied by the golden number to create nonperiodic
tilings of the plane. This helped spawn a nice little industry, the study
of "quasicrystals" with 5fold symmetry. Here's a good introduction for
mathematicians:
9) Andre Katz, A short introduction to quasicrystallography, in From
Number Theory to Physics, eds. M. Waldschmit et al, Springer, Berlin,
1992, pp. 496537.
By the way, this same book has some nice stuff on the role of the
golden number in KAM theory and the theory of iterated maps from
the circle to itself:
10) Predrag Cvitanovic, Circle maps: irrationally winding, in From
Number Theory to Physics, eds. M. Waldschmit et al, Springer, Berlin,
1992, pp. 631658.
11) JeanChristophe Yoccoz, Introduction to small divisors problems,
in From Number Theory to Physics, eds. M. Waldschmit et al, Springer,
Berlin, 1992, pp. 659679.
Conway and Sloane are also pulling their weight. Starting from the
relation between the golden ratio, the isosahedron, and the 4dimensional
big brother of the icosahedron (the "600cell"), they've described how
to construct the coolest lattices in 8 and 24 dimensions using "icosians" 
which are certain quaternions built using the golden ratio. I discussed
this circle of ideas in "week20", "week59" and "week155".
But if you want some really scary formulas involving the golden ratio,
Ramanujan is the one to go to. Check these out:
1

1 + exp(2pi)

1 + exp(4pi) = exp(2pi/5) [sqrt(G sqrt(5))  G]

1 + exp(6pi)

1 + exp(8pi)

.
.
.
and
1 + exp(2pi sqrt(5))

1 + exp(4pi sqrt(5))

1 + exp(6pi sqrt(5))

1 + exp(8pi sqrt(5))

.
.
.
sqrt(5)
= exp(2pi/5) [   G]
1 + [5^{3/4} (G  1)^{5/2}  1]^{1/5}
These are special cases of a monstrosity called the RogersRamanujan
continued fraction, which is a kind of "qdeformation" of the continued
fraction for the golden ratio. For details, start here:
12) Eric W. Weisstein, RogersRamanujan Continued Fraction,
http://mathworld.wolfram.com/RogersRamanujanContinuedFraction.html
It's these two formulas, and one other like it, that led Hardy to
write the famous lines:
A single look at them is enough to show that they could only be
written down by a mathematician of the highest class. They must
be true because, if they were not true, no one would have had the
imagination to invent them.
For more by Hardy on these continued fractions, see section 1
and section 6.17 of his book:
13) G. H. Hardy, Ramanujan: Twelve Lectures on Subjects Suggested
by His Life and Work, Chelsea Publishing Co., New York, 1959.
The golden number also shows up in the theory of quantum groups.
I talked about this in "week22" so I won't explain it again here.
But, I can't resist mentioning that Freedman, Larsen and Wang have
subsequently shown that a certain topological quantum field theory
called ChernSimons theory, built using the quantum group SU_q(2),
can serve as a universal quantum computer when the parameter q is a
fifth root of unity. And, this is exactly the case where the spin1/2
representation of SU_q(2) has quantum dimension equal to the golden number!
14) Michael Freedman, Michael Larsen, Zhenghan Wang,
A modular functor which is universal for quantum computation,
available at quantph/0001108.
But don't get the wrong idea: it's not that some magic feature of the
golden number is required to build a universal quantum computer! It's
just that the 5 seems to be the *smallest* number n such that SU_q(2)
ChernSimons theory is computationally universal when q is an nth root of 1.
That's pretty much everything I know about the golden number. So now,
what about this "Golden Object" puzzle?
Basically, the problem was to find an object that acts like the golden
number. The golden number has G^2 = G + 1, so we want to find
a object G equipped with a nice isomorphism between G^2 and G + 1.
If G is just a set, this means we want a nice onetoone correspondence
between pairs of elements of G, and elements of G together with one other
It doesn't matter what that other thing is, so let's call it "@".
(You may be wondering about the word "nice". The point is, the problem
is too easy if we don't demand that the solution be nice in some way 
some way that I don't feel like making precise.)
Here's Robin Houston's answer:
Define a "bit" to be either 0 or 1. Define a "golden tree" to be a
(planar) binary tree with leaves labelled by 0, 1, or *, where every
node has at most one bitchild. For example:
/\ is a golden tree, but /\ is not.
/\ 1 /\ *
0 * 0 1
Let G be the set of golden trees. We define an isomorphism
f: G^2 > G + {@}
as follows. First we define f(X, Y) when both X and Y are golden
trees with just one node, this node being labelled by a bit. We
can identify such a tree with a bit, and doing this we set
f(0, 0) = 0
f(0, 1) = 1
f(1, 0) = *
f(1, 1) = @
In the remaining case, where the golden trees X and Y are not just bits,
we set
f(X, Y) = /\
X Y
There are different ways to show this function f is a onetoone
correspondence, but the best way is to see how Houston came up with
this answer! He didn't just pull it out of a hat; he tackled the
problem systematically, and that's why his solution counts as "nice".
It's easy to find a set S equipped with an isomorphism
S = P(S)
where P is some polynomial with natural number coefficients. You
just use the fixedpoint principle described in "week108". Namely,
you start with the empty set, keep hitting it with P forever, and take
a kind of limit. This is how I built the set of binary trees last week,
as a solution of T = T^2 + 1.
The problem is that the isomorphism we seek now:
G^2 = G + 1 (1)
is not of this form. So, what Houston does is to make a substitution:
G = H + 2
Given this, we'd get (1) if we had
H^2 + 4H + 4 = H + 3 (2)
and we'd get (2) if we had
H^2 + 4H + 1 = H (3)
which is of the desired form.
We can rewrite (3) as
H = 1 + H^2 + 2H + H2
and in English this says "an element of H is either a *, or
a pair consisting of two guys that are either bits or elements
of H  but not both bits". So, a guy in H is a golden tree!
But, if it has just one node, that node can only be labelled
by a *, not a 0 or 1. This means there are precisely 2 golden trees
not in H. So, G = H + 2 is the set of all golden trees, and our
calculation above gives an isomorphism G^2 = G + 1.
Voila!
Note that to derive (3) from (1) we need to subtract, which in general
is not allowed in this game. Here we are subtracting constants, and
Houston says that's allowed by the "GarsiaMilne involution theorem".
I don't know this theorem, so I'll make a note to myself to learn it.
But luckily, we don't really need it here: we only need to derive (1)
from (3), and that involves addition, so it's fine.
Part of what makes Houston's solution "nice" is that it suggests a
general method for turning polynomial equations into recursive definitions
of the form S = P(S). Another nice thing is that his trick delivers
a structure type G(X) that reduces to G when X = 1. To get this, first
use the fixedpoint method to construct a structure type H(X) with an
isomorphism
H(X) = (H(X) + X)^2 + 2H(X)
Then, define
G(X) = H(X) + X + 1
and note that this gives
G(X)^2 = G(X) + X
which reduces to G^2 = G + 1 when X = 1.
As if this weren't enough, Houston also gave another solution to the
puzzle. He showed that James Propp's proposed Golden Object, described
last week, really is a Golden Object! Maybe Propp already knew this,
but I sure didn't.
The idea of the proof is pretty general. Suppose we've got a category
that's a "2rig" in the sense of "week191". And, suppose we've got an
object X equipped with an isomorphism
X = 1 + 2X (4)
so that X acts like "1". For example, following Schanuel and Propp,
we can take the category of "sigmapolytopes" and let X be the open
interval: then isomorphism (4) says
(0,1) = (0,1/2) + {1/2} + (1/2,1)
Or, following Houston, we can take the category of sets and let X be
the set of finite bitstrings. Then (4) says "a finite bitstring is
either the empty bitstring, or a bit followed by a finite bitstring".
The relation between these two examples is puzzling to me  if anyone
understands it, let me know! But anyway, either one works.
Now let G be the object of "binary trees with Xlabelled leaves":
G = X + X^2 + 2X^3 + 5X^4 + 14X^5 + 42X^6 + ...
where the coefficients are Catalan numbers. Let's show that G is a
Golden Object. To do this, we'll use (4) and this isomorphism:
G = G^2 + X (5)
which says "a binary tree with Xlabelled leaves is a pair of such
trees, or a degenerate tree with just one Xlabelled node". The formula
for G involving Catalan numbers is really just the fixedpoint solution
to this!
Here is Houston's fiendishly clever argument. Suppose Z is any type
equipped with an isomorphism
Z = Z' + X
for some Z'. Then
Z + X + 1 = Z' + 2X + 1
= Z' + X
= Z
This applies to Z = G^2, since
G^2 = (X + G^2)^2 = (2X + 1 + G^2)^2
has a X term in it when you multiply it out, so it's of the form Z' + X.
Therefore we have an isomorphism
G^2 = G^2 + X + 1
But we also have an isomorphism G + 1 = G^2 + X + 1 by (5). Composing
these, we get our isomorphism
G^2 = G + 1.
Golden! I'll stop here.
Quote of the week:
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and people, hurricanes and galaxies, and the heart of musical scales and
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Pyramid centuries before, man has employed the Golden Ratio to create his
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Cardas Audio speaker cable advertisement

Addendum: The computer scientist Sebastiano Vigna pointed out this paper:
14) Paolo Boldi, Massimo Santini, and Sebastiano Vigna, Measuring with
jugs, or: what if mathematicians were asked to defuse bombs?, Theoret.
Comput. Sci. 2 (2002). Also available at http://vigna.dsi.unimi.it/papers.php
which shows that if you want to approximately measure an arbitrary amount
of water using only two jugs, it's best if they have capacity 1 and G.
This paper cites a a charming result by Swierczkowski which picks up where
a famous theorem due to Dedekind leaves off. Dedekind showed that if x is
any irrational number, the numbers nx mod 1 are uniformly distributed in
the interval [0,1]. But if x = 1/G, these numbers have an especially
nice property: each new point in the sequence (nx mod 1) lands in one
of the longest intervals not containing a previous point! And, it chops
this interval in a golden way.
Stephen Schanuel said some things about "week203" on the category
theory mailing list, so I'll include his post here along with various
replies, concluding with my own.
...........................................................................
From: Stephen Schanuel
Subject: categories: mystification and categorification
Date: Thu, 4 Mar 2004 00:44:46 0500
I was unable to understand John Baez' golden object problem, nor his
description of the solutions. He refuses to tell us what 'nice' means,
but let me at least propose that to be 'tolerable' a solution must be an
object in a category, and John doesn't tell us what category is involved
in either of the solutions; at least I couldn't find a specification of
the objects, nor the maps, so I found the descriptions 'intolerable', in
the technical sense defined above. He is very generous, allowing one to
use a category with both plus and times as extra monoidal structures.
(Does anyone know an example of interest in which the plus is not
coproduct?) This freedom is unnecessary; a little algebra plus Robbie
Gates' theorem provides a solution G to G^2=G+1 which satisfies no
additional equations, in an extensive category (with coproduct as plus,
cartesian product as times).
Briefly, here it is. A primitive fifth root of unity z is a root of
the polynomial 1+X+X^2+X^3+X^4, hence satisfies 1+z+z^2+z^3+z^4+z=z,
which is of the 'fixed point' form p(z)=z with p in N[X] and p(0) not
0. Gates' theorem then says that the free distributive category
containing an object Z and an isomorphism from p(Z) to Z is extensive,
and its Burnside rig B (of isomorphism classes of objects) is, as one
would hope, N[X]/(p(X)=X); that is, Z satisfies no unexpected
equations. Since the degree of p is greater than 1, an easy general
theorem tells us (from the joint injectivity of the Euler and dimension
homomorphisms) that two polynomials agree at the object Z if and only if
either they are the same polynomial or both are nonconstant and they
agree at the number z.Now the 'algebra': the golden number is 1+z+z^4.
So G=1+Z+Z^4 satisfies G^2=G+1, as desired. It satisfies no
unexpected equations, because the relation X^2=X+1 reduces any
polynomial in N[X] to a linear polynomial, and these reduced forms have
distinct Euler characteristics, i.e. differ at z. Thus the homomorphism
from N[X]/(X^2=X+1) to B (sending X to G) is injective, and that's all
I wanted.
Since in the category of sets, any nasty old infinite set satisfies
the golden equation and many others, I have taken the liberty of
interpreting 'nice' to mean at least 'satisfying no unexpected
equations'. One could ask for more; the construction above has produced
a distributive, but not extensive, category whose Burnside rig is
N[X]/(X^2=X+1), the full subcategory with objects polynomials in G.
(If it were extensive, it would be closed under taking summands, but
every object in the larger category is a summand of G.) I don't know
whether there is an extensive category with N[X]/(X^2=X+1) as its full
Burnside rig; perhaps one or both of the examples John mentioned would
do, if I knew what they were.
While I'm airing my confusions, can anyone tell me what
'categorification' means? I don't know any such process; the simplest
exanple, 'categorifying' natural numbers to get finite sets, seems to me
rather 'remembering the finite sets and maps which gave rise to natural
numbers by the abstraction of passing to isomorphism classes'.
Finally, a note to John: While you're trying to give your audience
some feeling for the virtues of ncategories, couldn't you give them a
little help with n=1, by being a little more precise about objects and
maps?
Greetings to all, and thanks for your patience while I got this stuff
off my chest,
Steve Schanuel
..........................................................................
From: David Yetter
Subject: categories: Re: mystification and categorification
Date: Fri, 5 Mar 2004 10:55:26 0600
Categorification is a bit like quantization: it isn't a construction so much
as a desideratum for a relationship between one thing and another (in the
case of categorification an (n+1)categorical structure and an ncategorical
structure; in the case of quantization a quantum mechanical system and
a classical mechanical system).
Categorification wants to find a higherdimensional categorical structure
corresponding to a lowerdimensional one by weakening equations to
natural isomorphisms and imposing new, sensible, coherence conditions.
In general, for the original purpose for which it was proposedconstructions
of TQFT's and models of quantum gravityone wants the highest categorical
level to have a linear structure (hence Baez wanting tensor product
and a sum it distributes over, rather than cartesian product and coproduct).
Specific lowerdimensional categories with structure are 'categorified' by
finding a higherdimensional category with the new structure which 'lies over'
the lower dimensional one in the way an additive monoidal category lies
over its Grothendieck rig.
For instance any (klinear) monoidal category with monoid of isomorphism
classes M is a categorification of M, and more generally (klinear) monoidal
categories are a categorification of monoids.
A simple example shows why it is not a construction: commutative monoids
(as rather special categories with one object) admit two different
categorifications: symmetric monoidal categories and braided monoidal
categories (each regarded as a kind of bicategory with one object).
There is a good argument for regarding braided monoidal categories
as the 'correct' categorification: the EckmannHilton theorem ('a group
in GROUPS is an abelian group' or, really as the proof shows, 'a monoid
in MONOIDS is a commutative monoid') 'categorifies' to: A monoidal category
in MONCAT is a braided monoidal category.
..........................................................................
From: Vaughan Pratt
Subject: categories: Re: mystification and categorification
Date: Fri, 05 Mar 2004 22:49:56 0800
>While I'm airing my confusions, can anyone tell me what
>'categorification' means? I don't know any such process; the simplest
>exanple, 'categorifying' natural numbers to get finite sets, seems to me
>rather 'remembering the finite sets and maps which gave rise to natural
>numbers by the abstraction of passing to isomorphism classes'.
A fair question. I attended John's Coimbra lectures on this stuff in 1999
but a lot of it leaked out afterwards. If I had to guess I'd say he was
categorifying the free monoid on one generator to make it a monoidal category,
but then how did the monoid end up as coproduct and the generator as the
final object? One suspects some free association thereJohn, how *do*
you make that connection?
With regard to categorification in general, sets seem to play a central
role in at least one development of category theory. The homobjects of
"ordinary" categories are homsets. (In that sense I guess "ordinary" must
entail "locally small.") 2categories are what you get if instead you let
them be homcats, suitably elaborated.
Going in the other direction, if you take homsets to be vacuous, not
in the sense that they are empty but rather that they are all the same,
then you get sets. One more step in that direction makes everything look
the same, which may have something to do with the Maharishi Yogi hiring
category theorists for the math dept. of his university in Fairfield, Iowa.
(When I spoke last with the MY's "Minister of World Health," an MD who like
Ross Street was a classmate of mine but eight years earlier starting in 1957,
the entire conversation seemed to be largely a skirting of the minefield
of the sameness of everything, which may subconsciously have been behind my
obscure reply to Peter Freyd's posting a while back about unique existence
going back to Descartes, where I tried to oneup him by claiming it went
*much* further back.)
Categorification isn't the only way to get to 2categories, which can be
understood instead in terms of the interchange law as a twodimensional
associativity principle. However John has got a lot of mileage out of
the categorification approach, which one can't begrudge in an era where
mileage and minutes are as integral to a balanced life as one's checkbook.
(Q: How many minutes in a month? A: Depends on your plan.)
>Since in the category of sets, any nasty old infinite set satisfies
>the golden equation and many others, I have taken the liberty of
>interpreting 'nice' to mean at least 'satisfying no unexpected
>equations'.
Quite right. I would add to this "and satisfying the expected equations."
The "nasty sets" of which Steve speaks fail to satisy such expected
equations as 2^2^X ~ X. (The power set of a set is a Boolean algebra,
for heaven's sake. Why on earth forget that structure prior to taking the
second exponentiation? Set theorists seem to think that they can simply
forget structure without paying for it, but in the real world it costs
kT/2 joules per element of X to forget that structure. If set theorists
aren't willing to pay realworld prices in their modeling, why should we
taxpayers pay them realworld salaries? Large cardinals are a figment of
their overactive imaginations, and the solution to consistency concerns is
not to go there.)
Vaughan Pratt
.........................................................................
From: Tom Leinster
Subject: Re: categories: mystification and categorification
Date: 07 Mar 2004 20:50:39 +0000
Steve Schanuel wrote:
> a category with both plus and times as extra monoidal structures.
> (Does anyone know an example of interest in which the plus is not
> coproduct?)
Here are two examples that I've come across previously of rig categories
in which the plus is not coproduct:
(i) the category of finite sets and bijections, with + and x inherited
from the category of sets;
(ii) discrete rig categories, which are of course the same thing as
rigs.
> This freedom is unnecessary; a little algebra plus Robbie
> Gates' theorem provides a solution G to G^2=G+1 which satisfies no
> additional equations, in an extensive category (with coproduct as plus,
> cartesian product as times).
If you *do* allow yourself the freedom to use any rig category then an
even simpler solution exists, also satisfying no additional equations:
just take the rig freely generated by an element G satisfying G^2 = G +
1 and regard it as a discrete rig category.
> Since in the category of sets, any nasty old infinite set satisfies
> the golden equation and many others, I have taken the liberty of
> interpreting 'nice' to mean at least 'satisfying no unexpected
> equations'.
I'd interpret "nice" differently. (Apart from anything else, the
trivial example in my previous paragraph would otherwise make the golden
object problem uninteresting.) "Nice" as I understand it is not a
precise term  at least, I don't know how to make it precise. Maybe I
can explain my interpretation by analogy with the equation T = 1 + T^2.
A nice solution T would be the set of finite, binary, planar trees
together with the usual isomorphism T ~> 1 + T^2; a nasty solution
would be a random infinite set T with a random isomorphism to 1 + T^2.
(Both these solutions are in the rig category Set with its standard +
and x.) I regard the first solution as nice because I can see some
combinatorial content to it (and maybe, at the back of my mind, because
it has a constructive feel), and the second as nasty because I can't.
I'm not certain what I think of the solution given by the set of
notnecessarilyfinite binary planar trees (nice?), or by the set of
binary planar trees of cardinality at most aleph_5 (probably nasty).
Maybe the finding of a "nice" solution is similar in spirit to the
finding of a "concrete interpretation" of a combinatorial identity. As
an extremely simple example, consider the identity saying that each
entry in Pascal's triangle is the sum of the two above it,
(n+1 choose r) = (n choose r1) + (n choose r).
This is a doddle to prove, but all the same you'd be missing something
if you didn't know the standard "concrete interpretation": choosing r
objects out of n+1 objects amounts to EITHER choosing the first one and
then choosing r1 of the remaining n OR ... . Even if the challenge of
finding a "nice solution" or "concrete interpretation" isn't made
precise, I think there is a shared sense of what would count as an
answer, and finding an answer is in general not straightforward.
Best wishes,
Tom
..........................................................................
From: John Baez
Subject: golden objects
Date: Sun, 7 Mar 2004 12:50:29 0800 (PST)
Dear Categorists 
Sorry to take a while to respond. People at UCR have been unable to
receive posts on the category theory mailing list, due to problems with
our internet connection.
I'd asked for some nice examples of an object G in a rig category
equipped with an isomorphism from G^2 to G + 1. Steve Schanuel replied:
>I was unable to understand John Baez' golden object problem, nor his
>description of the solutions. He refuses to tell us what 'nice' means, [...]
The problem was deliberately openended, but you seem to have
understood it perfectly, since you've given a nice solution,
including a precise specification of what you consider "nice".
Let me repeat the two solutions given by Robin Houston:
1) The first solution works in any rig category having an object H
equipped with an isomorphism to H^2 + 4H + 1. The solution is to take
G = H + 2.
I described how Houston uses the isomorphism H > H^2 + 4H + 1 to
construct an isomorphism G^2 > G + 1.
What's nice about this is that it reduces a problem that's not
obviously of fixedpoint form to one that is.
2) Houston's second solution works in any monoidal cocomplete category,
tensor product distributing over colimits, that contains an object X
equipped with an isomorphism to 2X + 1. The solution is to let G be
the object of "binary planar rooted trees with Xlabelled leaves", i.e.
G = X + X^2 + 2X^3 + 5X^4 + 14X^5 + 42X^6 + ...
where the coefficients are Catalan numbers. He uses the obvious
isomorphism G > G^2 + X to construct an isomorphism G^2 > G + 1.
What's nice about this is that it shows Propp's originally proposed
golden object really is one: just take the category of sigmapolytopes
with its cartesian product, and let X be the open interval! And,
it makes precise the sense in which the alternating sum of Catalan
numbers equals the golden ratio.
Steve writes:
>I don't know whether there is an extensive category with N[X]/(X^2=X+1)
>as its full Burnside rig; perhaps one or both of the examples John
>mentioned would do, if I knew what they were.
I think example 1) does the job if we take the free distributive
category on an object H equipped with an isomorphism to H^2 + 4H + 1.
Right?
Steve also writes:
>He is very generous, allowing one to use a category with both plus
>and times as extra monoidal structures. (Does anyone know an example
>of interest in which the plus is not coproduct?) This freedom is
>unnecessary [...]
It's unnecessary, but handy: I think there's also an golden object in
the rig category of reps of quantum SU(2) at a suitable value of q.
Here the tensor product is not cartesian.
In the lingo of quantum group theory, this object has "quantum dimension"
equal to the golden number. It's interesting how such nonintegral but
algebraic "dimensions" show up naturally in quantum group theory,
just as nonintegral but algebraic "cardinalities" show up in the theory
of distributive categories.
I don't know any golden objects in rig categories where the plus is
not coproduct, and I agree that such rig categories arise less often
than those where times is not product. But, if you use the obvious
way of making the groupoid of finite sets into a rig category, + isn't
coproduct, nor is x product.
> While I'm airing my confusions, can anyone tell me what
> 'categorification' means? I don't know any such process; the simplest
> exanple, 'categorifying' natural numbers to get finite sets, seems to me
> rather 'remembering the finite sets and maps which gave rise to natural
> numbers by the abstraction of passing to isomorphism classes'.
You're right: categorification is not a systematic process!
I explained this idea back in "week121", and also in my paper
"Categorification", at http://www.arXiv.org/abs/math.QA/9802029.
Here's what I said.
If one studies categorification one soon discovers an amazing fact: many
deepsounding results in mathematics are just categorifications of facts
we learned in high school! There is a good reason for this. All along,
we have been unwittingly `decategorifying' mathematics by pretending
that categories are just sets. We `decategorify' a category by
forgetting about the morphisms and pretending that isomorphic objects
are equal. We are left with a mere set: the set of isomorphism classes
of objects.
To understand this, the following parable may be useful. Long ago, when
shepherds wanted to see if two herds of sheep were isomorphic, they
would look for an explicit isomorphism. In other words, they would line
up both herds and try to match each sheep in one herd with a sheep in
the other. But one day, along came a shepherd who invented
decategorification. She realized one could take each herd and `count'
it, setting up an isomorphism between it and some set of `numbers',
which were nonsense words like `one, two, three, ...' specially
designed for this purpose. By comparing the resulting numbers, she
could show that two herds were isomorphic without explicitly
establishing an isomorphism! In short, by decategorifying the category
of finite sets, the set of natural numbers was invented.
According to this parable, decategorification started out as a stroke
of mathematical genius. Only later did it become a matter of dumb
habit, which we are now struggling to overcome by means of
categorification. While the historical reality is far more
complicated, categorification really has led to tremendous progress
in mathematics during the 20th century. For example, Noether
revolutionized algebraic topology by emphasizing the importance of
homology groups. Previous work had focused on Betti numbers, which
are just the dimensions of the rational homology groups. As with
taking the cardinality of a set, taking the dimension of a vector
space is a process of decategorification, since two vector spaces are
isomorphic if and only if they have the same dimension. Noether noted
that if we work with homology groups rather than Betti numbers, we can
solve more problems, because we obtain invariants not only of spaces,
but also of maps.
In modern language, the nth rational homology is a *functor* defined
on the *category* of topological spaces, while the nth Betti number is
a mere *function*, defined on the *set* of isomorphism classes of
topological spaces. Of course, this way of stating Noether's insight
is anachronistic, since it came before category theory. Indeed, it
was in Eilenberg and Mac Lane's subsequent work on homology that
category theory was born!
Decategorification is a straightforward process which typically
destroys information about the situation at hand. Categorification,
being an attempt to recover this lost information, is inevitably
fraught with difficulties.
>Finally, a note to John: While you're trying to give your audience
>some feeling for the virtues of ncategories, couldn't you give them a
>little help with n=1, by being a little more precise about objects and
>maps?
I hope it's clearer now.
Best,
jb

Previous issues of "This Week's Finds" and other expository articles on
mathematics and physics, as well as some of my research papers, can be
obtained at
http://math.ucr.edu/home/baez/
For a table of contents of all the issues of This Week's Finds, try
http://math.ucr.edu/home/baez/twf.html
A simple jumpingoff point to the old issues is available at
http://math.ucr.edu/home/baez/twfshort.html
If you just want the latest issue, go to
http://math.ucr.edu/home/baez/this.week.html
ns, in an extensive category (with coproduct as plus,
> cartesian product as times).
If you *do* allow yourself the freedom to use any rig category then an
even simpler solution exists, also satisfying no additional equations:
just take the rig freely generated by an element G satisfying G^2 = G +
1 and regard it as a discrete rig category.
> Since in the category of sets, any nasty old infinite set satisfies
> the golden equation and many others, I have taktwf_ascii/week122000064400020410000157000000350531015243335300141250ustar00baezhttp00004600000001Also available at http://math.ucr.edu/home/baez/week122.html
June 24, 1997
This Week's Finds in Mathematical Physics  Week 122
John Baez
In summertime, academics leave the roost and fly hither and thither,
seeking conferences and conversations in farflung corners of the
world. At the end of May, everyone started leaving the Center
for Gravitational Physics and Geometry: Lee Smolin for the Santa Fe
Institute, Abhay Ashtekar for Uruguay and Argentina, Kirill Krasnov for
his native Ukraine, and so on. It got so quiet that I could actually
get some work done, were it not for the fact that I, too, flew the coop:
first for Chicago, then Portugal, and then to one of the most isolated,
technologically backwards areas on earth: my parents' house. Connected
to cyberspace by only the thinnest of threads, writing new issues of This
Week's Finds became almost impossible....
I did, however, read some newsgroups, and by this means Jim Carr
informed me that an article on spin foam models of quantum gravity had
appeared in Science News. I can't resist mentioning it, since it
quotes me:
1) Ivars Peterson, Loops of gravity: calculating a foamy quantum
spacetime, Science News, June 13, 1998, Vol. 153, No. 24, 376377.
It gives a little history of loop quantum gravity, spin networks,
and the new burst of interest in spin foams. Nothing very technical 
but good if you're just getting started. If you want something more
detailed, but still userfriendly, try Rovelli's new paper:
2) Carlo Rovelli and Peush Upadhya, Loop quantum gravity and quanta of
space: a primer, preprint available as grqc/9806079.
I haven't read it yet, since I'm still in a rather lowtech portion of
the globe, but it gives simplified derivations of some of the basic
results of loop quantum gravity, like the formula for the eigenvalues
of the area operator. As explained in "week110", one of the main
predictions of loop quantum gravity is that geometrical observables
such as the area of any surface take on a discrete spectrum of values,
much like the energy levels of a hydrogen atom. At first the
calculation of the eigenvalues of the area operator seemed rather
complicated, but by now it's wellunderstood, so Rovelli and Upadhya
are able to give a simpler treatment.
While I'm talking about the area operator, I should mention another
paper by Rovelli, in which he shows that its spectrum is not affected
by the presence of matter (or more precisely, fermions):
3) Carlo Rovelli and Merced Montesinos, The fermionic contribution
to the spectrum of the area operator in nonperturbative quantum
gravity, preprint available at grqc/9806120.
This is especially interesting because it fits in with other pieces
of evidence that fermions could simply be the ends of wormholes  an
old idea of John Wheeler (see "week109").
I should also mention some other good review articles that have turned
up recently. Rovelli has written a survey comparing string theory,
the loop representation, and other approaches to quantum gravity,
which is very good because it points out the flaws in all these approaches,
which their proponents are usually all too willing to keep quiet about:
4) Carlo Rovelli, Strings, loops and others: a critical survey of the
present approaches to quantum gravity. Plenary lecture on quantum
gravity at the GR15 conference, Pune, India, preprint available as
grqc/9803024.
Also, Loll has written a review of approaches to quantum gravity that
assume spacetime is discrete. It does *not* discuss the spin foam approach,
which is too new; instead it mainly talks about lattice quantum gravity,
the Regge calculus, and the dynamical triangulations approach. In lattice
quantum gravity you treat spacetime as a fixed lattice, usually a
hypercubical one, and work with discrete versions of the usual fields
appearing in general relativity. In the Regge calculus you triangulate
your 4dimensional spacetime  i.e., chop it into a bunch of 4dimensional
simplices  and use the lengths of the edges of these simplices as your
basic variables. (For more details see "week120".) In the dynamical
triangulations approach you also triangulate spacetime, but not in a
fixed way  you consider all possible triangulations. However, you
assume all the edges of all the simplices have the same length  the
Planck length, say. Thus all the information about the geometry of
spacetime is in the triangulation itself  hence the name "dynamical
triangulations". Everything becomes purely combinatorial  there are
no real numbers in our description of spacetime geometry anymore. This
makes the dynamical triangulations approach great for computer
simulations. Computer simulations of quantum gravity! Loll reports
on the results of a lot of these:
5) Renate Loll, Discrete approaches to quantum gravity in four dimensions,
preprint available as grqc/9805049, also available as a webpage on
Living Reviews in Relativity at
http://www.livingreviews.org/Articles/Volume1/199813loll/
By the way, "Living Reviews in Relativity" is a cool website run by
the AEI, the Albert Einstein Institute for gravitational physics,
located in Potsdam, Germany. The idea is that experts will write
review articles on various subjects and *keep them up to date* as new
developments occur. You can find this as follows:
6) Living Reviews in Relativity, http://www.livingreviews.org
Here are some other good places to learn about the dynamical triangulations
approach to quantum gravity:
7) J. Ambjorn, Quantum gravity represented as dynamical triangulations,
Class. Quant. Grav. 12 (1995) 20792134.
8) J. Ambjorn, M. Carfora, and A. Marzuoli, The Geometry of Dynamical
Triangulations, SpringerVerlag, Berlin, 1998. Also available
electronically as hepth/9612069  watch out, this is 166 pages long!
I can't resist pointing out an amusing relationship between dynamical
triangulations and mathematical logic, which Ambjorn mentions in his
review article. In computer simulations using the dynamical triangulations
approach, one wants to compute the average of certain quantities over all
triangulations of a fixed compact manifold  e.g., the 4dimensional sphere,
S^4. The typical way to do this is to start with a particular triangulation
and then keep changing it using various operations  "Pachner moves" 
that are guaranteed to eventually take you from any triangulation of a
compact 4dimensional manifold to any other.
Now here's where the mathematical logic comes in. Markov's theorem says
there is no algorithm that can decide whether or not two triangulations
are triangulations of the same compact 4dimensional manifold. (Technically,
by "the same" I mean "piecewise linearly homeomorphic", but don't worry
about that!) If they *are* triangulations of the same manifold, blundering
about using the Pachner moves will eventually get you from one to the other,
but if they are *not*, you may never know for sure.
On the other hand, S^4 may be special. It's an open question whether or
not S^4 is "algorithmically detectable". In other words, it's an open
question whether or not there's an algorithm that can decide whether or
not a triangulation is a triangulation of the 4dimensional sphere.
Now, suppose S^4 is *not* algorithmically detectable. Then the maximum
number of Pachner moves it takes to get between two triangulations of
the 4sphere must grow really fast: faster than any computable function!
After all, if it didn't, we could use this upper bound to know when to
give up when using Pachner moves to try to reduce our triangulation
to a known triangulation of S^4. So there must be "bottlenecks" that
make it hard to efficiently explore the set of all triangulations of S^4
using Pachner moves. For example, there must be pairs of triangulations
such that getting from one to other via Pachner moves requires going through
triangulations with a *lot* more 4simplices.
However, computer simulations using triangulations with up to 65,536
4simplices have not yet detected such "bottlenecks". What's going on?
Well, maybe S^3 actually *is* algorithmically recognizable. Or perhaps
it's not, but the bottlenecks only occur for triangulations that have
more than 65,536 4simplices to begin with. Interestingly, one dimension
up, it's known that the 5dimensional sphere is *not* algorithmically
detectable, so in this case bottlenecks *must* exist  but computer
simulations still haven't seen them.
I should emphasize that in addition to this funny computability stuff,
there is also a whole lot of interesting *physics* coming out of the
dynamical triangulations approach to quantum gravity. Unfortunately
I don't have the energy to explain this now  so read those review
articles, and check out that nice book by Ambjorn, Carfora and Marzuoli!
On another front... Ambjorn and Loll, who are both hanging out at the
AEI these days, have recently teamed up to study causality in a lattice
model of 2dimensional Lorentzian quantum gravity:
9) J. Ambjorn and R. Loll, Nonperturbative Lorentzian quantum
gravity, causality and topology change, preprint available as
hepth/9805108.
I'll just quote the abstract:
We formulate a nonperturbative lattice model of
twodimensional Lorentzian quantum gravity by performing the
path integral over geometries with a causal structure. The
model can be solved exactly at the discretized level. Its
continuum limit coincides with the theory obtained by
quantizing 2d continuum gravity in propertime gauge, but it
disagrees with 2d gravity defined via matrix models or
Liouville theory. By allowing topology change of the compact
spatial slices (i.e. baby universe creation), one obtains
agreement with the matrix models and Liouville theory.
And now for something completely different...
I've been hearing rumbles off in the distance about some interesting
work by Kreimer relating renormalization, Feynman diagrams, and Hopf
algebras. A friendly student of Kreimer named Mathias Mertens
handed me a couple of the basic papers when I was in Portugal:
10) Dirk Kreimer, Renormalization and knot theory, Journal of Knot
Theory and its Ramifications, 6 (1997), 479581. Preprint available
as qalg/9607022  beware, this is 103 pages long!
Dirk Kreimer, On the Hopf algebra structure of perturbative quantum
field theories, preprint available as qalg/9707029.
I'm looking through them but I don't really understand them yet.
The basic idea seems to be something like this. In quantum field
theory you compute the probability for some reaction among particles
by doing integrals which correspond in a certain way to pictures
called Feynman diagrams. Often these integrals give infinite answers,
which forces you to do a trick called renormalization to cancel the
infinities and get finite answers. Part of why this trick works is
that while your integrals diverge, they usually diverge at a welldefined
rate. For example, you might get something asymptotic to a constant
times 1/d^k, where d is the spatial cutoff you put in to get a finite
answer. And the constant you get here can be explicitly computed.
For example, it often involves numbers like zeta(n), where zeta is the
Riemann zeta function, much beloved by number theorists:
zeta(n) = 1/1^n + 1/2^n + 1/3^n + ....
Kreimer noticed that if you take the Feynman diagram and do some
tricks to turn it into a drawing of a knot or link, the constant
you get is related in interesting ways to the topology of this knot
or link! More complicated knots or links give fancier constants,
and there are all sorts of suggestive patterns. He worked out a
bunch of examples in the first paper cited above, and since then
people have worked out lots more, which you can find in the
references.
Apparently the secret underlying reason for these patterns comes
from the combinatorics of renormalization, which Kreimer was able
to summarize in a certain algebraic structure called a Hopf algebra.
Hopf algebras are important in both combinatorics and physics, so
perhaps this shouldn't be surprising. But there is still a lot of
mysterious stuff going on, at least as far as I can tell.
What's really intriguing about all this is *which* quantum field
theories Kreimer was studying when he discovered this stuff: *not*
topological quantum field theories like ChernSimons theory, which
already have wellunderstood relationship to knot theory, but instead,
field theories that ordinary particle physicists have been thinking
about for decades, like quantum electrodynamics, phi^4 theory in 4
dimensions, and phi^3 theory in 6 dimensions  field theories where
renormalization is a deadly serious business, thanks to nasty problems
like "overlapping divergences".
The idea that knot theory is relevant to *these* field theories
is exciting but also somewhat puzzling, since they don't live in
3dimensional spacetime the way ChernSimons theory does. People
familiar with ChernSimons theory have already been seeing fascinating
patterns relating knot theory, quantum field theory and number theory.
Is this new stuff related? Or is it something completely different?
Kreimer seems to think it's related.
According to Kirill Krasnov, the famous mathematician Alain Connes is
going around telling people to learn about this stuff. Apparently
Connes is now writing a paper on it with Kreimer, and it was Connes
who got the authors of this paper interested in the subject:
11) Thomas Krajewski and Raimar Wulkenhaar, On Kreimer's Hopf algebra
structure of Feynman graphs, preprint available as hepth/9805098.
Since I haven't plunged in yet, I'll just quote the abstract:
We reinvestigate Kreimer's Hopf algebra structure of perturbative
quantum field theories. In Kreimer's original work, overlapping
divergences were first disentangled into a linear combination
of disjoint and nested ones using the SchwingerDyson
equation. The linear combination then was tackled by the Hopf
algebra operations. We present a formulation where the
coproduct itself produces the linear combination, without
reference to external input.
With any luck, mathematicians will study this stuff and finally understand
renormalization!

Previous issues of "This Week's Finds" and other expository articles on
mathematics and physics, as well as some of my research papers, can be
obtained at
http://math.ucr.edu/home/baez/
For a table of contents of all the issues of This Week's Finds, try
http://math.ucr.edu/home/baez/twf.html
A simple jumpingoff point to the old issues is available at
http://math.ucr.edu/home/baez/twfshort.html
If you just want the latest issue, go to
http://math.ucr.edu/home/baez/this.week.html
mical triangulations approach great for computer
simulations. Computer simulations of quantum gravity! Loll reports
on the results of a lot of these:
5) Renate Loll, Discrete approaches to quantum gravity in four dimensions,
preprint available as grqc/9805049, also available as a webpage on
Living Reviews in Relativity at
http://www.livingreviews.org/Articles/Volume1/199813loll/
By the way, "Living Reviews in Relativity" is a cool website run by
the AEI, thtwf_ascii/week123000064400020410000157000000554661017751322000141370ustar00baezhttp00004600000001Also available at http://math.ucr.edu/home/baez/week123.html
September 19, 1998
This Week's Finds in Mathematical Physics (Week 123)
John Baez
It all started out as a joke. Argument for argument's sake. Alison
and her infuriating heresies.
"A mathematical theorem," she'd proclaimed, "only becomes true when a
physical system tests it out: when the system's behaviour depends in
some way on the theorem being *true* or *false*.
It was June 1994. We were sitting in a small paved courtyard,
having just emerged from the final lecture in a onesemester
course on the philosophy of mathematics  a bit of light relief
from the hard grind of the real stuff. We had fifteen minutes to
to kill before meeting some friends for lunch. It was a social
conversation  verging on mild flirtation  nothing more. Maybe
there were demented academics, lurking in dark crypts somewhere,
who held views on the nature of mathematical truth which they were
willing to die for. But were were twenty years old, and we *knew*
it was all angels on the head of a pin.
I said, "Physical systems don't create mathematics. Nothing
*creates* mathematics  it's timeless. All of number theory would
still be exactly the same, even if the universe contained nothing
but a single electron."
Alison snorted. "Yes, because even *one electron*, plus a space
time to put it in, needs all of quantum mechanics and all of
general relativity  and all the mathematical infrastructure they
entail. One particle floating in a quantum vacuum needs half the
major results of group theory, functional analysis, differential
geometry  "
"OK, OK! I get the point. But if that's the case... the events in
the first picosecond after the Big Bang would have `constructed'
every last mathematical truth required by *any* physical system,
all the way to the Big Cruch. Once you've got the mathematics
which underpins the Theory of Everything... that's it, that's all
you ever need. End of story."
"But it's not. To *apply* the Theory of Everything to a particular
system, you still need all the mathematics for dealing with *that
system*  which could include results far beyond the mathematics
the TOE itself requires. I mean, fifteen billion years after the
Big Bang, someone can still come along and prove, say... Fermat's
Last Theorem." Andrew Wiles at Princeton had recently announced
a proof of the famous conjecture, although his work was still being
scrutinised by his colleagues, and the final verdict wasn't yet in.
"Physics never needed *that* before."
I protested, "What do you mean, `before'? Fermat's Last Theorem
never has  and never will  have anything to do with any branch
of physics."
Alison smiled sneakily. "No *branch*, no. But only because the
class of physical systems whose behaviour depend on it is so
ludicrously specific: the brains of mathematicians who are trying
to validate the Wiles proof."
"Think about it. Once you start trying to prove a theorem, then
even if the mathematics is so `pure' that it has no relevance to
any other object in the universe... you've just made it relevant
to *yourself*. You have to choose *some* physical process to test
the theorem  whether you use a computer, or a pen and paper... or
just close your eyes and shuffle *neurotransmitters*. There's no
such thing as a proof which doesn't rely on physical events, and
whether they're inside or outside your skull doesn't make them
any less real."
And this is just the beginning... the beginning of Greg Egan's tale of
an inconsistency in the axioms of arithmetic  a "topological defect"
left over in the fabric of mathematics, much like the cosmic strings or
monopoles hypothesized by certain physicists thinking about the early
universe  and the mathematicians who discover it and struggle to
prevent a large corporation from exploiting it for their own nefarious
purposes. This is the title story of his new collection, "Luminous".
I should also mention his earlier collection of stories, named after a
sophisticated class of mindaltering nanotechnologies, the "axiomatics",
that affect specific beliefs of anyone who uses them:
1) Greg Egan, Axiomatic, Orion Books, 1995.
Greg Egan, Luminous, Orion Books, 1998.
Some of the stories in these volumes concern math and physics, such as
"The Planck Dive", about some farfuture explorers who send copies of
themselves into a black hole to study quantum gravity firsthand. One
nice thing about this story, from a pedant's perspective, is that Egan
actually works out a plausible scenario for meeting the technical
challenges involved  with the help of a little 23rdcentury technology.
Another nice thing is the further exploration of a world in which
everyone has long been uploaded to virtual "scapes" and can easily
modify and copy themselves  a world familiar to readers of his novel
"Diaspora" (see "week115"). But what I really like is that it's not
just a hardscience extravaganza; it's a meditation on mortality. You
can never really know what it's like to cross an event horizon unless
you do it....
Other stories focus on biotechnology and philosophical problems of
identity. The latter sort will especially appeal to everyone who
liked this book:
2) Daniel C. Dennett and Douglas R. Hofstadter, The Mind's I: Fantasies
and Reflections on Self and Soul, Bantam Books, 1982.
Among these, one of my favorite is called "Closer". How close can you
be to someone without actually *being them*? Would temporarily merging
identities with someone you loved help you understand them better?
Luckily for you pennypinchers out there, this particular story is
available free at the following website:
3) Greg Egan, Closer, http://www.eidolon.net/old_site/issue_09/09_closr.htm
Whoops! I'm drifting pretty far from mathematical physics, aren't I?
Selfreference has a lot to do with mathematical logic, but.... To
gently drift back, let me point out that Egan has a website in which he
explains special and general relativity in a nice, nontechnical way:
4) Greg Egan, Foundations,
http://www.netspace.net.au/~gregegan/FOUNDATIONS/index.html
Also, here are some interesting papers:
5) Gordon L. Kane, Experimental evidence for more dimensions reported,
Physics Today, May 1998, 1316.
Paul M. Grant, Researchers find extraordinarily high temperature
superconductivity in bioinspired nanopolymer, Physics Today, May
1998, 1719.
Jack Watrous, Ribosomal robotics approaches critical experiments;
government agencies watch with mixed interest, Physics Today, May
1998, 2123.
What these papers have in common is that they are all works of science
fiction, not science. They read superficially like straight science
reporting, but they are actually the winners of Physics Today's "Physics
Tomorrow" essay contest!
For example, Grant writes:
"Little's concept involved replacing the phonons  characterized by the
Debye temperature  with excitons, whose much higher characteristic
energies are on the order of 2 eV, or 23,000 K. If excitons were to
become the electronpairing `glue', superconductors with T_c's as high
as 500 K might be possible, even under weak coupling conditions. Little
even proposed a possible realization of the idea: a structure composed
of a conjugated polymer chain (polyene) dressed with highly polarizable
molecule (aromatics) as side groups. Simply stated, the polyene chain
would be a normal metal with a single mobile electron per CH molecular
unit; electrons on separate units would be paired by interacting with
the exciton field on the polarizable side groups."
Actually, I think this part is perfectly true  William A. Little
suggested this way to achieve hightemperature superconductivity back in
the 1960s. The science fiction part is just the description, later on
in Grant's article, of how Little's dream is actually achieved.
Okay, enough science fiction! Time for some real science! Quantum
gravity, that is. (Stop snickering, you skeptics....)
6) Laurent Freidel and Kirill Krasnov, Spin foam models and the
classical action principle, preprint available as hepth/9807092.
I described the spin foam approach to quantum gravity in "week113". But
let me remind you how the basic idea goes. A good way to get a handle
on this idea is by analogy with Feynman diagrams. In ordinary quantum
field theory there is a Hilbert space of states called "Fock space".
This space has a basis of states in which there are a specific number of
particles at specific positions. We can visualize such a state simply
by imagining a bunch of points in space, with labels to say which
particles are which kinds: electrons, quarks, and so on. One of the
main jobs of quantum field theory is to let us compute the amplitude for
one such state to evolve into another as time passes. Feynman showed
that we can do it by computing a sum over graphs in spacetime. These
graphs are called Feynman diagrams, and they represent "histories". For
example,
\u e/
\ /
\__W__/
/ \
/ \
/d nu\
would represent a history in which an up quark emits a W boson and turns
into a down quark, with the W being absorbed by an electron, turning it
into a neutrino. Time passes as you march down the page. Quantum field
theory gives you rules for computing amplitudes for any Feyman diagram.
You sum these amplitudes over all Feynman diagrams starting at one state
and ending at another to get the total amplitude for the given transition
to occur.
Now, where do these rules for computing Feynman diagram amplitudes come
from? They are not simply postulated. They come from perturbation
theory. There is a general abstract formula for computing amplitudes in
quantum field theory, but it's not so easy to use this formula in
concrete calculations, except for certain very simple field theories
called "free theories". These theories describe particles that don't
interact at all. They are mathematically tractable but physically
uninteresting. Uninteresting, that is, *except* as a startingpoint for
studying the theories we *are* interested in  the socalled "interacting
theories".
The trick is to think of an interacting theory as containing parameters,
called "coupling constants", which when set to zero make it reduce
to a free theory. Then we can try to expand the transition amplitudes
we want to know as a Taylor series in these parameters. As usual,
computing the coefficients of the Taylor series only requires us to to
compute a bunch of derivatives. And we can compute these derivatives
using the free theory! Typically, computing the nth derivative of some
transition amplitude gives us a bunch of integrals which correspond to
Feynman diagrams with n vertices.
By the way, this means you have to take the particles you see in Feynman
diagrams with a grain of salt. They don't arise purely from the
mathematics of the interacting theory. They arise when we *approximate*
that theory by a free theory. This is not an idle point, because we can
take the same interacting theory and approximate it by *different* free
theories. Depending on what free theory we use, we may say different
things about which particles our interacting theory describes! In
condensed matter physics, people sometimes use the term "quasiparticle"
to describe a particle that appears in a free theory that happens to be
handy for some problem or other. For example, it can be helpful to
describe vibrations in a crystal using "phonons", or waves of tilted
electron spins using "spinons". Condensed matter theorists rarely worry
about whether these particles "really exist". The question of whether
they "really exist" is less interesting than the question of whether the
particular free theory they inhabit provides a good approximation for
dealing with a certain problem. Particle physicists, too, have
increasingly come to recognize that we shouldn't worry too much about
which elementary particles "really exist".
But I digress! My point was simply to say that Feynman diagrams arise
from approximating interacting theories by free theories. The details
are complicated and in most cases nobody has ever succeeded in making
them mathematically rigorous, but I don't want to go into that here.
Instead, I want to turn to spin foams.
Everything I said about Feynman diagrams has an analogy in this approach
to quantum gravity. The big difference is that ordinary "free theories"
are formulated on a spacetime with a fixed metric  usually Minkowski
spacetime, with its usual flat metric. Attempts to approximate quantum
gravity by this sort of free theory failed dismally. Perhaps the
fundamental reason is that general relativity doesn't presume that
spacetime has a fixed metric  au contraire, it's a theory in which
the metric is the main variable!
So the idea of Freidel and Krasnov is to approximate quantum graivty
with a very different sort of "free theory", one in which the metric is
a variable. The theory they use is called "BF theory". I said a lot
about BF theory in "week36", but here the main point is simply that it's
a topological quantum field theory, or TQFT. A TQFT is a quantum field
theory that does not presume a fixed metric, but of a very simple sort,
because it has no local degrees of freedom. I very much like the idea
that a TQFT might serve as a novel sort of "free theory" for the purposes
of studying quantum gravity.
Everything that Freidel and Krasnov do is reminscent of familiar quantum
field theory, but also very different, because their startingpoint is
BF theory rather than a free theory of a traditional sort. For example,
just as ordinary quantum field theory starts out with Fock space, in the
spin network approach to quantum gravity we start with a nice simple
Hilbert space of states. But this space has a basis consisting, not of
collections of 0dimensional particles sitting in space at specified
positions, but of 1dimensional "spin networks" sitting in space. (For
more on spin networks, see "week55" and "week110".) And instead of
using 1dimensional Feynman diagrams to compute transition amplitudes,
the idea is now to use 2dimensional gadgets called "spin foams". The
amplitudes for spin foams are easy to compute in BF theory, because there
are a lot of explicit formulas using the socalled "Kauffman bracket", which
is an easily computable invariant of spin networks. So then the trick
is to use this technology to compute spin foam amplitudes for quantum
gravity.
Now, I shouldn't give you the wrong impression here. There are lots of
serious problems and really basic open questions in this work, and the
whole thing could turn out to be fatally flawed somehow. Nonetheless,
something seems right about it, so I find it very interesting.
Anyway, on to some other papers. I'm afraid I don't have enough energy
for detailed descriptions, because I'm busy moving into a new house, so
I'll basically just point you at them....
7) Abhay Ashtekar, Alejandro Corichi and Jose Antonio Zapata,
Quantum theory of geometry III: Noncommutativity of Riemannian
structures, preprint available as grqc/9806041.
This is the longawaited third part of a series giving a mathematically
rigorous formalism for interpreting spin network states as "quantum
3geometries", that is, quantum states describing the metric on
3dimensional space together with its extrinsic curvature (as it sits
inside 4dimensional spacetime). Here's the abstract:
"The basic framework for a systematic construction of a quantum
theory of Riemannian geometry was introduced recently. The quantum
versions of Riemannian structures  such as triad and area operators 
exhibit a noncommutativity. At first sight, this feature is
surprising because it implies that the framework does not admit a
triad representation. To better understand this property and to
reconcile it with intuition, we analyze its origin in detail. In
particular, a careful study of the underlying phase space is made and
the feature is traced back to the classical theory; there is no
anomaly associated with quantization. We also indicate why the
uncertainties associated with this noncommutativity become negligible
in the semiclassical regime."
In case you're wondering, the "triad" field is more or less what
mathematicians would call a "frame field" or "soldering form"  and
it's the same as the "B" field in BF theory. It encodes the information
about the metric in Ashtekar's formulation to general relativity.
Moving on to matters ncategorical, we have:
8) Andre Hirschowitz, Carlos Simpson, Descente pour les nchamps
(Descent for nstacks), approximately 240 pages, in French, preprint
available as math.AG/9807049.
Apparently this provides a theory of "nstacks"  the ncategorical
generalization of sheaves. Ever since Grothendieck's 600page letter to
Quillen (see "week35"), this has been the holy grail of ncategory
theory. Unfortunately I haven't mustered sufficient courage to force my
way through 240 pages of French, so I don't really know the details!
For the following two ncategory papers, exploring some themes close
to my heart, I'll just quote the abstracts:
9) Michael Batanin, Computads for finitary monads on globular sets,
preprint available at http://www.ics.mq.edu.au/~mbatanin/papers.html
"This work arose as a reflection on the foundation of higher
dimensional category theory. One of the main ingredients of any
proposed definition of weak ncategory is the shape of diagrams
(pasting scheme) we accept to be composable. In a globular approach
[due to Batanin] each kcell has a source and target (k1)cell. In
the opetopic approach of Baez and Dolan and the multitopic approach of
Hermida, Makkai and Power each kcell has a unique (k1)cell as
target and a whole (k1)dimensional pasting diagram as source. In
the theory of strict ncategories both source and target may be a
general pasting diagram.
The globular approach being the simplest one seems too restrictive to
describe the combinatorics of higher dimensional compositions. Yet, we
argue that this is a false impression. Moreover, we prove that this
approach is a basic one from which the other type of composable
diagrams may be derived. One theorem proved here asserts that the
category of algebras of a finitary monad on the category of nglobular
sets is *equivalent* to the category of algebras of an appropriate
monad on the special category (of computads) constructed from the data
of the original monad. In the case of the monad derived from the
universal contractible operad this result may be interpreted as the
equivalence of the definitions of weak ncategories (in the sense of
Batanin) based on the `globular' and general pasting diagrams. It may
be also considered as the first step toward the proof of equivalence
of the different definitions of weak ncategory.
We also develop a general theory of computads and investigate some
properties of the category of generalized computads. It turned out,
that in a good situation this category is a topos (and even a presheaf
topos under some not very restrictive conditions, the property firstly
observed by S. Schanuel and reproved by A. Carboni and P. Johnstone
for 2computads in the sense of Street)."
10) Tom Leinster, Structures in higherdimensional category theory,
preprint available at http://www.dpmms.cam.ac.uk/~leinster
"This is an exposition of some of the constructions which have arisen
in higherdimensional category theory. We start with a review of the
general theory of operads and multicategories. Using this we give an
account of Batanin's definition of ncategory; we also give an
informal definition in pictures. Next we discuss Graycategories and
their place in coherence problems. Finally, we present various
constructions relevant to the opetopic definitions of ncategory.
New material includes a suggestion for a definition of lax cubical
ncategory; a characterization of small Graycategories as the small
substructures of 2Cat; a conjecture on coherence theorems in higher
dimensions; a construction of the category of trees and, more
generally, of npasting diagrams; and an analogue of the BaezDolan
slicing process in the general theory of operads."
Okay  now for something completely different. In "week122" I said how
Kreimer and Connes have teamed up to write a paper relating Hopf
algebras, renormalization, and noncommutative geometry. Now it's out:
11) Alain Connes and Dirk Kreimer, Hopf Algebras, Renormalization and
Noncommutative Geometry, preprint available as hepth/9808042.
Also, here's an introduction to Kreimer's work:
12) Dirk Kreimer, How useful can knot and number theory be for loop
calculations?, Talk given at the workshop "Loops and Legs in Gauge
Theories", preprint available as hepth/9807125.
Switching over to homotopy theory and its offshoots... when I visited
Dan Christensen at Johns Hopkins this spring, he introduced me to all
the homotopy theorists there, and Jack Morava gave me a paper which
really indicates the extent to which newfangled "quantum topology" has
interbred with good old fashioned homotopy theory:
12) Jack Morava, Quantum generalized cohomology, preprint available
as math.QA/9807058 and http://hopf.math.purdue.edu/
Again, I'll just quote the abstract rather than venturing my own
summary:
"We construct a ring structure on complex cobordism tensored with the
rationals, which is related to the usual ring structure as quantum
cohomology is related to ordinary cohomology. The resulting object
defines a generalized two dimensional topological field theory taking
values in a category of spectra."
Finally, Morava has a student who gave me an interesting paper on
operads and moduli spaces:
13) Satyan L. Devadoss, Tessellations of moduli spaces and the mosaic operad,
preprint available as math.AG/9807010.
"We construct a new (cyclic) operad of `mosaics' defined by polygons
with marked diagonals. Its underlying (aspherical) spaces are the sets
Mbar_{0,n}(R) of real points of the moduli space of punctured
Riemann spheres, which are naturally tiled by Stasheff associahedra.
We (combinatorially) describe them as iterated blowups and show that
their fundamental groups form an operad with similarities to the operad
of braid groups."

Some things are so serious that one can only jest about them.  Niels Bohr.

Previous issues of "This Week's Finds" and other expository articles on
mathematics and physics, as well as some of my research papers, can be
obtained at
http://math.ucr.edu/home/baez/
For a table of contents of all the issues of This Week's Finds, try
http://math.ucr.edu/home/baez/twf.html
A simple jumpingoff point to the old issues is available at
http://math.ucr.edu/home/baez/twfshort.html
If you just want the latest issue, go to
http://math.ucr.edu/home/baez/this.week.html
So the idea of Freidel and Krasnov is to approximate quantum graivty
with a very different sort of "free theory", one in which the metric is
a variable. The theory they use is called "BF theory". Itwf_ascii/week124000064400020410000157000000412721015243355600141340ustar00baezhttp00004600000001Also available at http://math.ucr.edu/home/baez/week124.html
October 23, 1998
This Week's Finds in Mathematical Physics (Week 124)
John Baez
I'm just back from Tucson, where I talked a lot with my friend Minhyong
Kim, who teaches at the math department of the University of Arizona.
I met Minhyong in 1986 when I was a postdoc and he was a grad student at
Yale. At the time, strings were all the rage. Having recently found 5
consistent superstring theories, many physicists were giddy with
optimism, some even suggesting that the Theory of Everything would be
completed before the turn of the century. A lot of mathematicians were
going along for the ride, delighted by the beautiful and intricate
mathematical infrastructure: conformal field theory, vertex operator
algebras, and so on. Minhyong was considering doing his thesis on one
of these topics, so we spent a lot of time talking about mathematical
physics.
However, he eventually decided to work with Serge Lang on arithmetic
geometry. This is a branch of algebraic geometry where you work over
the integers instead of a field  especially important for Diophantine
equations. Personally, I was a bit disappointed. Perhaps it was
because I thought physics was more important than the decadent pleasures
of pure mathematics  or perhaps it was because it made it much less
likely that we'd ever collaborate on a paper.
However, a lot of the math Minhyong learned when studying string theory
is also important in arithmetic geometry. An example is the theory of
elliptic curves. Roughly speaking, an elliptic curve is a torus formed
taking a parallelogram in the complex plane and identifying opposite
edges.
You might wonder why something basically doughnutshaped is called an
elliptic curve! Let's clear that up right away. The "elliptic" part
comes from a relationship to elliptic functions, which generalize the
familiar trig functions from circles to ellipses. The "curve" part
comes from the fact that it takes one complex number z = x+iy to
describe your location on a surface with two real coordinates (x,y), so
showoffs like to say that a torus is onedimensional  one *complex*
dimension, that is!  hence a "curve". In short, you have to already
understand elliptic curves to know why the heck they're called elliptic
curves.
Anyway, why are elliptic curves important? On the one hand, they show
up all over number theory, like in Wiles' proof of Fermat's last
theorem. On the other hand, in string theory, a string traces out a
surface in spacetime called the string worldsheet, and points on this
surface are conveniently described using a single complex number, so
it's what those showoffs call a "curve"  and among the simplest
examples are elliptic curves!
If you're interested to see how Fermat's last theorem was reduced to a
problem about elliptic curves  the socalled ShimuraTaniyamaWeil
conjecture  you can look at the textbooks on elliptic curves listed
in "week13". But I won't say anything about this, since I don't
understand it. Instead, I want to talk about how elliptic curves
show up in string theory. For more on how these two applications
fit together, try:
1) Yuri I. Manin, Reflections on arithmetical physics, in Conformal
Invariance and String Theory, eds. Petre Dita and Vladimir Georgescu,
Academic Press, 1989.
Let me just quote the beginning:
"The development of theoretical physics in the last quarter of the
twentieth century is guided by a very romantic system of values.
Aspiring to describe fundamental processes at the Planck scale,
physicists are bound to lose any direct connection with the
observable world. In this social context the sophisticated
mathematics emerging in string theory ceases to be only a technical
tool needed to calculate some measurable effects and becomes a
matter of principle.
Today at least some of us are again nurturing an ancient Platonic
feeling that mathematical ideas are somehow predestined to describe
the physical world, however remote from reality their origins seem
to be.
From this viewpoint one should perversely expect number theory to
become the most applicable branch of mathematics."
I think this remark wisely summarizes both the charm and the dangers
of physics that relies more heavily on criteria of mathematical
elegance than of experimental verification.
Anyway, I don't want to get too deep into the theory of elliptic curves;
just enough so we see why the number 24 is so important in string
theory. You may remember that bosonic string theory works best in 26
dimensions (while the physically more important superstring theory,
which includes spin1/2 particles, works best in 10). Why is this true?
Well, there are various answers, but one is that if you think of the
string as wiggling in the 24 directions perpendicular to its own
2dimensional surface  two *real* dimensions, that is!  various
magical properties of the number 24 conspire to make things work out.
What are these magical properties of the number 24? Well,
1^2 + 2^2 + 3^2 + ... + 24^2
is itself a perfect square, and 24 is the only integer with this
property besides silly ones like 0 and 1. As described in "week95", this
has some very profound relationships to string theory. Unfortunately, I
don't know any way to deduce from this that bosonic string theory *works
best* in 26 dimensions.
One reason bosonic string theory works best in 26 dimensions is that
1 + 2 + 3 + .... = 1/12
and 2 x 12 = 24. Of course, this explanation is unsatisfactory in many
ways. First of all, you might wonder what the above equation means!
Doesn't the sum diverge???
Actually this is the *least* unsatisfactory feature of the explanation.
Although the sum diverges, you can still make sense of it. The Riemann
zeta function is defined by
zeta(s) = 1/1^s + 1/2^s + 1/3^s + ....
whenever the real part of s is greater than 1, which makes the sum
converge. But you can analytically continue it to the whole complex
plane, except for a pole at 1. If you do this, you find that
zeta(1) = 1/12.
Thus we may jokingly say that 1 + 2 + 3 + .... = 1/12. But the
real point is how the zeta function shows up in string theory, and
quantum field theory in general. (It's also big in number theory.)
Unfortunately, the details quickly get rather technical; one has to do
some calculations and so on. That's the really unsatisfactory part. I
want something that clearly relates strings and the number 24, something
so simple even a child could understand it, and which, when you work out
all the implications, implies that bosonic string theory only makes
sense in 26 dimensions. I don't expect a child to be able to figure
out all the implications... but I want the essence to be childishly
simple.
Here it is. Suppose the string worldsheet is an elliptic curve. Then we
can make it by taking a "lattice" of parallelograms in the complex plane:
*
*
*
*
*
*
*
*
*
and identifying each point in each parallelogram with the corresponding
points on all the others. This rolls the plane up into a torus. Now,
two lattices are more symmetrical than the rest. One of them is the
square lattice:
* * * *
* * * *
* * * *
which has 4fold rotational symmetry. The other is the lattice with
lots of equilateral triangles in it:
* * * *
* * *
* * * *
which has 6fold rotational symmetry. The magic property of the number
24, which makes string theory work so well in 26 dimensions, is that
4 x 6 = 24 !!!
Okay, great. But if you're anything like me, at this point you're
wondering how the heck this actually helps. Why should string theory
care about these specially symmetrical lattices? And why should we
*multiply* 4 and 6? So far everything I've said has been flashy but
insubstantial. Next week I'll fill in some of the details. Of course,
I'll need to turn up the sophistication level a notch or two.
In the meantime, you can read a bit more about this stuff in the
following article on Richard Borcherds, who won the Fields medal for his
work relating bosonic string theory, the Leech lattice in 24 dimensions,
and the Monster group:
2) W. Wayt Gibbs, Monstrous moonshine is true, Scientific American, November
1998, 4041. Also available at
http://www.sciam.com/1998/1198issue/1198profile.html
Gibbs asked me to come up with a simple explanation of the j invariant
for elliptic curves; you can judge how well I succeeded. For a more
detailed attempt to do the same thing, see "week66", which also has more
references on the Monster group. By the way, John McKay didn't actually
make his famous discovery relating the j invariant and Monster while
reading a 19thcentury book on elliptic modular functions; he says
"It was du Val's Elliptic Functions book in which j is expanded
incorrectly as a qseries  very much a 20th century book." Apart from
that, the article seems accurate, as far as I can tell.
If you really want to understand how elliptic curves are related to strings,
you need to learn some conformal field theory. For that, try:
4) Phillippe Di Francesco, Pierre Mathieu, and David Senechal, Conformal
Field Theory, Springer, 1997.
This is a truly wonderful tour of the subject. It's 890 pages long, but
it's designed to be readable by both mathematicians and physicists, so
you can look at the bits you want. It starts out with a 60page
introduction to quantum field theory and a 30page introduction to
statistical mechanics. The reason is that when we perform the
substitution called the "Wick transform":
it/hbar > k/T,
quantum field theory turns into statistical mechanics, and a nice
Lorentzian manifold may turn into a Riemannian manifold  in other
words, "spacetime" turns into "space". And this gives conformal field
theory a double personality.
First, conformal field theory studies quantum field theories in 2
dimensions that are invariant under all conformal transformations 
transformations that preserve angles but not necessarily lengths. These
are important in string theory because we can think of them as
transformations of the string worldsheet that preserve its complex
structure.
Secondly, if we do a Wick transform, these quantum field theories become
2dimensional *statistical mechanics* problems that are invariant under
all conformal transformations. This may seem an esoteric concern, but
thin films of material can often be treated as 2dimensional for all
practical purposes, and conformal invariance is typical at "critical
points"  boundaries between two or more phases for which there is no
latent heat, such as the boundary between the magnetized and
unmagnetized phases of a ferromagnet. In 2 dimensions, one can use
conformal field theory to thoroughly understand these critical points.
After this warmup, the book covers the fundamentals of conformal field
theory proper, including:
a) the idea of conformal invariance (which is especially powerful in 2
dimensions because then the group of conformal transformations is
infinitedimensional),
b) the free boson and fermion fields,
c) operator product expansions,
d) the Virasoro algebra (which is closely related to the Lie algebra of
the group of conformal transformations, and has a representation on the
Hilbert space of states of any conformal field theory),
e) minimal models (roughly, conformal field theories whose Hilbert space
is built from finitely many irreducible representations of the Virasoro
algebra),
f) the Coulombgas formalism (a way to describe minimal models in terms
of the free boson and fermion fields),
g) modular invariance (the study of conformal field theory on tori 
this is where the elliptic curves start sneaking into the picture,
dragging along with them the wonderful machinery of elliptic functions,
theta functions, the Dedekind zeta function, and so forth),
h) critical percolation (applying conformal field theory to systems
where a substance is trying to ooze through a porous medium, with
special attention paid to the critical point when the holes are *just*
big enough to let it ooze all the way through),
i) the 2dimensional Ising model (applying conformal field theory to
ferromagnets, with special attention paid to the critical point when
the temperature is *just* low enough for ferromagnetism to set in)
By now we're at page 486. I'm getting tired just summarizing this thing!
Anyway, the book then turns to conformal field theories having Lie group
symmetries: in particular, the socalled WessZuminoWitten or "WZW"
models. Pure mathematicians are free to join here, even amateurs,
because we are now treated to a wonderful 78page introduction to simple
Lie algebras, starting from scratch and working rapidly through all
sorts of fun stuff, skipping all the yucky proofs. Then we get a
54page introduction to affine Lie algebras, which are infinite
dimensional generalizations of the simple Lie algebras, and play a
crucial role in string theory. Finally, we get a detailed 143page
course on WZW models  which are basically conformal field theories
where your field takes values in a Lie group  and coset models  where
your field takes values in a Lie group modulo a subgroup. It sounds
like all minimal models can be described as coset models, though I'm
not quite sure.
Whew! Believe it or not, the authors plan a second volume! Anyway,
this is a wonderful book to have around. I was just about to buy a copy
in Chicago last spring  on sale for a mere $50  when I discovered I'd
lost my credit card. Sigh. The big ones always get away....
There are various formalisms for doing conformal field theory that
aren't covered in the above text. For example, the theory of "vertex
operator algebras", or "vertex algebras" is really popular among
mathematicians studying conformal field theory and the Monster group.
The standard definition of a vertex operator algebra is long and
complicated: it summarizes a lot of what you'd want a conformal field
theory to be like, but it's hard to learn to love it unless you already
know some *other* approaches to conformal field theory. There's another
definition using operads that's much nicer, which will eventually
catch on  some people complain that operads are too abstract, but
that's just hogwash. But anyway, there is a definite need for more
elementary texts on the subject. Here's one:
5) Victor Kac, Vertex Algebras for Beginners, American Mathematical
Society, University Lecture Series vol. 10, 1997.
And then of course there is string theory proper. How do you learn
that? There's always the bible by Green, Schwarz and Witten (see
"week118"), but a lot of stuff has happened since that was written.
Luckily, Joseph Polchinski has come out with a "new testament"; I
haven't seen it yet but physicists say it's very good:
6) Joseph Polchinski, String Theory, 2 volumes, Cambridge U. Press,
1998.
There are also other textbooks, of course. Here's one that's free if
you print it out yourself:
7) E. Kiritsis, Introduction to Superstring Theory, 244 pages,
to be published by Leuven University Press, preprint available
as hepth/9709062.
For a more mathematical approach, you might want to try this when it
comes out:
8) Quantum Fields and Strings: A Course for Mathematicians, eds.
P. Deligne, P. Etinghof, D. Freed, L. Jeffrey, D. Kazhdan, D. Morrison
and E. Witten, American Mathematical Society, to appear.
Finally, when you get sick of all this newfangled stuff and want
to read about the good old days when physicists predicted new
particles that actually wound up being *observed*, you can turn to
this book about Dirac and his work:
9) Abraham Pais, Maurice Jacob, David I. Olive, and Michael F. Atiyah,
Review of Paul Dirac: The Man and His Work, Cambridge U. Press, 1998.

Previous issues of "This Week's Finds" and other expository articles on
mathematics and physics, as well as some of my research papers, can be
obtained at
http://math.ucr.edu/home/baez/
For a table of contents of all the issues of This Week's Finds, try
http://math.ucr.edu/home/baez/twf.html
A simple jumpingoff point to the old issues is available at
http://math.ucr.edu/home/baez/twfshort.html
If you just want the latest issue, go to
http://math.ucr.edu/home/baez/this.week.html
t a child to be able to figure
out all the implications... but I want the essence to be childishly
simple.
Here it is. Suppose the string worldsheet is an elliptic curve. Then we
can make it by taking a "lattice" of parallelograms in the complex plane:
*
twf_ascii/week125000064400020410000157000000473560774011336400141470ustar00baezhttp00004600000001Also available at http://math.ucr.edu/home/baez/week125.html
November 3, 1998
This Week's Finds in Mathematical Physics (Week 125)
John Baez
Last week I promised to explain some mysterious connections between
elliptic curves, string theory, and the number 24. I claimed that
it all boils down to the fact that there are two especially symmetric
lattices in the plane, namely the square lattice:
* * * *
* * * *
* * * *
with 4fold symmetry, and the hexagonal lattice:
* * * *
* * *
* * * *
with 6fold symmetry. Now it's time for me to start backing up those
claims.
First I need to talk a bit about lattices and SL(2,Z). As I explained
in "week66", a lattice in the complex plane consists of all points that
are integer linear combinations of two complex numbers, say omega_1 and
omega_2. However, we can change these numbers without changing the
lattice by letting
omega_1' = a omega_1 + b omega_2
omega_2' = c omega_1 + d omega_2
where
a b
c d
is an invertible 2x2 matrix of integers whose inverse again consists of
integers. Usually it's good to require that our transformation preserve
the handedness of the basis (omega_1, omega_2), which means that this
matrix should have determinant 1. Such matrices form a group called SL(2,Z).
In the context of elliptic curves it's also called the "modular group".
Now associated to the square lattice is a special element of SL(2,Z)
that corresponds to a 90 degree rotation. Everyone calls it S:
0 1
S =
1 0
Associated to the hexagonal lattice is a special element of SL(2,Z)
that corresponds to a 60 degree rotation. Everyone calls it ST:
0 1
ST =
1 1
(See, there's a matrix they already call T, and ST is the product of S
and that one.) Now, you may complain that the matrix ST doesn't look
like a rotation, but you have to be careful! What I mean is, if you
take the hexagonal lattice and pick a basis for it like this:
omega_2
* * * *
omega_1
* 0* *
* * * *
then in *this* basis the matrix ST represents a 60 degree rotation.
So far this is pretty straightforward, but now come some surprises.
First, it turns out that SL(2,Z) is *generated* by S and ST. In other
words, every 2x2 integer matrix with determinant 1 can be written as a
product of a bunch of copies of S, ST, and their inverses. Second, all
the relations satisfied by S and ST follow from these obvious ones:
S^4 = 1
(ST)^6 = 1
together with
S^2 = (ST)^3
which holds because both sides describe a 180 degree rotation.
Right away this implies that SL(2,Z) has a certain inherent "12ness" to
it. Let me explain. SL(2,Z) is a nonabelian group  this is how
someone with a Ph.D. says that matrix multiplication doesn't commute 
but suppose we abelianize it by imposing extra relations *forcing*
commutativity. Then we get a group generated by S and ST, satisfying
the above relations together with an extra one saying that S and ST
commute. This is the group Z/12, which has 12 elements!
This "12ness" has a lot to do with the magic properties of the number
24 in string theory. But to see how this "12ness" affects string
theory, we need to talk about elliptic curves a bit more. It will take
forever unless I raise the mathematical sophistication level a little.
So....
We can define an elliptic curve to be a torus C/L formed by taking the
complex plane C and modding out by a lattice L. Since C is an abelian
group and L is a subgroup, this torus is an abelian group, but in the
theory of elliptic curves we consider it not just as a group but also as
a complex manifold. Thus two elliptic curves C/L and C/L' are
considered isomorphic if there is a complexanalytic function from one
to the other that's also an isomorphism of groups. This happens
precisely when there is a nonzero number z such that zL = L', or in
other words, whenever L' is a rotated and/or dilated version of L.
There's a wonderful space called the "moduli space" of elliptic
curves: each point on it corresponds to an isomorphism class of
elliptic curves. In physics, we think of each point in it as
describing the geometry of a torusshaped string worldsheet. Thus in
the pathintegral approach to string theory we need to integrate over
this space, together with a bunch of other moduli spaces corresponding
to string worldsheets with different topologies. All these moduli
spaces are important and interesting, but the moduli space of elliptic
curves is a nice simple example when you're first trying to learn this
stuff. What does this space look like?
Well, suppose we have an elliptic curve C/L. We can take our lattice
L and describe it in terms of a righthanded basis (omega_1, omega_2).
For the purposes of classifying the describing the elliptic curve up
to isomorphism, it doesn't matter if we multiply these basis elements
by some number z, so all that really matters is the ratio
tau = omega_2/omega_1
Since our basis was righthanded, tau lives in the upper halfplane,
which people like to call H.
Okay, so now we have described our elliptic curve in terms of a
complex number tau lying in H. But the problem is, we could have
chosen a different righthanded basis for our lattice L and gotten a
different number tau. We've got to think about that. Luckily, we've
already seen how we can change bases without changing the lattice: we
just apply a matrix in SL(2,Z), getting a new basis
omega_1' = a omega_1 + b omega_2
omega_2' = c omega_1 + d omega_2
This has the effect of changing tau to
tau' = (a tau + b)/(c tau + d)
If you don't see why, figure it out  you've gotta understand this to
understand elliptic curves!
Anyway, two numbers tau and tau' describe isomorphic elliptic curves
if and only if they differ by the above sort of transformation. So
we've figured out the moduli space of elliptic curves: it's the
quotient space H/SL(2,Z), where SL(2,Z) acts on H as above!
Now, the quotient space H/SL(2,Z) is not a smooth manifold, because
while the upper halfplane H is a manifold and the group SL(2,Z) is
discrete, the action of SL(2,Z) on H is not free: i.e., certain points
in H don't move when you hit them with certain elements of SL(2,Z).
If you don't see why this causes trouble, think about a simpler
example, like the group G = Z/n acting as rotations of the complex
plane, C. Most points in the plane move when you rotate them, but the
origin doesn't. The quotient space C/G is a cone with its tip
corresponding to the origin. It's smooth everywhere except the tip,
where it has a "conical singularity". The moral of the story is that
when we mod out a manifold by a group of symmetries, we get a space
with singularities corresponding to especially symmetrical points in
the original manifold.
So we expect that H/SL(2,Z) has singularities corresponding to points in
H corresponding to especially symmetrical lattices. These, of course,
are our friends the square and hexagonal lattices!
But let's be a bit more careful. First of all, *nothing* in H moves when
you hit it with the matrix 1. But that's no big deal: we can just
replace the group SL(2,Z) by
PSL(2,Z) = SL(2,Z)/{+1}
Since 1 doesn't move *any* points of H, the action of SL(2,Z) on H
gives an action of PSL(2,Z), and the moduli space of elliptic curves is
H/PSL(2,Z).
Now most points in H aren't preserved by any element of PSL(2,Z).
However, certain points are! The point
tau = i
corresponding to the square lattice, is preserved by S and all its
powers. And the point
tau = exp(2 pi i/3)
corresponding to the hexagonal lattice, is preserved by ST and all its
powers. These give rise to two conical singularities in the moduli
space of elliptic curves. Away from these points, the moduli space is
smooth.
Lest you get the wrong impression, I should hasten to reassure you that
the moduli space is not all that complicated: it looks almost like the
complex plane! There's a famous onetoone and onto function from the
moduli space to the complex plane: it's called the "modular function"
and denoted by j. So the moduli space is *topologically* just like the
complex plane; the only difference is that it fails to be *smooth* at
two points, where there are conical singularities.
This may seem a bit hard to visualize, but it's actually not too hard.
Here's one way. Start with the region in the upper halfplane outside
the unit circle and between the vertical lines x = 1/2 and x = 1/2.
It looks sort of like this:
.....................
.....................
.....................
.....................
.....................
.....................
.....................
.....................
..........A..........
..... .....
... ...
. .
B B'
Then glue the vertical line starting at B to the one starting at B',
and glue the arc AB to the arc AB'. We get a space that's smooth
everywhere except at the points A and B = B', where there are conical
singularities. The total angle around the point A is just 180 degrees
 half what it would be if the moduli space were smooth there. The
total angle around B is just 120 degrees  one third what it would be
if the moduli space were smooth there.
The reason this works is that the region shown above is a "fundamental
domain" for the action of PSL(2,Z) on H. In other words, every
elliptic curve is isomorphic to one where the parameter tau lies in
this region. The point A is where tau = i, and the point B is where
tau = exp(2 pi i/3).
Now let's see where the "12ness" comes into this picture. Minhyong
Kim explained this to me in a very nice way, but to tell you what he
said, I'll have to turn up the level of mathematical sophistication
another notch. (Needless to say, all the errors will be mine.)
So, I'll assume you know what a "complex line bundle" is  this is
just another name for a 1dimensional complex vector bundle. Locally
a section of a complex line bundle looks a lot like a complexvalued
function, but this isn't true globally unless your line bundle is
trivial. If you aren't careful, sometimes you may *think* you have a
function defined on a space, only to discover later that it's actually
a section of a line bundle. This sort of thing happens all the time
in physics. In string theory, when you're doing path integrals on
moduli space, you have to make sure that what you're integrating is
really a function! So it's important to understand all the line bundles
on moduli space.
Now, given any sort of space, we can form the set of all isomorphism
classes of line bundles over this space. This is actually an abelian
group, since when we tensor two line bundles we get another line
bundle, and when you tensor any line bundle with its dual, you get the
trivial line bundle, which plays the role of the multiplicative
identity for tensor products. This group is called the "Picard
group" of your space.
What's the Picard group of the moduli space of elliptic curves? Well,
when I said "any sort of space" I was hinting that there are all sorts
of spaces  topological spaces, smooth manifolds, algebraic varieties,
and so on  each one of which comes with its own particular notion of
line bundle. Thus, before studying the Picard group of moduli space
we need to decide what context we're going to work in! As a mere
*topological space*, we've seen that the moduli space of elliptic
curves is indistinguishable from the plane, and every *topological*
line bundle over the plane is trivial, so in *this* context the Picard
group is the trivial group  boring!
But the moduli space is actually much more than a mere topological
space. It's not a smooth manifold, but it's awfully close: it's the
quotient of the smooth manifold H by the discrete group SL(2,Z), and
its singularities are pretty mild in nature.
Somehow we should take advantage of this when defining the Picard
group of the moduli space. One way to do so involves the theory of
"stacks". Without getting into the details of this theory, let me
just vaguely sketch what it does for us here. For a much more careful
treatment, with more of an algebraic geometry flavor, try:
1) David Mumford, Picard groups of moduli problems, in Arithmetical
Algebraic Geometry, ed. O. F. G. Schilling, Harper and Row, New York,
1965.
Suppose a discrete group G acts on a smooth manifold X. A
"Gequivariant" line bundle on X is a line bundle equipped with an
action of G that gets along with the action of G on X. If G acts
freely on X, a line bundle on X/G is the same as a Gequivariant line
bundle on X. This isn't true when the action of G on X isn't free.
But we can still go ahead and *define* the Picard group of X/G to be
the group of isomorphism classes of Gequivariant line bundles on X.
Of course we should say something to let people know that we're using
this funny definition. In our example, people call it the Picard
group of the moduli *stack* of elliptic curves.
So what's this group, anyway?
Well, it turns out that you can get any SL(2,Z)equivariant line
bundle on H, up to isomorphism, by taking the trivial line bundle on H
and using a 1dimensional representation of SL(2,Z) to say how it acts
on the fiber. So we just need to understand 1dimensional
representations of SL(2,Z). The set of isomorphism classes of these
forms a group under tensor product, and this is the group we're after.
Well, a 1dimensional representation of a group always factors through
the abelianization of that group. We saw the abelianization of
SL(2,Z) was Z/12. But everyone knows that the group of 1dimensional
representations of Z/n is again Z/n  this is called Pontryagin duality.
So: the Picard group of the moduli stack of elliptic curves is Z/12.
So we see again an inherent "12ness" built into the theory of
elliptic curves! You may be wondering how this makes the number 24 so
important in string theory. In particular, where does that extra
factor of 2 come from? I'll say a little more about this next Week.
I may or may not manage to tie together the loose ends!
You may also be wondering about "stacks". In this you're not alone.
There's an amusing passage about stacks in the following book:
2) Joe Harris and Ian Morrison, Moduli of Curves, SpringerVerlag, New
York, 1998.
They write:
"Of course, here I'm working with the moduli stack rather than
with the moduli space. For those of you who aren't familiar with
stacks, don't worry: basically, all it means is that I'm allowed to
pretend that the moduli space is smooth and that there's a universal
family over it."
Who hasn't heard these words, or their equivalent, spoken in a
talk? And who hasn't fantasized about grabbing the speaker by
the lapels and shaking him until he says what  exactly  he means
by them? But perhaps you're now thinking that all that is in the
past, and that at long last you're going to learn what a stack is
and what they do.
Fat chance.
Actually Mumford's paper cited above gives a nice introduction to
the theory of stacks without mentioning the dreaded word "stack".
Alternatively, you can wait and read this book when it comes out:
3) K. Behrend, L. Fantechi, W. Fulton, L. Goettsche and A. Kresch,
An Introduction to Stacks, in preparation.
But let me just briefly say a bit about stacks and the moduli stack of
elliptic curves in particular. A stack is a weak sheaf of categories.
For this to make sense you must already know what a sheaf is! In the
simplest case, a sheaf over a topological space, the sheaf S gives you
a set S(U) for each open set U, and gives you a function S(U,V): S(U)
> S(V) whenever the open set U is contained in the open set V. These
functions must satisfy some laws. The notion of "stack" is just a
categorification of this idea. That is, a stack S over a topological
space gives you a *category* S(U) for each open set U, and gives you a
*functor* S(U,V): S(U) > S(V). These functors satisfy the same laws
as before, but *only up to specified natural isomorphism*. And these
natural isomorphisms must in turn satisfy some new laws of their own,
socalled coherence laws.
In the case at hand there's a stack over the moduli space of elliptic
curves. For any open set U in the moduli space, an object of S(U) is a
family of elliptic curves over U, such that each elliptic curve in the
family sits over the point in moduli space corresponding to its
isomorphism class. Similarly, a morphism in S(U) is a family of
isomorphisms of elliptic curves. This allows us to keep track of the
fact that some elliptic curves have more automorphisms than others! And
it takes care of the funny stuff that happens at the singular points in
the moduli space.
By the way, this watereddown summary leaves out a lot of the
algebraic geometry that you usually see when people talk about stacks.
Finally, one more thing  it looks like Kreimer and company are making
great progress on understanding renormalization in a truly elegant way.
D. J. Broadhurst and D. Kreimer, Renormalization automated by Hopf algebra,
preprint available as hepth/9810087.
Let me quote the abstract:
"It was recently shown that the renormalization of quantum field theory
is organized by the Hopf algebra of decorated rooted trees, whose
coproduct identifies the divergences requiring subtraction and whose
antipode achieves this. We automate this process in a few lines of
recursive symbolic code, which deliver a finite renormalized expression
for any Feynman diagram. We thus verify a representation of the operator
product expansion, which generalizes Chen's lemma for iterated
integrals. The subset of diagrams whose forest structure entails a
unique primitive subdivergence provides a representation of the Hopf
algebra H_R of undecorated rooted trees. Our undecorated Hopf algebra
program is designed to process the 24,213,878 BPHZ contributions to the
renormalization of 7,813 diagrams, with up to 12 loops. We consider 10
models, each in 9 renormalization schemes. The two simplest models
reveal a notable feature of the subalgebra of Connes and Moscovici,
corresponding to the commutative part of the Hopf algebra H_T of the
diffeomorphism group: it assigns to Feynman diagrams those weights which
remove zeta values from the counterterms of the minimal subtraction
scheme. We devise a fast algorithm for these weights, whose squares are
summed with a permutation factor, to give rational counterterms."

Previous issues of "This Week's Finds" and other expository articles on
mathematics and physics, as well as some of my research papers, can be
obtained at
http://math.ucr.edu/home/baez/
For a table of contents of all the issues of This Week's Finds, try
http://math.ucr.edu/home/baez/twf.html
A simple jumpingoff point to the old issues is available at
http://math.ucr.edu/home/baez/twfshort.html
If you just want the latest issue, go to
http://math.ucr.edu/home/baez/this.week.html
glue the arc AB to the arc AB'. We get a space that's smooth
everywhere except at the points A and B = B', where there are conical
singularities. The total angle around the point A is just 180 degrees
 half what it would be if the moduli space were smooth there. The
totwf_ascii/week126000064400020410000157000000427231106131450600141310ustar00baezhttp00004600000001Also available at http://math.ucr.edu/home/baez/week126.html
November 17, 1998
This Week's Finds in Mathematical Physics (Week 126)
John Baez
To round off some things I said in the previous two weeks, let me
say a bit more about string theory and Euler's mysterious equation
1 + 2 + 3 + .... = 1/12.
For this I'll need to assume a nodding acquaintance with quantum field
theory.
There are two complementary ways to attack almost any problem in
quantum field theory: the Lagrangian approach, also known as "path
integral quantization", and the Hamiltonian approach, also called
"canonical quantization". Let me describe string theory from both
viewpoints. I'll only talk about bosonic string theory, because my goal
is to sketch why it works best in 26dimensional spacetime, and because
it's simpler than superstring theory. Also, I'll only talk about
closed strings.
Classically, such a string is simply a map from a closed surface into
spacetime. In the Lagrangian approach to quantization, we start by
choosing a formula for the action. We use the simplest possibility,
namely the *area* of the surface. Of course, to define the area of a
surface in spacetime, we need the spacetime to have a metric. The
simplest thing is to work with ndimensional Minkowski spacetime, so
let's do that.
We find the equations of motion of the string by extremizing the action.
These equations imply that if we watch the string in space as time
passes, it acts like collection of loops made of perfectly elastic
material. These loops vibrate, split and join as time passes.
It's perhaps a bit easier to see how the strings vibrate if we go over
to the Hamiltonian approach. This is a bit subtle, because string theory
has an enormous amount of "gauge symmetry"  by which physicists mean
any symmetry that arises from the ability to switch between different
mathematical descriptions of what counts as the same physical situation.
There's a recipe to figure out the gauge symmetries of any theory
starting from the action. Applying this to string theory, it turns out
that two maps from a surface into spacetime count as "physically the
same" if they differ only by a reparametrization of the surface that's
being mapped into spacetime.
When going over to the Hamiltonian approach, we have to deal with this
gauge symmetry. There are different ways to deal with it  but we
can't just ignore it. Suppose we use the approach called "lightcone
gaugefixing". This amounts to choosing a parametrization of our
surface so that the 2 coordinates on it are related in a simple way to
2 of the coordinates on ndimensional Minkowski space. We can do this
because of the reparametrization gauge symmetry. But once we've done
it, we no longer have any more freedom to reparametrize our surface. In
short, we've squeezed all the juice out of our gauge symmetry: this is
what "gaugefixing" is all about.
We started by studying a map from a surface S into ndimensional
spacetime, which we can think of as field on S with n components.
However, in lightcone gauge, 2 components of this field are given by
simple formulas in terms of the rest. This lets us think of our string
as a field X with only n2 components. And when we do this, it
satisfies the simplest equation you could imagine! Namely, the wave
equation
(d^2/dt^2  d^2/dx^2) X(t,x) = 0.
This is same equation that describes an idealized violin string. The
only difference is that now, instead of a segment of violin string, we
have a bunch of closed loops of string. The energy, or Hamiltonian,
is also given by the usual wave equation Hamiltonian:
H = (1/2) integral [(dX/dt)^2 + (dX/dx)^2] dx.
The first term represents the kinetic energy of the string, while
the second represents its potential energy  the energy it has due
to being stretched.
Henceforth I'll ignore the fact that loops of string can split or join,
and only talk about the vibrations of a single loop of string. Using
the linearity of the wave equation, we can decompose any solution of the
wave equation into sine waves moving in either direction  socalled
"leftmovers" and "rightmovers"  together with a solution of the form
X(t,x) = A + Bt
which describes the motion of the string's center of mass. The
leftmovers and rightmovers don't interact with each other or
with the centerofmass motion, so we can learn a lot just by studying
the rightmovers.
For starters, suppose the field X has just one component. Then the
rightmoving vibrational modes look like
X(t,x) = A sin(ik(tx)) + B cos(ik(tx))
with frequencies k = 1,2,3,.... Abstractly, each of these vibrational
modes is just like a harmonic oscillator of frequency k, so we can think
of the string as a big collection of harmonic oscillators.
Now suppose we quantize our string  or more precisely, the rightmoving
modes. By what we've said, this just amounts to quantizing a bunch of
harmonic oscillators, one of frequency k for each natural number k. This
is great, since the harmonic oscillator is one of the easiest physical
systems to quantize!
As you may know, the quantum harmonic oscillator has discrete energy
levels with energies k/2, 3k/2, 5k/2,.... (Here I'm working in units
where hbar = 1; otherwise I'd need a factor of hbar.) In particular,
the energy of the lowestenergy state is called the "zeropoint energy"
or "vacuum energy". It usually doesn't hurt much to subtract this off
by redefining the Hamiltonian, but sometimes it's important.
Now, what's the total zeropoint energy of all the rightmoving modes?
To figure this out, we add up the zeropoint energy k/2 for all
frequencies k = 1,2,3,..., obtaining
(1 + 2 + 3 + .... )/2.
Of course this is divergent, but there are lots of sneaky tricks for
assigning values to divergent series, so let's not be disheartened!
Euler figured out such a trick for calculating the sum 1 + 2 + 3 + ....,
and he got the value 1/12. If we momentarily assume this makes sense,
then the total zeropoint energy works out to be
1/24 !!!
More generally, if we have a string in ndimensional Minkowski spacetime,
the field X has n2 components, so the total zeropoint energy is
(n2)/24
Now, for other reasons, it turns out that string theory works best when
this zeropoint energy is 1. This is a bit tricky to explain, but it
has to do with the subtleties of gaugefixing in quantum field theory.
Things that work nicely at the classical level can easily screw up at
the quantum level; in particular, symmetries of a classical theory can
be lost when you quantize. One has to really check that the lightcone
gauge fixing doesn't screw up the Lorentzinvariance of string theory.
It turns out that it *does* screw it up unless the zeropoint energy of
the rightmovers is 1. So bosonic string theory works best when
(n2)/24 = 1
or in other words, when n = 26.
You really shouldn't take my word for this stuff! You can find more
details around pages 9596 in volume 1 of the following book:
1) Michael B. Green, John H. Schwarz and Edward Witten, Superstring Theory,
2 volumes, Cambridge University Press.
There's a lot I should say to fill in the details, but the most urgent
matter is to explain Euler's mysterious formula
1 + 2 + 3 + .... = 1/12
As I said in "week124", this is an example of zeta function regularization.
The Riemann zeta function is defined by
zeta(s) = 1/1^s + 1/2^s + 1/3^s + ....
when the sum converges, but it analytically continues to values of s where
the sum doesn't converge. If we do the analytic continuation, we get
zeta(1) = 1/12.
Proving this rigorously is a bit of work. One way is to use the
"functional equation" for the Riemann zeta function, which says that
F(s) = F(1s)
where
F(s) = pi^{s/2} Gamma(s/2) zeta(s)
and Gamma is the famous function with Gamma(n) = (n1)! for n = 1,2,3,...
and Gamma(s+1) = s Gamma(s) for all s. Using
Gamma(1/2) = sqrt(pi)
and
zeta(2) = pi^2/6,
the functional equation implies zeta(1) = 1/12. But of course you have
to prove the functional equation! A nice exposition of this can be found
in:
2) Neal Koblitz, Introduction to Elliptic Curves and Modular Forms,
2nd edition, SpringerVerlag, 1993.
I don't know Euler's original argument that zeta(1) = 1/12. However,
Dan Piponi recently gave the following "physicist's proof" on the
newsgroup sci.physics.research. Let D be the differentiation operator:
D = d/dx
Then Taylor's formula says that translating a function to the left by
a distance c is the same as applying the operator e^{cD} to it, since
e^{cD} f = f + cf' + (c^2/2!)f" + ....
Using some formal manipulations we obtain
f(0) + f(1) + f(2) + .... = [(1 + e^D + e^{2D} + .... )f](0)
= [(1/(1  e^D)) f](0)
or if F is an integral of f, so that DF = f,
f(0) + f(1) + f(2) + .... = [(D/(1  e^D)) F](0)
This formula can be made rigorous in certain contexts, but now we'll
throw rigor to the winds and apply it to the function f(x) = x, obtaining
1 + 2 + 3 + .... = [(D/(1  e^D)) F](0)
where F(x) = x^2/2. To finish the job, we work out the beginning of
the Taylor series for D/(1  e^D). The coefficients of this are
closely related to the Bernoulli numbers, and this could easily lead us
into further interesting digressions, but all we need to know is
D/(1  e^D) = 1 + D/2  D^2/12 + ....
Applying this operator to F(x) = x^2/2 and evaluating the result at
x = 0, the only nonzero term comes from the D^2 term in the power
series, so we get
1 + 2 + 3 + .... = [(D^2/12) F](0) = 1/12
Voila!
By the way, after he came up with this proof, Dan Piponi found an
almost identical proof in the following book:
3) G. H. Hardy, Divergent Series, Chelsea Pub. Co., New York, 1991.
Now let me change gears. Besides the Riemann zeta function, there are a
lot of other special functions that show up in the study of elliptic
curves. Unsurprisingly, many of them are also important in string
theory. For example, consider the partition function of bosonic string
theory. What do I mean by a "partition function" here? Well, whenever
we have a quantum system with a Hamiltonian H, its partition function is
defined to be
Z(b) = trace(exp(bH))
where b > 0 is the inverse temperature. This function is fundamental to
statistical mechanics, for reasons that I'm too lazy to explain here.
Before tackling the bosonic string, let's work out the partition function
for a quantum harmonic oscillator. To keep life simple, let's subtract
off the zeropoint energy so the energy levels are 0, k, 2k, and so on.
Mathematically, these energy levels are just the eigenvalues of the
harmonic oscillator Hamiltonian, H. Thus the eigenvalues of exp(bH)
are 1, exp(bk), exp(2bk), etc. The trace of this operator is just
the sum of its eigenvalues, so we get
Z(b) = 1 + exp(bk) + exp(2bk) + ...
= 1/(1  exp(bk))
This was first worked out by Planck, who assumed the harmonic oscillator
had discrete, evenly spaced energy levels and computed its partition
function as part of his struggle to understand the thermodynamics of
the electromagnetic field.
Okay, now let's do the bosonic string. To keep life simple we again
subtract off the zeropoint energy. Also, we'll consider only the
rightmoving modes, and we'll start by assuming the field X describing
the vibrations of the string has only one component. As we saw before,
the string then becomes the same as a collection of quantum harmonic
oscillators with frequencies k = 1, 2, 3, and so on. We've seen that
the oscillator with frequency k has partition function 1/(1  exp(bk)).
To get the partition function of a quantum system built from a bunch of
noninteracting parts, you multiply the partition functions of the parts
(since the trace of a tensor product of operators is the product of their
traces). So the partition function of our string is
product_{k=1}^infinity 1/(1  exp(bk))
So far, so good. But now suppose we take the zeropoint energy into
account. We do this by subtracting 1/24 from the Hamiltonian of the
string, which has the effect of multiplying its partition function by
exp(b/24). Thus we get
Z(b) = exp(b/24) product_{k=1}^infinity 1/(1  exp(bk))
Lo and behold: the reciprocal of the Dedekind eta function!
What's that, you ask? It's a very important function in the theory of
elliptic curves. People usually write it as a function of q = exp(b),
like this:
eta(q) = q^{1/24} product_{k=1}^infinity (1  q^k)
But to see the relation to elliptic curves we should switch variables
yet again and write q = exp(2 pi i tau). I already talked about the
variable tau in "week125", where we were studying the elliptic curve
formed by curling up a parallelogram like this in the complex plane:
tau tau + 1
* *
* *
0 1
In physics, this elliptic curve is just one possibility for the shape of
a surface traced out by a string. The number 1 says how far the surface
goes in the *space* direction before it loops around, and the number tau
says how far it goes in the *time* direction before it loops around!
(The idea of "looping around in time" may seem bizarre, but it's very
important in physics. It turns out that studying the statistical
mechanics of a system at a given inverse temperature is the same as
studying Euclidean quantum field theory on a spacetime where time is
periodic with a given period. This idea is what relates the variables
b and tau.)
Now as I explained in "week13", the above elliptic curve is not just
an abstract torusshaped thingie. We can also think of it as the set of
complex solutions of the following cubic equation in two variables:
y^2 = 4x^3  g_2 x  g_3
where the numbers g_2 and g_3 are certain functions of tau. Moreover,
this equation defines an elliptic curve whenever the polynomial on
the righthand side doesn't have repeated roots. So among other things,
elliptic curves are really just a way of studying cubic equations!
But when does 4x^3  g_2 x  g_3 have repeated roots? Precisely
when the "discriminant"
Delta = g_2^3  27 g_3^2
equals zero. This is just the analog for cubics of the more familiar
discriminant for quadratic equations.
Now for the cool part: there's an explicit formula for the discriminant
in terms of the variable tau. And it involves the 24th power of the
Dedekind eta function! Here it is:
Delta = (2 pi)^{12} eta^24 !!!
If you haven't seen this before, it should seem *amazing* that the
discriminant of a cubic equation can be computed using the 24th power of
a partition function that shows up in string theory. Of course that's
not how it went historically: Dedekind discovered his eta function long
before strings came along. What's really happening is that string
theory is exploiting special features of complex curves.
If I have the energy, next time I'll give you a better explanation of
why bosonic string theory works best in 26 dimensions, using some
special properties of the Dedekind eta function.
Meanwhile, if you want to see some pictures of the Dedekind eta function,
together with some cool formulas it satisfies, try these:
4) Mathworld, Dedekind eta function,
http://mathworld.wolfram.com/DedekindEtaFunction.html
5) Wikipedia, Dedekind eta function,
http://en.wikipedia.org/wiki/Dedekind_eta_function

Quote of the Week:
Dear Sir,
I am very much gratified on perusing your letter of the 8th
February 1913. I was expecting a reply from you similar to the one
which a Mathematics Professor at London wrote asking me to study
carefully Bromwich's Infinite Series and not fall into the pitfall
of divergent series. I have found a friend in you who views
my labors sympathetically. This is already some encouragement to
me to proceed with an onward course. I find in many a place in your
letter rigourous proofs are required and so on and you ask me to
communicate the method of proof. If I had given you my methods of
proof I am sure you will follow the London Professor. But as a fact
I did not give him any proof but made some assertions as the following
under my new theory. I told him that the sum of an infinite number
of terms in the series 1 + 2 + 3 + 4 + ... = 1/12 under my theory.
If I tell you this you will at once point out to me the lunatic
asylum as my goal.
 Srinivasa Ramanujan's second letter to G. H. Hardy

Previous issues of "This Week's Finds" and other expository articles on
mathematics and physics, as well as some of my research papers, can be
obtained at
http://math.ucr.edu/home/baez/
For a table of contents of all the issues of This Week's Finds, try
http://math.ucr.edu/home/baez/twf.html
A simple jumpingoff point to the old issues is available at
http://math.ucr.edu/home/baez/twfshort.html
If you just want the latest issue, go to
http://math.ucr.edu/home/baez/this.week.html
zeta(s) = 1/1^s + 1/2^s + 1/twf_ascii/week127000064400020410000157000000555421034052324300141340ustar00baezhttp00004600000001Also available at http://math.ucr.edu/home/baez/week127.html
November 30, 1998
This Week's Finds in Mathematical Physics (Week 127)
John Baez
If you like pi, take a look at this book:
1) Lennart Berggren, Jonathan Borwein and Peter Borwein, Pi: A Source
Book, SpringerVerlag, New York, 1997.
It's full of reprints of original papers about pi, from the Rhind
Papyrus right on up to the 1996 paper by Bailey, Borwein and Plouffe in
which they figured out how to compute a given hexadecimal digit of pi
without computing the previous digits  see "week66" for more about
that. By the way, Colin Percival has recently used this technique to
compute the 5 trillionth binary digit of pi! (It's either zero or one,
I forget which.) Percival is a 17year old math major at Simon Fraser
University, and now he's leading a distributed computation project to
calculate the quadrillionth binary digit of pi. Anyone with a Pentium
or faster computer using Windows 95, 98, or NT can join. For more
information, see:
2) PiHex project, http://www.cecm.sfu.ca/projects/pihex/pihex.html
Anyway, the above book is *full* of fun stuff, like a onepage proof
of the irrationality of pi which uses only elementary calculus, due
to Niven, and the following weirdly beautiful formula due to Euler,
which unfortunately is not explained:
3 5 7 11 13 17 19
pi/2 =  x  x  x  x  x  x  x ...
2 6 6 10 14 18 18
Here the numerators are the odd primes, and the denominators are the
closest numbers of the form 4n+2.
Since I've been learning about elliptic curves and the like lately, I
was also interested to see a lot of relations between pi and modular
functions. For example, the book has a reprint of Ramanujan's paper
"Modular equations and approximations to pi", in which he derives a
bunch of bizarre formulas for pi, some exact but others approximate,
like this:
12
pi ~  ln[ (2 sqrt(2) + sqrt(10)) (3 + sqrt(10)) ]
sqrt(190)
which is good to 18 decimal places. The strange uses to which genius
can be put know no bounds!
Okay, now I'd like to wrap up my little story about why bosonic string
theory works best in 26 dimensions. This time I want to explain how
you do the path integral in string theory. Most of what I'm about to
say comes from some papers that my friend Minhyong Kim recommended
to me when I started pestering him about this stuff:
3) JeanBenoit Bost, Fibres determinants, determinants regularises, et
mesures sur les espaces de modules des courbes complexes, Asterisque
152153 (1987), 113149.
4) A. A. Beilinson and Y. I. Manin, The Mumford form and the Polyakov
measure in string theory, Comm. Math. Phys. 107 (1986), 359376.
For a more elementary approach, try chapters IX and X.4 in this book:
5) Charles Nash, Differential Topology and Quantum Field Theory,
Academic Press, New York, 1991.
As I explained in "week126", a string traces out a surface in spacetime
called the "string worldsheet". Let's keep life simple and assume the
string worldsheet is a torus and that spacetime is Euclidean R^n. Then
to figure out the expectation value of any physical observable, we need
to calculate its integral over the space of all maps from a torus to R^n.
To make this precise, let's use X to denote a map from the torus to R^n.
Then a physical observable will be some function f(X), and its expectation
value will be
(1/Z) integral f(X) exp(S(X)) dX
Here S(X) is the action for string theory, which is just the *area* of
the string worldsheet. The number Z is a normalizing factor called the
partition function:
Z = integral exp(S(X)) dX
But there's a big problem here! As usual in quantum field thoery, the
space we're trying to integrate over is infinitedimensional, so the
above integrals have no obvious meaning. Technically speaking, the
problem is that there's no Lebesgue measure "dX" on an infinite
dimensional manifold.
Mathematicians might throw up our hands in despair and give up at this
point. But physicists take a more pragmatic attitude: they just keep
massaging the problem, breaking rules here and there if necessary, until
they get something manageable. Physicists call this "calculating the
path integral", but from a certain viewpoint what they're really doing
is *defining* the path integral, since it only has a precise meaning
after they're done.
In the case at hand, it was Polyakov who figured out the right massage:
6) A. M. Polyakov, Quantum geometry of bosonic strings, Phys. Lett. B103
(1981), 207.
He rewrote the above integral as a double integral: first an integral
over the space of metrics g on the torus, and then inside, for each
metric, an integral over maps X from the torus into spacetime. Of
course, any such map gives a metric on the torus, so this double
integral is sort of redundant. However, introducing this redundancy
turns out not to hurt. In fact, it helps!
To keep life simple, let's just talk about the partition function
Z = integral exp(S(X)) dX
If we can handle this, surely we can handle the integral with the
observable f(X) in it. Polyakov's trick turns the partition function
into a double integral:
Z = integral (integral exp() dX) dg
where "Laplacian" is the Laplacian on the torus and the angle brackets
stand for the usual inner product of R^nvalued functions, both defined
using the metric g.
At first glance Polyakov's trick may seem like a step backwards: now we
have two illdefined integrals instead of one! However, it's actually
a step forward. Now we can do the inside integral by copying the
formula for the integral of a Gaussian of finitely many variables  a
standard trick in quantum field theory. Ignoring an infinite constant
that would cancel later anyway, the inside integral works out to be:
(det Laplacian)^{1/2}
But wait! The Laplacian here is a linear operator on the vector space
of R^nvalued functions on the torus. This is an infinitedimensional
vector space, so we can't blithely talk about determinants the way we
can in finite dimensions. In finite dimensions, the determinant of a
selfadjoint matrix is the product of its eigenvalues. But the Laplacian
has infinitely many eigenvalues, which keep getting bigger and bigger.
How do we define the product of all its eigenvalues?
Of course the lowest eigenvalue of the Laplacian is zero, and you might
think that would settle it: the product of them all must be zero! But
that would make the above expression meaningless, and we are not going to
give up so easily. Instead, we will simply ignore the zero eigenvalue!
That way, we only have to face the product of all the *rest*.
(Actually there's something one can do which is slightly more careful
than simply ignoring the zero eigenvalue, but I'll talk about that later.)
Okay, so now we have a divergent product to deal with. Well, in
"week126" I used a trick called zeta function regularization to make
sense of a divergent sum, and we can use that trick here to make sense
of our divergent product. Suppose we have a selfadjoint operator A
with a discrete spectrum consisting of positive eigenvalues. Then the
"zeta function" of A is defined by:
Zeta(s) = tr(A^{s})
To compute Zeta(s) we just take all the eigenvalues of A, raise them to
the s power, and add them up. For example, if A has eigenvalues
1,2,3,..., then Zeta(s) is just the usual Riemann zeta function, which
we already talked about in "week126".
This stuff doesn't quite apply if A is the Laplacian on a compact
manifold, since one of its eigenvalues is zero. But we have already
agreed to throw out the zero eigenvalue, so let's do that when defining
Zeta(s) as a sum over eigenvalues. Then it turns out that the sum
converges when the real part of s is positive and large. Even better,
there's a theorem saying that Zeta(s) can be analytically continued to
s = 0. Thus we can use the following trick to define the determinant of
the Laplacian.
Suppose that A is a selfadjoint matrix with positive eigenvalues. Then
Zeta(s) = tr(exp(s ln A))
Differentiating gives
Zeta'(s) = tr(ln A exp(s ln A))
and setting s to zero we get
Zeta'(0) = tr(ln A).
But there's a nice little formula saying that
det(A) = exp(tr(ln A))
so we get
det(A) = exp(Zeta'(0)).
Now we can use this formula to *define* the determinant of the Laplacian
on a compact manifold! Sometimes this is called a "regularized determinant".
So  where are we? We used Polyakov's trick to write the partition
function of our torusshaped string as
Z = integral (integral exp() dX) dg,
then we did the inside integral and got this:
Z = integral (det Laplacian)^{1/2} dg
and then we figured out a meaning for the determinant here.
What next? Well, since the Laplacian on R^nvalued functions is the
direcct sum of n copies of the Laplacian on realvalued functions, we
expect that
(det Laplacian)^{1/2} = (det laplacian)^{n/2}
where "laplacian" stands for the Laplacian on ordinary realvalued
functions on the torus. One can actually check this rigorously using
the definition in terms of zeta functions. That's reassuring: at
least *one* step of our calculation is rigorous! So we get
Z = integral (det laplacian)^{n/2} dg
Great. But we are not out of the woods yet. We still have an integral
over the space of metrics to do  another nasty infinitedimensional
integral.
Time for another massage!
Look at this formula again:
Z = integral (integral exp() dX) dg
The Laplacian depends on the metric g, and so does the inner product.
However, the combination depends only on the "conformal
structure"  i.e., it doesn't change if we multiply the metric by a
positiondependent scale factor. It also doesn't change under
diffeomorphisms.
Now the space of conformal structures on a torus modulo diffeomorphisms
is something familiar: it's just the moduli space of elliptic curves!
We figured out what this space looks like in "week125". It's
finitedimensional and there's a nice way to integrate over it, called
the WeilPetersson measure. So we can hope to replace the outside
integral  the integral over metrics  by an integral over this moduli
space.
Indeed, we could hope that
Z = integral (integral exp() dX) d[g] [hope!]
where now the outside integral is over moduli space and d[g] is
the WeilPetersson measure. The hope, of course, is that the
stuff on the inside is welldefined as a function on moduli space.
Actually this hope is a bit naive. Even though
doesn't change if we recale the metric, the whole inside integral
integral exp() dX
*does* change. This may seem odd, but remember, we did a lot of
hairraising manipulations before we even got this integral to mean
anything! We basically wound up *defining* it to be
(det Laplacian)^{1/2},
and one can check that this *does* change when we rescale the metric.
This problem is called the "conformal anomaly".
Are we stuck? No! Luckily, there is *another* problem, which cancels
this one when n = 26. They say two wrongs don't make a right, but with
anomalies that's often the only way to get things to work....
So what's this other problem? It's that we shouldn't just replace the
measure dg by the measure d[g] as I did in my naive formula for the
partition function. We need to actually figure out the relation between
them. Of course this is hard to do, because the measure dg doesn't
exist, rigorously speaking! Still, if we do a bunch more hairraising
heuristic manipulations, which I will spare you, we can get a formula
relating dg and and d[g], and using this we get
Z = integral (integral exp() dX) f(g) d[g]
where f(g) is some function of the metric. There's a perfectly explicit
formula for this function, but your eyeballs would fall out if I showed
it to you. Anyway, the real point is that IN 26 DIMENSIONS AND ONLY IN 26
DIMENSIONS, the integrand
(integral exp() dX) f(g)
is invariant under rescalings of the metric (as well as diffeomorphisms).
In other words, the conformal anomaly in
integral exp() dX
is precisely canceled by a similar conformal anomaly in f(g), so their
product is a welldefined function on moduli space, so it makes sense
to integrate it against d[g]. We can then go ahead and figure out the
partition function quite explicitly.
By now, if you're a rigorous sort of pure mathematician, you must be
suffering from grave doubts about the sanity of this whole procedure.
But physicists regard this chain of reasoning, especially the miraculous
cancellation of anomalies at the end, as a real triumph. And indeed,
it's far *better* than *most* of what happens in quantum field theory!
I've heard publishers of science popularizations say that each equation
in a book diminishes its readership by a factor of 2. I don't know if
this applies to This Week's Finds, but normally I try very hard to keep
the equations to a minimum. This week, however, I've been very bad, and
if my calculations are correct, by this point I am the only one reading
this. So I might as well drop all pretenses of expository prose and
write in a way that only I can follow! The real reason I'm writing
this, after all, is to see if I understand this stuff.
Okay, so now I'd like to see if I understand how one explicitly
calculates this integral:
integral exp() dX
Since we're eventually going to integrate this (times some stuff) over
moduli space, we might as well assume the metric on our torus is gotten
by curling up the following parallelogram in the complex plane:
tau tau + 1
* *
* *
0 1
There are at least two ways to do the calculation. One is to actually
work out the eigenvalues of the Laplacian on this torus and then do the
zeta function regularization to compute its determinant. Di Francesco,
Mathieu, and Senechal do this in the textbook I talked about in "week124".
They get
integral exp() dX = 1/(sqrt(Im(tau)) eta(tau)^2)
where "eta" is the Dedekind eta function, regarded as function of tau.
But the calculation is pretty brutal, and it seems to me that there
should be a much easier way to get the answer. The lefthand side is
just the partition function for an massless scalar field on the torus,
and we basically did that back in "week126". More precisely, we
considered just the rightmoving modes and we got the following
partition function:
1/eta(tau)
How about the leftmoving modes? Well, I'd guess that their partition
function is just the complex conjugate,
1/eta(tau)*
since rightmovers correspond to holomorphic functions and leftmovers
correspond to antiholomorphic functions in this Euclidean picture. It's
just a guess! And finally, what about the zerofrequency mode? I have
no idea. But we should presumably multiply all three partition
functions together to get the partition function of the whole system 
that's how it usually works. And as you can see, we *almost* get the
answer that Di Francesco, Mathieu, and Senechal got. It would work out
*perfectly* if the partition function of the zerofrequency mode were
1/sqrt(Im(tau)). By the way, Im(tau) is just the *area* of the torus.
As evidence that something like this might work, consider this: the
zerofrequency mode is presumably related to the zero eigenvalue of the
Laplacian. We threw that out when we defined the regularized
determinant of the Laplacian, but as I hinted, more careful calculations
of
integral exp() dX
don't just ignore the zero eigenvalue. Instead, they somehow use it to
get an extra factor of 1/sqrt(Im(tau)). Admittedly, the calculations are
not particularly convincing: a more obvious guess would be that it gives a
factor of infinity. Di Francesco, Mathieu, and Senechal practically
admit that they *need* this factor just to get modular invariance, and
that they'll do whatever it takes to get it. Nash just sticks in the
factor of 1/sqrt(Im(tau)), mutters something vague, and hurriedly moves on.
Clearly the reason people want this factor is because of how the eta
function transforms under modular transformations. In "week125" I said
that the group PSL(2,Z) is generated by two elements S and T, and if you
look at the formulas there you'll see they act in the following way on
tau:
S: tau > 1/tau
T: tau > tau + 1
The Dedekind eta function satisfies
eta(1/tau) = sqrt(tau/i) eta(tau)
eta(tau + 1) = exp(2 pi i / 24) eta(tau)
The second one is really easy to see from the definition; the first
one is harder. Anyway, using these facts it's easy to see that
1/(sqrt(Im(tau)) eta(tau)^2)
is invariant under PSL(2,Z), so it's really a function on moduli
space  but only if that factor of 1/sqrt(Im(tau)) is in there!
Finally, I'd like to say something about why the conformal anomalies
cancel in 26 dimensions. When I began thinking about this stuff I
was hoping it'd be obvious from the transformation properties of the
eta function  since they have that promising number "24" in them 
but right now I do *not* see anything like this going on. Instead,
it seems to be something like this: in the partition function
Z = integral (integral exp() dX) f(g) d[g]
the mysterious function f is basically just the determinant of the
Laplacian on *vector fields* on the torus. So ignoring those
darn zero eigenvalues the whole integrand here is
det(laplacian)^{n/2} det(laplacian')
where "laplacian" is the Laplacian on realvalued functions and
"laplacian'" is the Laplacian on vector fields. Now these determinants
aren't welldefined functions on the space of conformal structures;
they're really sections of certain "determinant bundles". But in this
situation, the determinant bundle for the Laplacian on vector fields JUST
SO HAPPENS to be the 13th tensor power of the determinant bundle for the
Laplacian on functions  so the whole expression above is a welldefined
function on the space of conformal structures, and thence on moduli space,
precisely when n = 26!!!
Now this "just so happens" cannot really be a coincidence. There *are*
no coincidences in mathematics. That's why it pays to be paranoid when
you're a mathematician: nothing is random, everything fits into a grand
pattern, it's all just staring you in the face if you'd only notice it.
(Chaitin has convincingly argued otherwise using Goedel's theorem, and
certainly some patterns in mathematics seem "purely accidental", but
right now I'm just waxing rhapsodic, expressing a feeling one sometimes
gets....)
Indeed, look at the proof in Nash's book that one of these determinant
bundles is the 13th tensor power of the other  I think this result is
due to Mumford, but Nash's proof is easy to read. What does he do? He
works out the first Chern class of both bundles using the index theorem
for families, and he gets something involving the Todd genus  and the
Todd genus, as we all know, is defined using the same function
x/(1  e^x) = 1 + x/2  x^2/12 + ....
that we talked about in "week126" when computing the zeropoint energy
of the bosonic string! And yet again, it's that darn 1/12 in the power
series expansion that makes everything tick. That's where the 13 comes
from! It's all an elaborate conspiracy!
But of course the conspiracy is far grander than I've even begun
to let on. If we keep digging away at it, we're eventually led to
nothing other than....
MONSTROUS MOONSHINE!!!
But I don't have the energy to talk about *that* now. For more, try:
7) Richard E. Borcherds, What is moonshine?, talk given upon winning
the Fields medal, preprint available as math.QA/9809110.
8) Peter Goddard, The work of R. E. Borcherds, preprint available as
math.QA/9808136.
Okay, if you've actually read this far, you deserve a treat. Let's
calculate the determinant of an operator A whose eigenvalues are the
numbers 1, 2, 3, .... You can think of this operator as the Hamiltonian
for the wave equation on the circle, where we only keep the rightmoving
modes. As I already said, the zeta function of this operator is the
Riemann zeta function. This function has zeta'(0) = ln(2 pi)/2, so
using our cute formula relating determinants and zeta functions, we get
det(A) = exp(zeta'(0)) = sqrt(2 pi).
Just for laughs, if we pretend that the determinant of A is the product
of its eigenvalues as in the finitedimensional case, we get:
1 x 2 x 3 x ... = sqrt(2 pi)
or if you really want to ham it up,
infinity! = sqrt(2 pi).
Cute, eh? Dan Piponi told me this, as well as some of the other things
I've been talking about. You can also find it in Bost's paper.

Notes and digressions:
a. In all of the above, I put a minus sign into my Laplacian, so that it
has nonnegative eigenvalues. This is common among erudite mathematical
physics types, who like "positive elliptic operators".
b. The zeta function trick for defining the determinant of the Laplacian
works for any positive elliptic operator on a compact manifold. A huge
amount has been written about this trick. It's all based on the fact
that the zeta function of a positive elliptic operator analytically
continues to s = 0. This fact was proved by Seeley:
9) R. T. Seeley, Complex powers of an elliptic operator, Proc. Symp.
Pure Math. 10 (1967), 288307.
c. Why is the Polyakov action conformally invariant?
Because the Laplacian has dimensions of 1/length^2, while the integral
used to define the inner product has dimensions of length^2, since the
torus is 2dimensional. This is the magic of 2 dimensions! The path
integral for higherdimensional "branes" has not yet been made precise,
because this magic doesn't happen there.

Previous issues of "This Week's Finds" and other expository articles on
mathematics and physics, as well as some of my research papers, can be
obtained at
http://math.ucr.edu/home/baez/
For a table of contents of all the issues of This Week's Finds, try
http://math.ucr.edu/home/baez/twf.html
A simple jumpingoff point to the old issues is available at
http://math.ucr.edu/home/baez/twfshort.html
If you just want the latest issue, go to
http://math.ucr.edu/home/baez/this.week.html
of anomalies at the end, as a real triumph. And indeed,
it's far *better* than *most* of what happens in quantum field theory!
I've heard publishers of scietwf_ascii/week128000064400020410000157000000544671125754676000141630ustar00baezhttp00004600000001Also available at http://math.ucr.edu/home/baez/week128.html
January 4, 1999
This Week's Finds in Mathematical Physics (Week 128)
John Baez
This week I'd like to catch you up on the latest developments
in quantum gravity. First, a book that everyone can enjoy:
1) John Archibald Wheeler and Kenneth Ford, Geons, Black Holes, and
Quantum Foam: A Life in Physics, Norton, New York, 1998.
This is John Wheeler's autobiography. If Wheeler's only contribution to
physics was being Bohr's student and Feynman's thesis advisor, that in
itself would have been enough. But he did much more. He played a
crucial role in the Manhattan project and the subsequent development of
the hydrogen bomb. He worked on nuclear physics, cosmic rays, muons and
other elementary particles. And he was also one of the earlier people
to get really excited about the more outlandish implications of general
relativity. For example, he found solutions of Einstein's equation that
correspond to regions of gravitational field held together only by their
own gravity, which he called "geons". He was not the first to study
black holes, but he was one of the first people to take them seriously,
and he invented the term "black hole". And the reason he is *my* hero
is that he took seriously the challenge of reconciling general
relativity and quantum theory. Moreover, he recognized how radical the
ideas needed to accomplish this would be  for example, the idea that
spacetime might not be truly be a continuum at short distance scales,
but instead some sort of "quantum foam".
Anyone interested in the amazing developments in physics during the 20th
century should read this book! Here is the story of how he first met
Feynman:
Dick Feynman, who had earned his bachelor's degree at MIT,
showed up at my office door as a brash and appealing twentyone
yearold in the fall of 1939 because, as a new student with a
teaching assistantship, he had been assigned to grade papers for
me in my mechanics course. As we sat down to talk about the course
and his duties, I pulled out and placed on the table between us
a pocket watch. Inspired by my father's keenness for timeand
motion studies, I was keeping track of how much time I spent on
teaching and teachingrelated activities, how much on research,
and how much on departmental or university chores. This meeting
was in the category of teachingrelated. Feynman may have been
a little taken aback by the watch but he was not one to be
intimidated. He went out and bought a dollar watch (as I learned
later), so he would be ready for our next meeting. When we got
together again, I pulled out my watch and put it on the table
between us. Without cracking a smile, Feynman pulled out his
watch and put it on the table next to mine. His theatrical sense
was perfect. I broke down laughing, and soon he was laughing as
hard as I, until both of us had tears in our eyes. It took quite
a while for us to sober up and get on with our discussion. This
set the tone for a wonderful friendship that endured for the rest
of his life.
Next for something a wee bit more technical:
2) Steven Carlip, Quantum Gravity in 2+1 Dimensions, Cambridge
University Press, 1998. ISBN 0521564085.
If you want to learn about quantum gravity in 2+1 dimensions this is
the place to start, because Carlip is the world's expert on this subject,
and he's pretty good at explaining things.
(By the way, physicists write "2+1 dimensions", not because they can't
add, but to emphasize that they are talking about 2 dimensions of space
and 1 dimension of time.)
Quantum gravity in 2+1 dimensions is just a warmup for what physicists
are really interested in  quantum gravity in 3+1 dimensions. Going
down a dimension really simplifies things, because Einstein's equations
in 2+1 dimensions say that the energy and momentum flowing through a
given point of spacetime completely determine the curvature there,
unlike in higher dimensions. In particular, spacetime is *flat*
in the vacuum in 2+1 dimensions, so there's no gravitational radiation.
Nonetheless, quantum gravity in 2+1 dimensions is very interesting, for
a number of reasons. Most importantly, we can solve the equations
exactly, so we can use it as a nice testingground for all sorts of
ideas people have about quantum gravity in 3+1 dimensions.
Quantum gravity is hard for various reasons, but most of all it's hard
because, unlike traditional quantum field theory, it's a "backgroundfree"
theory. What I mean by this is that there's no fixed way of measuring
times and distances. Instead, times and distances must be measured with
the help of the geometry of spacetime, and this geometry undergoes
quantum fluctuations. That throws most of our usual methods for doing
physics right out the window! Quantum gravity in 2+1 dimensions gives
us, for the first time, an example of a backgroundfree theory where we
can work out everything in detail.
Here's the table of contents of Carlip's book:
1. Why (2+1)dimensional gravity?
2. Classical general relativity in 2+1 dimensions
3. A field guide to the (2+1)dimensional spacetimes
4. Geometric structures and ChernSimons theory
5. Canonical quantization in reduced phase space
6. The connection representation
7. Operator algebras and loops
8. The WheelerDeWitt equation
9. Lorentzian path integrals
10. Euclidean path integrals and quantum cosmology
11. Lattice methods
12. The (2+1)dimensional black hole
13. Next steps
A. Appendix: The topology of manifolds
B. Appendix: Lorentzian metrics and causal structure
C. Appendix: Differential geometry and fiber bundles
And now for some stuff that's available online. First of all, anyone
who wants to keep up with research on gravity should remember to read
"Matters of Gravity". I've talked about it before, but here's the
latest edition:
3) Jorge Pullin, editor, Matters of Gravity, vol. 12, available at
grqc/9809031 and at http://vishnu.nirvana.phys.psu.edu/mog.html
There's a lot of good stuff in here. Quantum gravity buffs will
especially be interested in Gary Horowitz's article "A nonperturbative
formulation of string theory?" and Lee Smolin's "Neohistorical
approaches to quantum gravity". The curious title of Smolin's article
refers to *new* work on quantum gravity involving a sum over *histories*
 or in other words, spin foam models.
Even if you can't go to a physics talk, these days you can sometimes
find it on the worldwide web. Here's one by John Barrett:
4) John W. Barrett, State sum models for quantum gravity, Penn State
relativity seminar, August 27, 1998, audio and text of transparencies
available at http://vishnu.nirvana.phys.psu.edu/relativity.seminars.html
Barrett and Crane have a theory of quantum gravity, which I've also
worked on; I discussed it last in "week113" and "week120". Before I
describe it I should warn the experts that this theory deals with
Riemannian rather than Lorentzian quantum gravity (though Barrett and
Crane are working on a Lorentzian version, and I hear Friedel and
Krasnov are also working on this). Also, it only deals with vacuum
quantum gravity  empty spacetime, no matter.
In this theory, spacetime is chopped up into 4simplices. A 4simplex
is the 4dimensional analog of a tetrahedron. To understand what I'm
going to say next, you really need to understand 4simplices, so let's
start with them.
It's easy to draw a 4simplex. Just draw 5 dots in a kind of circle and
connect them all to each other! You get a pentagon with a pentagram
inscribed in it. This is a perspective picture of a 4simplex
projected down onto your 2dimensional paper. If you stare at this
picture you will see the 4simplex has 5 tetrahedra, 10 triangles,
10 edges and 5 vertices in it.
The shape of a 4simplex is determined by 10 numbers. You can take
these numbers to be the lengths of its edges, but if you want to be
sneaky you can also use the areas of its triangles. Of course, there
are some constraints on what areas you can choose for there to *exist* a
4simplex having triangles with those areas. Also, there are some
choices of areas that fail to make the shape *unique*: for one of these
bad choices, the 4simplex can flop around while keeping the areas of
all its triangles fixed. But generically, this nonuniqueness doesn't
happen.
In Barrett and Crane's theory, we chop spacetime into 4simplices and
describe the geometry of spacetime by specifying the area of each
triangle. But the geometry is "quantized", meaning that the area
takes a discrete spectrum of possible values, given by
sqrt(j(j+1))
where the "spin" j is a number of the form 0, 1/2, 1, 3/2, etc. This
formula will be familiar to you if you've studied the quantum mechanics
of angular momentum. And that's no coincidence! The cool thing about
this theory of quantum gravity is that you can discover it just by
thinking a long time about general relativity and the quantum mechanics
of angular momentum, as long as you also make the assumption that
spacetime is chopped into 4simplices.
So: in Barrett and Crane's theory the geometry of spacetime is described
by chopping spacetime into 4simplices and labelling each triangle with
a spin. Let's call such a labelling a "quantum 4geometry". Similarly,
the geometry of space is described by chopping space up into tetrahedra
and labelling each triangle with a spin. Let's call this a "quantum
3geometry".
The meat of the theory is a formula for computing a complex number
called an "amplitude" for any quantum 4geometry. This number plays the
usual role that amplitudes do in quantum theory. In quantum theory, if
you want to compute the probability that the world starts in some state
psi and ends up in some state psi', you just look at all the ways the
world can get from psi to psi', compute an amplitude for each way, add
them all up, and take the square of the absolute value of the result.
In the special case of quantum gravity, the states are quantum 3geometries,
and the ways to get from one state to another are quantum 4geometries.
So, what's the formula for the amplitude of a quantum 4geometry? It
takes a bit of work to explain this, so I'll just vaguely sketch how it
goes. First we compute amplitudes for each 4simplex and multiply all
these together. Then we compute amplitudes for each triangle and
multiply all these together. Then we multiply these two numbers.
(This is analogous to how we compute amplitudes for Feynman diagrams
in ordinary quantum field theory. A Feynman diagram is a graph whose
edges have certain labellings. To compute its amplitude, first we
compute amplitudes for each edge and multiply them all together. Then
we compute amplitudes for each vertex and multiply them all together.
Then we multiply these two numbers. One goal of work on "spin
foam models" is to more deeply understand this analogy with Feynman
diagrams.)
Anyway, to convince oneself that this formula is "good", one would like
to relate it to other approaches to quantum gravity that also involve
4simplices. For example, there is the Regge calculus, which is a
discretized version of *classical* general relativity. In this approach
you chop spacetime into 4simplices and describe the shape of each
4simplex by specifying the lengths of its edges. Regge invented a
formula for the "action" of such a geometry which approaches the usual
action for classical general relativity in the continuum limit. I
explained the formula for this "Regge action" in "week120".
Now if everything were working perfectly, the amplitude for a 4simplex
in the BarrettCrane model would be close to exp(iS), where S is the
Regge action of that 4simplex. This would mean that the BarrettCrane
model was really a lot like a path integral in quantum gravity. Of
course, in the BarrettCrane model all we know is the areas of the triangles
in each 4simplex, while in the Regge calculus we know the lengths of
its edges. But we can translate between the two, at least generically,
so this is no big deal.
Recently, Barrett came up with a nice argument saying that in the limit
where the triangles have large areas, the amplitude for a 4simplex in
the BarrettCrane theory is proportional, not to exp(iS), but to cos(S):
5) John W. Barrett and Ruth M. Williams, The asymptotics of an amplitude
for the 4simplex, preprint available as grqc/9809032.
This argument is not rigorous  it uses a stationary phase approximation
that requires further justification. But Regge and Ponzano used a
similar argument to show the same sort of thing for quantum gravity in 3
dimensions, and their argument was recently made rigorous by Justin
Roberts, with a lot of help from Barrett:
6) Justin Roberts, Classical 6jsymbols and the tetrahedron, preprint
available as mathph/9812013.
So one expects that with work, one can make Barrett's argument rigorous.
But what does it mean? Why does he get cos(S) instead of exp(iS)?
Well, as I said, the same thing happens one dimension down in the
socalled PonzanoRegge model of 3dimensional Riemannian quantum
gravity, and people have been scratching their heads for decades trying
to figure out why. And by now they know the answer, and the same
answer applies to the BarrettCrane model.
The problem is that if you describe 4simplex using the areas of its
triangles, you don't *completely* know its shape. (See, I lied to you
before  that's why you gotta read the whole thing.) You only know it
*up to reflection*. You can't tell the difference between a 4simplex
and its mirrorimage twin using only the areas of its triangles! When
one of these has Regge action S, the other has action S. The Barrett
Crane model, not knowing any better, simply averages over both of them,
getting
(1/2)(exp(iS) + exp(iS)) = cos(S)
So it's not really all that bad; it's doing the best it can under
the circumstances. Whether this is good enough remains to be seen.
(Actually I didn't really *lie* to you before; I just didn't tell you
my definition of "shape", so you couldn't tell whether mirrorimage
4simplices should count as having the same shape. Expository prose
darts between the Scylla of overwhelming detail and the Charybdis of
vagueness.)
Okay, on to a related issue. In the BarrettCrane model one describes a
quantum 4geometry by labelling all the triangles with spins. This
sounds reasonable if you think about how the shape of a 4simplex is
almost determined by the areas of its triangles. But if you actually
examine the derivation of the model, it starts looking more odd. What
you really do is take the space of geometries of a *tetrahedron*
embedded in R^4, and use a trick called geometric quantization to get
something called the "Hilbert space of a quantum tetrahedron in 4
dimensions". You then build your 4simplices out of these quantum
tetrahedra.
Now the Hilbert space of a quantum tetrahedron has a basis labelled by
the eigenvalues of operators corresponding to the areas of its 4
triangular faces. In physics lingo, it takes 4 "quantum numbers" to
describe the shape of a quantum tetrahedron in 4 dimensions.
But classically, the shape of a tetrahedron is *not* determined by the
areas of its triangles: it takes 6 numbers to specify its shape, not
just 4. So there is something funny going on.
At first some people thought there might be more states of the quantum
tetrahedron than the ones Barrett and Crane found. But Barbieri came up
with a nice argument suggesting that Barrett and Crane had really found
all of them:
7) Andrea Barbieri, Space of the vertices of relativistic
spin networks, preprint available as grqc/9709076.
While convincing, this argument was not definitive, since it
assumed something plausible but not yet proven  namely, that the
"6j symbols don't have too many exceptional zeros". Later,
Mike Reisenberger came up with a completely rigorous argument:
8) Michael P. Reisenberger, On relativistic spin network vertices,
preprint available as grqc/9809067.
But while this settled the facts of the matter, it left open the
question of "why"  why does it take *6* numbers to describe the
shape of classical tetrahedron in 4 dimensions but only *4* numbers
to describe the shape of a quantum one? John Barrett and I have
almost finished a paper on this, so I'll give away the answer.
Not surprisingly, the key is that in quantum mechanics, not all
observables commute. You only use the eigenvalues of *commuting*
observables to label a basis of states. The areas of the quantum
tetrahedron's faces commute, and there aren't any other independent
commuting observables. It's a bit like how in classical mechanics
you can specify both the position and momentum of a particle, but
in quantum mechanics you can only specify one.
This isn't news, of course. And indeed, people knew perfectly well
that for this reason, it takes only *5* numbers to describe the shape of
a quantum tetrahedron in 3 dimensions. The real puzzle was why it takes
even fewer numbers when your quantum tetrahedron lives in 4 dimensions!
It seemed bizarre that adding an extra dimension would reduce the number
of degrees of freedom! But it's true, and it's just a spinoff of the
uncertainty principle. Crudely speaking, in 4 dimensions the fact that
you know your tetrahedron lies in some hyperplane makes you unable to
know as much about its shape.
Here are some other talks available on the web:
9) Abhay Ashtekar, Chris Beetle and Steve Fairhurst, Mazatlan lectures
on black holes, slides available at
http://vishnu.nirvana.phys.psu.edu/online/Html/Conferences/Mazatlan/
These explain a new concept of "nonrotating isolated horizon"
which allow one to formulate and prove the zeroth and first laws
of black hole mechanics in a way that only refers to the geometry
of spacetime near the horizon. For more details try:
10) Abhay Ashtekar, Chris Beetle and S. Fairhurst, Isolated horizons:
a generalization of black hole mechanics, preprint available as
grqc/9812065.
This concept also serves as the basis for a forthcoming 2part paper
where Ashtekar, Corichi, Krasnov and I compute the entropy of a quantum
black hole (see "week112" for more on this).
Finally, here are a couple more papers. I don't have time to say much
about them, but they're both pretty neat:
11) Matthias Arnsdorf and R. S. Garcia, Existence of spinorial states in
pure loop quantum gravity, preprint available as grqc/9812006.
I'll just quote the abstract:
We demonstrate the existence of spinorial states in a theory of
canonical quantum gravity without matter. This should be regarded
as evidence towards the conjecture that bound states with particle
properties appear in association with spatial regions of
nontrivial topology. In asymptotically trivial general relativity
the momentum constraint generates only a subgroup of the spatial
diffeomorphisms. The remaining diffeomorphisms give rise to the
mapping class group, which acts as a symmetry group on the phase
space. This action induces a unitary representation on the loop
state space of the Ashtekar formalism. Certain elements of the
diffeomorphism group can be regarded as asymptotic rotations of
space relative to its surroundings. We construct states that
transform nontrivially under a 2 pi rotation: gravitational
quantum states with fractional spin.
14) Steve Carlip, Black hole entropy from conformal field theory in any
dimension, preprint available as hepth/9812013.
Again, here's the abstract:
When restricted to the horizon of a black hole, the 'gauge'
algebra of surface deformations in general relativity contains a
physically important Virasoro subalgebra with a calculable central
charge. The fields in any quantum theory of gravity must transform
under this algebra; that is, they must admit a conformal field
theory description. With the aid of Cardy's formula for the
asymptotic density of states in a conformal field theory, I use
this description to derive the BekensteinHawking entropy. This method
is universal  it holds for any black hole, in any dimension, and
requires no details of quantum gravity  but it is also explicitly
statistical mechanical, based on the counting of microscopic
states.
On Thursday I'm flying to Schladming, Austria to attend a workshop on
geometry and physics organized by Harald Grosse and Helmut Gausterer.
Some cool physicists will be there, like Daniel Kastler and Julius Wess.
If I understand what they're talking about I'll try to explain it here.
Happy new year!

Addendum: Above I wrote:
Recently, Barrett and Williams came up with a nice argument saying
that in the limit where the triangles have large areas, the
amplitude for a 4simplex in the BarrettCrane theory is
proportional, not to exp(iS), but to cos(S)....
This argument is not rigorous  it uses a stationary phase
approximation that requires further justification. But similar
argument to show the same sort of thing for quantum gravity in 3
dimensions, and their argument was recently made rigorous by Justin
Roberts, with a lot of help from Barrett....
So one expects that with work, one can make Barrett and Williams'
argument rigorous.
In fact one can't make it rigorous: it's wrong! In the limit of large
areas the amplitude for a 4simplex in the BarrettCrane model is
wildly different from cos(S), or exp(iS), or anything like that. Dan
Christensen, Greg Egan and I showed this in a couple of papers that I
discuss in "week170" and "week198". Our results were confirmed by
John Barrett, Chris Steel, Laurent Friedel and David Louapre.
By now  I'm writing this in 2009  it's generally agreed that the
BarrettCrane model is wrong and another model is better. To read
about this new model, see "week280".

Previous issues of "This Week's Finds" and other expository articles on
mathematics and physics, as well as some of my research papers, can be
obtained at
http://math.ucr.edu/home/baez/
For a table of contents of all the issues of This Week's Finds, try
http://math.ucr.edu/home/baez/twf.html
A simple jumpingoff point to the old issues is available at
http://math.ucr.edu/home/baez/twfshort.html
If you just want the latest issue, go to
http://math.ucr.edu/home/baez/this.week.html
rint available as grqc/9809032.
This argument is not rigorous  it uses a stationary phase approximation
that requires further justification. But Regge and Ponzano used a
similar argument to show thtwf_ascii/week129000064400020410000157000000465560774011336400141540ustar00baezhttp00004600000001Also available at http://math.ucr.edu/home/baez/week129.html
February 15, 1999
This Week's Finds in Mathematical Physics (Week 129)
John Baez
For the last 38 years the Austrians have been having winter workshops on
nuclear and particle physics in a little Alpine ski resort town called
Schladming. This year it was organized by Helmut Gausterer and Hermann
Grosse, and the theme was "Geometry and Quantum Physics":
1) Geometry and Quantum Physics lectures, 38th Internationale
Universitaetswochen fuer Kern und Teilchenphysik,
http://physik.kfunigraz.ac.at/utp/iukt/iukt_99/iukt99lect.html
I was invited to give some talks about spin foam models, and the other
talks looked interesting, so I decided to leave my warm and sunny home
for the chilly north. I flew out to Salzburg in early January and took
a train to Schladming from there. Jetlagged and exhausted, I almost
slept through my train stop, but I made it and soon collapsed into my
hotel bed.
The next day I alternately slept and prepared my talks. The workshop
began that evening with a speech by Helmut Grosse, a speech by the town
mayor, and a reception featuring music by a brass band. The last two
struck me as a bit unusual  there's something peculiarly Austrian about
drinking beer and discussing quantum gravity over loud oompah music!
This was also the first conference I've been to that featured skiing and
bowling competitions.
Anyway, there were a number of 4hour minicourses on different subjects,
which should eventually appear as articles in this book:
2) Geometry and Quantum Physics, proceedings of the 38th Int.
Universitaetswochen fuer Kern und Teilchenphysik, Schladming, Austria,
Jan. 916, 1999, eds. H. Gausterer, H. Grosse and L. Pittner, to appear
in Lecture Notes in Physics, SpringerVerlag, Berlin.
Right now they exist in the form of lecture notes:
Anton Alekseev Symplectic and noncommutative geometry of systems with symmetry
John Baez Spin foam models of quantum gravity
Cesar Gomez Duality and Dbranes
Daniel Kastler Noncommutative geometry and fundamental physical interactions
John Madore An introduction to noncommutative geometry
Rudi Seiler Geometric properties of transport in quantum Hall systems
Julius Wess Physics on noncommutative spacetime structures
All these talks were about different ways of combining quantum theory
and geometry. Quantum theory is so strange that ever since its
invention there has been a huge struggle to come to terms with it at all
levels. It took a while for it to make its full impact in pure
mathematics, but now you can see it happening all over: there are lots
of papers on quantum topology, quantum geometry, quantum cohomology,
quantum groups, quantum logic... even quantum set theory! There are
even some fascinating attempts to apply quantum mechanics to unsolved
problems in number theory like the Riemann hypothesis... will they bear
fruit? And if so, what does this mean about the world? Nobody really
knows yet; we're in a period of experimentation  a bit of a muddle.
I don't have the energy to summarize all these talks so I'll concentrate
on part of Alekseev's  just a tiny smidgen of it, actually! But first,
let me just quickly say a word about each speaker's topic.
Alekseev talked about some ideas related to the stationary phase
approximation. This is one of the main tools linking classical
mechanics to quantum mechanics. It's a trick for approximately
computing the integral of a function of the form exp(iS(x)) knowing only
S(x) and its 2nd derivative at points where its first derivative
vanishes. In physics, people use it to compute path integrals in the
semiclassical limit where what matters most is paths near the classical
trajectories. Alekseev discussed problems where the stationary phase
approximation gives the exact answer. There's a wonderful thing called
the DuistermaatHeckman formula which says that this happens in certain
situations with circular symmetry. There are also generalizations to
more complicated symmetry groups. These are related to `equivariant
cohomology'  more on that later.
I talked about the spin foam approach to quantum gravity. I've already
discussed this in "week113", "week114", "week120", and "week128", so
there's no need to say more here.
Cesar Gomez gave a wonderful introduction to string theory, starting
from scratch and rapidly working up to Tduality and Dbranes. The idea
behind Tduality is very simple and pretty. Basically, if you have
closed strings living in a space with one dimension curled up into a
circle of radius R, there is a symmetry that involves replacing R by 1/R
and switching two degrees of freedom of the string, namely the number of
times it winds around the curledup direction and its momentum in the
curledup direction. Both these numbers are integers. Dbranes are
something that shows up when you consider the consequences of this symmetry
for *open* strings.
String theory is rather conservative in that, at least until recently,
it usually treated spacetime as a manifold with a fixed geometry and
only applied quantum mechanics to the description of the strings
wiggling around *in* spacetime. In spin foam models, by contrast,
spacetime itself is modelled quantummechanically as a kind of
higherdimensional version of a Feynman diagram. There are also other
ideas about how to treat spacetime quantummechanically. One of them is
to treat the coordinates on spacetime as noncommuting variables. In
this approach, called noncommutative geometry, the uncertainty
principle limits our ability to simultaneously know all the coordinates
of a particle's position, giving spacetime a kind of quantum "fuzziness".
Personally I don't find noncommutative geometry convincing as a theory
of physical spacetime, because there are no clues that spacetime actually
has this sort of fuzziness. But I find it quite interesting as mathematics.
Daniel Kastler talked about Alain Connes' theories of physics based on
noncommutative geometry. He discussed both the original ConnesLott
version of the Standard Model and newer theories that include gravity.
Kastler is a real character! As usual, his talks lauded Connes to the
heavens and digressed all over the map in a frustrating but entertaining
manner. Throughout the conference, he kept us wellfed with anecdotes,
bringing back the aura of heroic bygone days. A random example: Pauli
liked to work long into the night  so when a student asked "Could I
meet you at your office at 9 a.m.?" he replied "No, I can't possibly
stay that late".
One nice idea mentioned by Kastler came from this paper:
3) Alain Connes, Noncommutative geometry and reality, J. Math. Phys.
36 (1995), 6194.
The idea is to equip spacetime with extra curledup dimensions shaped
like the quantum group SU_q(2) where q is a 3rd root of unity. A
quantum group is actually a kind of noncommutative algebra, but using
Connes' ideas you can think of it as a kind of "space". If you mod out
this particular algebra by its nilradical, you get the algebra M_1(C) +
M_2(C) + M_3(C), where M_n(C) is the algebra of n x n complex matrices.
This has a tantalizing relation to the gauge group of the Standard
Model, namely U(1) x SU(2) x SU(3).
John Madore also spoke about noncommutative geometry, but more on the
general theory and less on the applications to physics. He concentrated
on the notion of a "differential calculus"  a structure you can equip
an algebra with in order to do differential geometry thinking of it as a
kind of "space".
Julius Wess also spoke on noncommutative geometry, focussing on a
qdeformed version of quantum mechanics. The process of
"qdeformation" is something you can do not only to groups like SU(2)
but also other spaces. You get noncommutative algebras, and these often
have nice differential calculi that let you go ahead and do
noncommutative geometry. Wess had a nice humorous way of defusing tense
situations. When one questioner pointedly asked him whether the
material he was presenting was useful in physics or merely a pleasant
game, he replied "That's a very good question. I will try to answer
that later. For now you're just like students in calculus: you don't
know why you're learning all this stuff...." And when Kastler and other
mathematicians kept hassling him over whether an operator was
selfadjoint or merely hermitian, he begged for mercy by saying "I would
like to be a physicist. That was my dream from the beginning."
Anyway, I hope that from these vague descriptions you get some
sense of the ferment going on in mathematical physics these days.
Everyone agrees that quantum theory should change our ideas about
geometry. Nobody agrees on how.
Now let me turn to Alekseev's talk. In addition to describing his own
work, he explained many things I'd already heard about. But he did it
so well that I finally understood them! Let me talk about one of these
things: equivariant deRham cohomology. For this, I'll assume you know
about deRham cohomology, principal bundles, connections and curvature.
So I assume you know that given a manifold M, we can learn a lot about
its topology by looking at differential forms on M and figuring out the
space of closed pforms modulo exact ones  the socalled pth deRham
cohomology of M. But now suppose that some Lie group G acts on M in a
smooth way. What can differential forms tell us about the topology of
this group action?
All sorts of things! First suppose that G acts freely on M  meaning
that gx is different from x for any point x of M and any element g of G
other than the identity. Then the quotient space M/G is a manifold.
Even better, the map M > M/G gives us a principal Gbundle with total
space M and base space M/G.
Can we figure out the deRham cohomology of M/G? Of course if we were
smart enough we could do it by working out M/G and then computing its
cohomology. But there's a sneakier way to do it using the differential
forms on M. The map M > M/G lets us pull back any form on M/G to get a
form on M. This lets us think of forms on M/G as forms on M satisfying
certain equations  people call them "basic" differential forms because
they come from the base space M/G.
What are these equations? Well, note that each element v of the Lie
algebra of G gives a vector field on M, which I'll also call v. This
give two operations on the differential forms on M: the Lie derivative L_v
and the interior product i_v. It's easy to see that any basic differential
form is annihilated by these operations for all v. The converse is true
too! So we have some nice equations describing the basic forms.
If we now take the space of closed basic pforms modulo the exact basic
pforms, we get the deRham cohomology of M/G! This lets us study the
topology of M/G using differential forms on M. It's very convenient.
If the action of G on M isn't free, the quotient space M/G might not be
a manifold. This doesn't stop us from defining "basic" differential
forms on M just as before. We can also define some cohomology groups by
taking the closed basic pforms modulo the exact ones. But topologists
know from long experience that another approach is often more useful.
Group actions that aren't free are touchy, sensitive creatures  a real
nuisance to work with. Luckily, when you have an action that's not
free, you can tweak it slightly to make it free. This involves "puffing
up" the space that the group acts on  replacing it by a bigger space
that the group acts on freely.
For example, suppose you have a group G acting on a onepoint space.
Unless G is trivial, this action isn't free. In fact, it's about as far
from free as you can get! But we can "puff it up" and get a space
called EG. Like the onepoint space, EG is contractible, but G acts
freely on it. Actually there are various spaces with these two
properties, and it doesn't much matter which one we use  people call
them all EG. People call the quotient space EG/G the "classifying
space" of G, and they denote it by BG.
More generally, suppose we have *any* action of G on a manifold M. How
can we puff up M to get a space on which G acts freely? Simple: just
take its product with EG. Since G acts on M and EG, it acts on the
product M x EG in an obvious way. Since G acts freely on EG, its action
on M x EG is free. And since EG is contractible, the space M x EG is a
lot like M, at least as far as topology goes. More precisely, it has
the same homotopy type!
Actually the last 2 paragraphs can be massively generalized at no extra
cost. There's no need for G to be a Lie group or for M to be a manifold.
G can be any topological group and M can be any topological space! But
since I want to talk about *deRham* cohomology, I don't need this extra
generality here.
Anyway, now we know the right substitute for the quotient space M/G when
the action of G on M isn't free: it's the quotient space (M x EG)/G.
So now let's figure out how to compute the pth deRham cohomology of
(M x EG)/G. Since G acts freely on M x EG, this should be just the
closed basic pforms on M x EG modulo the exact ones, where "basic" is
defined as before. In fact this is true. We call the resulting space
the pth "equivariant deRham cohomology" of the space M. It's a kind of
wellbehaved substitute for the deRham cohomology of M/G in the case
when M/G isn't a manifold.
There's only one slight problem: the space EG is very big, so it's not
easy to deal with differential forms on M x EG!
You'll note that I didn't say much about what EG looks like. All I said
is that it's some contractible space on which G acts freely. I didn't
even say it was a manifold, so it's not even obvious that "differential
forms on EG" makes sense! If you are smart you can choose your space EG
so that it's a manifold. However, you'll usually need it to be
infinitedimensional.
Differential forms make perfect sense on infinitedimensional
manifolds, but they can be a bit tiresome when we're trying to do
explicit calculations. Luckily there is a small subalgebra of the
differential forms on EG that's sufficient for the purpose of computing
equivariant cohomology! This is called the "Weil algebra", WG.
To guess what this algebra is, let's just list all the obvious
differential forms on EG that we can think of. Well, I guess none of
them are obvious unless we know a few more facts! First of all, since
the action of G on EG is free, the quotient map EG > BG gives us a
principal Gbundle with total space EG and base space BG. This bundle
is very interesting. It's called the "universal" principal Gbundle.
The reason is that any other principal Gbundle is a pullback of this
one.
(I guess I'm upping the sophistication level again here: I'm assuming
you know how to pull back bundles!)
Even better, if we choose our space EG so that it's a manifold, then
there is a godgiven connection on the bundle EG > BG, and any other
principal Gbundle *with connection* is a pullback of this one.
(And now I'm assuming you know how to pull back connections! However,
this pullback stuff is not necessary in what follows, so just ignore it
if you like.)
Okay, so how can we get a bunch of differential forms on EG just using
the fact that it's the total space of a Gbundle equipped with a connection?
Well, whenever we have a Gbundle E > B, we can think of a connection on
it as a 1form on E taking values in the Lie algebra of G. Let's see what
differential forms on E this gives us! Let's call the connection A. If
we pick a basis of the Lie algebra, we can take the components of A in
this basis, and we get a bunch of 1forms A_i on E. We also get a bunch
of 2forms dA_i. We also get a bunch of 2forms A_i ^ A_j. And so on.
In general, we can form all possible linear combinations of wedge products
of the A_i's and the dA_i's. We get a big fat algebra. In the case when
our bundle is EG > BG, equipped with its godgiven connection, we define
this algebra to be the Weil algebra, WG!
Great. But let's try to define WG in a purely algebraic way, so we can
do computations with it more easily. We're starting out with the
1forms A_i and taking all linear combinations of wedge products of them
and their exterior derivatives. There are in fact no relations except
the obvious ones, so WG is just "the supercommutative differential
graded algebra freely generated by the variables A_i". Note: all the
mumbojumbo about supercommutative differential graded algebras is a way
of mentioning the *obvious* relations.
Warning: people don't usually describe the Weil algebra quite this way.
They usually seem describe it in terms of the connection 1forms and
curvature 2forms. However, the curvature is related to the connection by
the formula F = dA + A ^ A, and if you use this you can go from the usual
description of the Weil algebra to mine  I think.
(Actually, people often describe the Weil algebra as an algebra generated
by a bunch of things of degree 1 and a bunch of things of degree 2, without
telling you that the things of degree 1 are secretly components of a
connection 1form and the things of degree 2 are secretly components of
a curvature 2form! That's why I'm telling you all this stuff  so that
if you ever study this stuff you'll have a better chance of seeing what's
going on behind all the murk.)
Okay, so here is the upshot. Say we want to compute the equivariant deRham
cohomology of some manifold M on which G acts. In other words, we want
to compute the deRham cohomology of (M x EG)/G. On the one hand, we can
start with the differential forms on M x EG, figure out the "basic" pforms,
and take the space of closed basic pforms modulo exact ones. But remember:
up to details of analysis, the algebra of differential forms on M x EG is
just the tensor product of the algebra of forms on M and the algebra of forms
on EG. And we have this nice small "substitute" for the algebra of forms
on EG, namely the Weil algebra WG. So let's take the algebra of differential
forms on M and just tensor it with WG. We get a differential graded algebra
with Lie derivative operations L_v and interior product operations i_v defined
on it. We then proceed as before: we take the space of closed basic
elements of degree p modulo exact ones. Voila! This is something one
can actually compute, with sufficient persistence. And it gives the same
answer, at least when G is connected.
There are all sorts of other things to say. For example, if we take the
simplest posssible case, namely when M is a single point, this gives a
nice trick for computing the deRham cohomology of EG/G = BG. Guys in
this cohomology ring are called "characteristic classes", and they're
really important in physics. Since any principal Gbundle is a pullback
of EG > BG, and cohomology classes pull back, these characteristic
classes give us cohomology classes in the base space of any principal
Gbundle  thus helping us classify Gbundles. But if I started explaining
this now, we'd be here all night.
Also sometime I should say more about how to construct EG.

Previous issues of "This Week's Finds" and other expository articles on
mathematics and physics, as well as some of my research papers, can be
obtained at
http://math.ucr.edu/home/baez/
For a table of contents of all the issues of This Week's Finds, try
http://math.ucr.edu/home/baez/twf.html
A simple jumpingoff point to the old issues is available at
http://math.ucr.edu/home/baez/twfshort.html
If you just want the latest issue, go to
http://math.ucr.edu/home/baez/this.week.html
gx is different from x for any point x of M and any element g of G
other than the identity. Then the quotient space M/G is a manifold.
Even betwf_ascii/week13000064400020410000157000000333420774011336400140510ustar00baezhttp00004600000001Also available at http://math.ucr.edu/home/baez/week13.html
This Week's Finds in Mathematical Physics (Week 13)
John Baez
Well, folks, this'll be the last "This Week's Finds" for a while, since
I'm getting rather busy preparing for my conference on knots and quantum
gravity, and I have a paper that seems to be taking forever to finish.
1. Elliptic Curves by Anthony W. Knapp, Mathematical Notes, Princeton
University Press, 1992.
This is a shockingly userfriendly introduction to a subject that can
all too easily seem intimidating. I'm certainly no expert but maybe
just for that reason I should sketch a brief "introduction to the
introduction" that may lure some of you into studying this beautiful
subject.
What I will say will perhaps appeal to people who like complex analysis
or mathematical physics, but Knapp concentrates on the aspects related
to number theory. For other approaches one might try
2. Elliptic Functions by Serge Lang, SpringerVerlag, 2nd edition, 1987.
3. Elliptic Curves by Dale Husemoeller, SpringerVerlag, 1987.
Okay, where to start? Well, how about this: the sine function is an
analytic function on the complex plane with the property that
sin(z + 2 pi) = sin z
It also satisfies a nice differential equation
(sin' z)^2 = 1  (sin z)^2
and for this reason, we could, if we hadn't noticed the sine function
otherwise, have run into when we tried to integrate
(1  u^2)^{1/2}
The differential equation above implies that the integral is nice to do
by the substitution u = sin z, and we get the answer arcsin u. If the
sine function  or more generally, trig functions  didn't exist yet, we
would have invented them when we tried to do integrals involving square roots
of quadratic polynomials.
Elliptic functions are a beautiful generalization of all of this stuff.
Say we wanted, just for the heck of it, an analytic function that was
periodic not just in one direction on the complex plane, like the sine
function, but in *two* directions. For example, we might want some
function P(z) with
P(z + 2 pi) = P(z)
and also
P(z + 2 pi i) = P(z)
This function would look just the same on each 2piby2pi square:
x x x x x
x x x x x
x x x x x
so if we wanted, we could think of it as being a function on the torus
formed by taking one of these squares and identifying its top side with
its bottom side, and its left side with its right side.
More generally  while we're fantasizing about this wonderful
doublyperiodic function  we could ask for one that was periodic in any
old two directions.. That is, fixing two numbers omega_1 and omega_2
that aren't just realvalued multiples of each other, we could hope to
find an analytic function on the complex plane with omega_1 and omega_2
as periods:
P(z + omega_1) = P(z)
P(z + omega_2) = P(z).
Then P(z) would be the same at all points on the "lattice" of points
n omega_1 + m omega_2, which might look like the square above or might
be like
x
x
x x
x x
x x
x x
x x
x x
x x
x x
x
x
or some such thing.
Let's think about this nice function P(z) we are fantasizing
about. Alas, if it were analytic on the whole plane (no poles), it
would be bounded on each little parallelogram, and since it's doubly
periodic, it would be a bounded analytic function on the complex plane,
hence CONSTANT by Liouville's theorem. Well, so a constant function has
all the wonderful properties we want  but that's too boring!
So let's allow it to have poles! But let's keep it as nice as possible,
so let's have the only poles occur at the lattice points
L = { n omega_1 + m omega_2 }
And let's make the poles as nice as possible. Can we have each pole
be of order one? That is, can we make P(z) blow up like 1/(z  omega) at
each lattice point omega in L? No, because if it did, the integral of
P around a nicely chosen parallelogram around the pole would be zero, because
the contributions from opposite sides of the parallelogram would cancel
by symmetry. (A fun exercise.) But by the Cauchy residue formula this
means that the residue of the pole vanishes, so it can't be of order
one.
Okay, try again. Let's try to make the pole at each lattice point be of
order two. How can we cook up such a function? We might try something
obvious: just sum up, for all omega in the lattice L, the functions
1/(z  omega)^2. This is clearly periodic and has poles like
1/(z  omega)^2 at each lattice point omega. But there's a big problem
 the sum doesn't converge! (Another fun exercise.)
Oh well, try again. Let's act like physicists and RENORMALIZE the sum
by subtracting off an infinite constant! Just subtract the sum over all
omega in L of 1/omega^2. Well, all omega except zero, anyway. This
turns out to work, but we really should be careful about the order of
summation here: really, we should let P(z) be 1/z^2 plus the sum for all
nonzero omega in the lattice L of 1/(z  omega)^2  1/omega^2. This sum
does converge and the limit is a function P(z) that's analytic except for
poles of order two at the lattice points. This is none other than the
Weierstrass elliptic function, usually written with a fancy Gothic P to
intimidate people. Note that it really depends on the two periods
omega_1 and omega_2, not just z.
Now, it turns out that P(z) really *is* a cool generalization of the
sine function. Namely, it satisfies a differential equation like the
one the sine does, but fancier:
(P'(z))^2 = 4 (P(z))^3  g_2 P(z)  g_3
where g_2 and g_3 are some constants that depend on the periods omega_1
and omega_2. Just as with the sine function we can use the *inverse* of
Weierstrass P function to do some integrals, but this time we can do
integrals involving square roots of cubic polynomials! If you look in
big nasty books of special functions or tables of integrals, you will
see that there's a big theory of this kind of thing that was developed
in the 1800's  back when heavyduty calculus was hip.
There are, however, some other cool ways of thinking about what's going
on here. First of all, remember that we can think of P(z) as a function
on the torus. We can think of this torus as being "coordinatized"  I
use the word loosely  by P(z) and its first derivative P'(z). I.e., if
we know x = P(z) and y = P'(z) we can figure out where the point z is on
the torus. But of course x and y can't be any old thing; the
differential equation above says they have to satisfy
y^2 = 4x^3  g_2 x  g_3.
Here x and y are complex numbers of course. But look what this means:
it means that if we look at the pairs of complex numbers (x,y)
satisfying the cubic equation y^2 = 4x^3  g_2 x  g_3, we get something
that looks just like a torus! This is called an elliptic curve, since
for algebraic geometers a "curve" is the set of solutions (x,y) of some
polynomial in two *complex* variables  not two real variables.
So  an "elliptic curve" is basically just the solutions of a cubic equation in
two variables. Actually, we want to rule out curves that have
singularities, that is, places where there's no unique tangent line to
the curve, as in y^2 = x^3 or y^2 = x^2(x+1)  draw these in the real
plane and you'll see what I mean. Anyway, all elliptic curves can, by
change of variables, be made to look like our favorite one,
y^2 = 4x^3  g_2 x  g_3.
There are lots of more fancy ways of thinking about elliptic curves, and
one is to think of the fact that they look like a torus as the key part.
In a book on algebraic geometry you might see an elliptic curve as a
curve with genus one (i.e., with one "handle," like a torus has). One
nice thing about a torus is that is a group. That is, we know how to
add complex numbers, and we can add modulo elements of the lattice L,
so the torus becomes a group with addition mod L as the group operation.
This is simple enough, but it means that when we look at the solutions
of
y^2 = 4x^3  g_2 x  g_3
they must form a group somehow, and viewed this way it's not at all
obvious! Nonetheless, there is a beautiful geometric description of the
group operation in these terms  I'll leave this for Knapp to explain..
Let me wrap this up  the story goes on and on, but I'm getting tired 
with a bit about what it has to do with number theory. It has a lot to
do with Diophantine equations, where one wants integer, or rational
solutions to a polynomial equation. Suppose that g_2 and g_3 are
rational, and one has some solutions to the equation
y^2 = 4x^3  g_2 x  g_3.
Then it turns out that one can use the group operation on the elliptic
curve to get new solutions! Actually, it seems as if Diophantus knew
this way back when in some special cases. For example, for the problem
y(6  y) = x^3  x
Diophantus could start with the trivial solution (x,y) = (1,0), do some
mysterious stuff, and get the solution (17/9,26/27). Knapp explains how
this works in the Overview section, but then more deeply later.
Basically, it uses the fact that this curve is an elliptic curve, and
uses the group structure.
In fact, one can solve mighty hardseeming Diophantine problems using
these ideas. Knapp talks a bit about a problem Fermat gave to Mersenne
in 1643  this increased my respect for Fermat a bit. He asked, find a
Pythagorean triple (X,Y,Z), that is:
X^2 + Y^2 = Z^2,
such that Z is a square number and X + Y is too! One can solve this
using elliptic curves. I don't know if Mersenne got it  the answer is
at the end of this post, but heavyduty number theorists out there
might enjoy trying this one if they don't know it already.
Some more stuff:
2) Closed string field theory, strong homotopy Lie algebras and the
operad actions of moduli spaces, by Jim Stasheff, preprint available as
hepth/9304061.
One conceptually pleasing approach to string theory is closed string field
theory, where one takes as the basic object unparametrized maps from
circle into a manifold M representing "space", i.e., elements of
Maps(S^1,M)/Diff^+(S^1).
A state of closed string field theory would be roughly a function on the
above set. Then one tries to define all sorts of operations on these
states, in order to define write down ways the strings can interact.
For example, there is a "convolution product" on these functions which
almost defines a Lie algebra structure. However, the Jacobi identity
only holds "up to homotopy," so we have an algebraic structure called a
homotopy Lie algebra. Physicists would say that the Jacobi identity
holds modulo a BRST exact term. This is just the beginning of quite a
big bunch of mathematics being developed by Stasheff, Zwiebach, Getzler,
Kapranov and many others. My main complaint with the physics is that
all these structures seem to depend on choosing a Riemannian metric on
M  a socalled "background metric." Since string theory is supposed to
include a theory of quantum gravity it is annoying to have this
Godgiven background metric stuck in at the very start. Perhaps I just
don't understand this stuff. I am looking around for stuff on
backgroundindependent closed string field theory, since I have lots of
reason to believe that it's related to the loop representation of
quantum gravity. Unfortunately, I scarcely know the subject  I had
hoped Stasheff's work would help me, but it seems that this metric
always enters.
3) A geometrical presentation of the surface mapping class group and
surgery, by Sergey Matveev and Michael Polyak, preprint.
This paper shows how to express the mapping class group of a surface in
terms of tangles. This gives a nice relationship between two approaches
to 3d TQFTs (topological quantum field theories): the Heegard
decomposition approach, and the surgery on links approach.
4) Invariants of 3manifolds and conformal field theories, by Micheal
Polyak, preprint.
The main good thing about this paper in my opinion is that it
simplifies the definition of a modular tensor category. Recall that
Moore and Seiberg showed how any string theory (more precisely, any
rational conformal field theory) gave rise to a modular tensor category,
and then Crane showed that any modular tensor category gave rise to a 3d
TQFT. Unfortunately a modular tensor category seems initially to be a
rather baroque mathematical object. In this paper Polyak shows how to
get lots of the structure of a modular tensor category from just the
"fusion" and "braiding" operators, subject to some mild conditions. I
have a conjecture that all nonnegative link invariants (in the sense
of my paper on tangles and quantum gravity) give rise to modular tensor
categories, and this simplifies things to the point where maybe I might
eventually be able to prove it. There are lots of nice pictures here,
too, by the way.

Previous editions of "This Week's Finds," and other expository posts
on mathematical physics, are available by anonymous ftp from
math.princeton.edu, thanks to Francis Fung. They are in the directory
/pub/fycfung/baezpapers. The README file contains lists of the
papers discussed in each week of "This Week's Finds."
Please don't ask me about hepth and grqc; instead, read the
sci.physics FAQ or the file "preprint.info" in /pub/fycfung/baezpapers.
Answer to puzzle:
X = 1061652293520
Y = 4565486027761
Z = 4687298610289
ence CONSTANT by Liouville's theorem. Well, so a constant function has
all the wonderful properties we want  but that's too boring!
So let's allow it to have poles! But let's keep it as nice as possible,
so let's have the only poles occur at the lattice points
L = { n omega_1 +twf_ascii/week130000064400020410000157000000475310774011336400141360ustar00baezhttp00004600000001Also available at http://math.ucr.edu/home/baez/week130.html
February 27, 1999
This Week's Finds in Mathematical Physics (Week 130)
John Baez
All sorts of cool stuff is happening in physics  and I don't mean
mathematical physics, I mean real live experimental physics! I feel
slightly guilty for not mentioning it on This Week's Finds. Let me
atone.
Here's the big news in a nutshell: we may have been wrong about four
fundamental constants of nature. We thought they were zero, but maybe
they're not! I'm talking about the masses of the neutrinos and the
cosmological constant.
Let's start with neutrinos.
There are three kinds of neutrinos: electron, muon, and tau neutrinos.
They are closely akin to the charged particles whose names they borrow 
the electron, muon and tau  but unlike those particles they are
electrically neutral and very light. They are rather elusive, since
they interact only via the weak force and gravity. I'm sure you've all
heard how a neutrino can easily make it through hundreds of light years
of lead without being absorbed.
But despite their ghostly nature, neutrinos play a very real role in
physics, since radioactive decay often involves a neutron turning into a
proton while releasing an electron and an electron antineutrino. (In
fact, Pauli proposed the existence of neutrinos in 1930 to account for a
little energy that went missing in this process. They were only
directly observed in 1956.) Similarly, in nuclear fusion, a proton may
become a neutron while releasing a positron and an electron neutrino.
For example, when a type II supernova goes off, it emits so many
neutrinos that if you're anywhere nearby, they'll kill you before
anything else gets to you! Indeed, in 1987 a supernova in the Large
Magellanic Cloud, about 100,000 light years away, was detected by four
separate neutrino detectors.
I said neutrinos were "very light", but just how light? So far most
work has only given upper bounds. In the 1980s, the Russian ITEP group
claimed to have found a nonzero mass for the electron neutrino, but this
was subsequently blamed on problems with their apparatus. As of now,
laboratory experiments give upper bounds of 4.4 eV for the electron
neutrino mass, .17 MeV for the muon neutrino, and 18 MeV for the tau
neutrino. By contrast, the electron's mass is .511 MeV, the muon's is
106 MeV, and the tau's is a whopping 1771 MeV.
For this reason, the conventional wisdom used to be that neutrinos were
massless. After all, the electron neutrino is definitely far lighter
than any known particle except the photon  which is massless. The
larger upper bounds on the other neutrino's masses are mainly due to
the greater difficulty in doing the experiments.
Having neutrinos be massless would also nicely explain their most
stunning characteristic, namely that they're only found in a lefthanded
form. What I mean by this is that they spin counterclockwise when
viewed headon as they come towards you. It turns out that this
violation of leftright symmetry comes fairly easily to massless
particles, but only with more difficulty to massive ones. The reason is
simple: massless particles move at the speed of light, so you can't
outrun them. Thus everyone, regardless of their velocity, agrees on
what it means for such a particle to be spinning one way or another as
it comes towards them. This is not the case for a massive particle!
There was, however, a fly in the ointment. Since the sun is powered by
fusion, it should emit lots of neutrinos. In fact, the standard solar
model predicts that here on earth we are bombarded by 60 billion solar
neutrinos per square centimeter per second! So in the late 1960s, a
team led by Ray Davis set out to detect these neutrinos by putting a
tank of 100,000 gallons of perchloroethylene down into a gold mine in
Homestake, South Dakota. Lots of different nuclear reactions are going
on in the sun, producing neutrinos of different energies. The Homestake
experiment can only detect the most energetic ones  those produced when
boron8 decays into beryllium8. These neutrinos have enough energy to
turn chlorine37 in the tank into argon37. Being a noble gas, the
argon can be separated out and measured. This is not easy  one only
expects about 4 atoms of argon a day! So the experiment required
extreme care and went on for decades.
They only saw about a quarter as many neutrinos as expected.
Of course, with an experiment as delicate as this, there are always many
possibilities for error, including errors in the standard solar model.
So a Japanese group decided to use a tank of 2,000 tons of water in a
mine in Kamioka to look for solar neutrinos. This "Kamiokande"
experiment used photomultiplier tubes to detect the Cherenkov radiation
formed by electrons that happen to be hit by neutrinos. Again it was
sensitive only to highenergy neutrinos.
After 5 years, they started seeing signs of a correlation between
sunspot activity and their neutrino count. Interesting. But more
interesting still, they didn't see as many neutrinos as expected.
Only about half as many, in fact.
Starting in the 1990s, various people began to build detectors that
could detect lowerenergy neutrinos  including those produced in the
dominant fusion reactions powering the sun. For this it's good to use
gallium71, which turns to germanium71 when bombarded by neutrinos.
The GALLEX detector in Italy uses 30 tons of gallium in the form of
gallium chloride dissolved in water. The SAGE detector, located in a
tunnel in the Caucasus mountains, uses 60 tons of molten metallic
gallium. This isn't quite as scary as it sounds, because gallium has a
very low melting point  it melts in your hand! But still, of course,
these experiments are very difficult.
Again, these experiments didn't see as many neutrinos as expected.
By this point, the theorists had worked themselves into a full head of
steam trying to account for the missing neutrinos. Currently the most
popular theory is that some of the electron neutrinos have turned into
muon and tau neutrinos by the time they reach earth. These other
neutrinos would be not be registered by our detectors.
Folks call this hypothetical process "neutrino oscillation". For it
to happen, the neutrinos need to have a nonzero mass. After all,
a massless particle moves at the speed of light, so it doesn't experience
any passage of time  thanks to relativistic time dilation. Only particles
with mass can become something else while they are whizzing along minding
their own business.
If in fact you posit a small mass for the neutrinos, oscillations happen
automatically as long as the "mass eigenstates" are different from the
"flavor eigenstates". By "flavor" we mean whether the neutrino is an
electron, muon or tau neutrino. For simplicity, imagine that the state
of a neutrino at rest is given by a vector whose 3 components are the
amplitudes for it to be these three different flavors. If all but one
of these components are zero we have a neutrino with a definite
flavor  a "flavor eigenstate". On the other hand, the energy of a
particle at rest is basically just its mass. Thus in the present
context the energy of the neutrino is described by a 3 x 3 selfadjoint
matrix H, the "Hamiltonian", whose eigenvectors are called "mass
eigenstates". These may or may not be the same as the flavor
eigenstates! Schroedinger's equation says that any state psi of the
neutrino evolves as follows:
d psi/dt = iH psi
Thus if psi starts out being a mass eigenstate it stays a mass eigenstate.
But if it starts out being a flavor eigenstate, it won't stay a flavor
eigenstate  unless the mass and flavor eigenstates coincide! Instead, it
will oscillate.
I bet you were wondering when the math would start. Don't worry, there
won't be much this time.
Anyway, for other particles, like quarks, it's wellknown that the mass
and flavor eigenstates *don't* coincide. So we shouldn't be surprised
at neutrino oscillations, at least if neutrinos actually have nonzero
mass.
Actually things are more complicated than I'm letting on. In addition
to oscillating in empty space, it's possible that neutrinos oscillate
*more* as they are passing through the sun itself, thanks to something
called the MSW effect  named after Mikheyev, Smirnov and Wolfenstein.
And there are two different ways for neutrinos to have mass, depending
on whether they are Dirac spinors or Majorana spinors (see "week93").
But I don't want to get caught up in theoretical nuances here! I want
to talk about experiments, and I haven't even gotten to the new stuff
yet  the stuff that's getting everybody *really* confused!
First of all, there's now some laboratory evidence for neutrino
oscillations coming from the Liquid Scintillator Neutrino Detector at
Los Alamos. What these folks do is let positively charged pions decay
into antimuons and muon neutrinos. Then they check to see if any muon
neutrinos become electron neutrinos. They claim that they do! They
also claim to see evidence of muon antineutrinos becoming electron
antineutrinos.
Secondly, and more intriguing still, there are a bunch of experiments
involving atmospheric neutrinos: SuperKamiokande, Soudan 2, IMB, and
MACRO. You see, when cosmic rays smack into the upper atmosphere, they
produce all sorts of particles, including electron and muon neutrinos
and their corresponding antineutrinos. Cosmic ray experts think they
know how many of each sort of neutrino should be produced. But the
experimenters down on the ground are seeing different numbers!
Again, this could be due to neutrino oscillations. But what's REALLY
cool is that the numbers seem to depend on where the neutrinos are
coming from: from the sky right above the detector, from right below the
detector  in which case they must have come all the way through the
earth  or whatever. Neutrinos coming from different directions take
different amounts of time to get from the upper atmosphere to the
detector. Thus an obvious explanation for the experimental results is
that we're actually seeing the oscillation process AS IT TAKES PLACE.
If this is true, we can try to get detailed information about the
neutrino mass matrix from the numbers these experiments are measuring!
And this is exactly what people have been doing. But they're finding
something very strange. If all the experiments are right, and nobody is
making any mistakes, it seems that NO choice of neutrino mass matrix
really fits all the data! To fit all the data, folks need to do
something drastic  like posit a 4th kind of neutrino!
Now, it's no light matter to posit another neutrino. The known
neutrinos couple to the weak force in almost identical ways. This
allows one to create equal amounts of neutrinoantineutrino pairs of
all 3 flavors by letting Z bosons decay  the Z being the neutral
carrier of the weak force. When a Z boson seemingly decays into
"nothing", we can safely bet that it has decayed into a neutrino
antineutrino pair. In 1989, an elegant and famous experiment at CERN
showed that Z bosons decay into "nothing" at exactly the rate one would
expect if there were 3 flavors of neutrino. Thus there can only be
extra flavors of neutrino if they are very massive, if they couple very
differently to the weak force, or if some other funny business is going
on.
Now, electron or muon neutrinos are unlikely to oscillate into a very
*massive* sort of neutrino  basically because of energy conservation.
So if we want an extra neutrino to explain the experimental results
we find ourselves stuck with, it'll have to be one that couples to the
weak force very differently from the ones we know. A simple, but
drastic, possibility is that it not interact via the weak force at all!
Folks call this a "sterile" neutrino.
Now, sterile neutrinos would blow a big hole in the Standard Model,
much more so than plain old *massive* neutrinos. So things are getting
very interesting.
Wilczek recently wrote a nice easytoread paper describing arguments
that *massive* neutrinos fit in quite nicely with the possibility that
the Standard Model is just part of a bigger, better theory  a "Grand
Unified Theory". I sketched the basic ideas of the SU(5) and SO(10)
grand unified theories in "week119". Recall that in the SU(5) theory,
the lefthanded parts of all fermions of a given generation fit into two
irreducible representations of SU(5)  a 5dimensional rep and a
10dimensional rep. For example, for the first generation, the
5dimensional rep consists of the lefthanded down antiquark (which
comes in 3 colors), the lefthanded electron, and the lefthanded electron
neutrino. The 10dimensional rep consists of the lefthanded up quark,
down quark, and up antiquark (which come in 3 colors each), together
with the lefthanded positron.
In the SO(10) theory, all these particles AND ONE MORE fit into a single
16dimensional irreducible representation of SO(10). What could this
extra particle be?
Well, since this extra particle transforms trivially under SU(5), it
must not feel the electromagnetic, weak or strong force! Thus it's
tempting to take this missing particle to be the lefthanded electron
antineutrino. Of course, we don't see such a particle  we only see
antineutrinos that spin clockwise. But if neutrinos are massive Dirac
spinors there must be such a particle, and having it not feel the
electromagnetic, weak or strong force would nicely explain *why* we
don't see it.
Grotz and Klapdor consider this possibility in their book on the weak
interaction (see below), but unfortunately, it seems this theory would
make the electron neutrino have a mass of about 5 MeV  much too big!
Sigh. So Wilczek, following the conventional wisdom, assumes the missing
particle is very massive  he calls it the "N". And he summarizes some
arguments that this massive particle could help give the neutrinos very
small masses, via something called the "seesaw mechanism". Unfortunately
I don't have the energy to describe this now, so for more you should look
at his paper (referred to below).
To wrap up, let me just say one final thing about the cosmic significance
of the neutrino. Massive neutrinos could account for some of the "missing
mass" that cosmologists are worrying about. So there's an indirect
connection between the neutrino mass and the cosmological constant!
The cosmological constant is essentially the energy density of the vacuum.
It was long assumed to be zero, but now there are some glimmerings of
evidence that it's not. In fact, some people are quite convinced that
it's not. The fate of the universe hangs in the balance....
Unfortunately I am too tired now to say much more about this.
So let me just give you a nice easy startingpoint:
1) Special Report: Revolution in Cosmology, Scientific American,
January 1999. Includes the articles "Surveying spacetime with
supernovae" by Craig J. Horgan, Robert P. Kirschner and Nicholoas B.
Suntzeff, "Cosmological antigravity" by Lawrence M. Krauss, and
"Inflation in a lowdensity universe" by Martin A. Bucher and David N.
Spergel.
How can you learn more about neutrinos? It can't hurt to start here:
2) Nikolas Solomey, The Elusive Neutrino, Scientific American Library,
1997.
If you want to dig in deeper, you need to learn about the weak force,
since we've only seen neutrinos via their weak interaction with other
particles. The following book is a great place to start:
3) K. Grotz and H. V. Klapdor, The Weak Interaction in Nuclear, Particle
and Astrophysics, Adam Hilger, Bristol, 1990.
Then you'll be ready for this book, which examines every aspect of
neutrinos in detail  complete with copies of historical papers:
4) Klaus Winter, ed., Neutrino Physics, Cambridge U. Press, Cambridge,
1991.
And then, if you want to study the possibility of *massive* neutrinos,
you should try this:
5) Felix Boehm and Petr Vogel, Physics of Massive Neutrinos, Cambridge
U. Press, Cambridge, 1987.
But neutrino physics is moving fast, and lots of the new stuff hasn't
made its way into books yet, so you should also look at other stuff.
For links to lots of great neutrino websites, including websites for
most of the experiments I mentioned, try:
6) The neutrino oscillation industry,
http://www.hep.anl.gov/NDK/hypertext/nu_industry.html
For some recent general overviews, try these:
7) Paul Langacker, Implications of neutrino mass,
http://dept.physics.upenn.edu/neutrino/jhu/jhu.html
8) Boris Kayser, Neutrino mass: where do we stand, and where are we
going?, preprint available as hepph/9810513
For information on various experiments, try these:
9) GALLEX collaboration, GALLEX solar neutrino observations: complete
results for GALLEX II, Phys. Lett. B357 (1995), 237247.
Final results of the CR51 neutrino source experiments in GALLEX,
Phys. Lett. B420 (1998), 114126.
GALLEX solar neutrino observations: results for GALLEX IV, Phys.
Lett. B447 (1999), 127133.
11) SAGE collaboration, Results from SAGE, Phys. Lett B328 (1994),
234248.
The RussianAmerican gallium experiment (SAGE) CR neutrino source
measurement, Phys. Rev. Lett. 77 (1996), 47084711.
12) LSND collaboration, Evidence for neutrino oscillations from
muon decay at rest, Phys. Rev. C54 (1996) 26852708, preprint available
as nuclex/9605001.
Evidence for antimuonneutrino > antielectronneutrino oscillations
from the LSND experiment at LAMPF, Phys. Rev. Lett 77 (1996), 30823085,
preprint available as nuclex/9605003.
Evidence for nu_mu > nu_e oscillations from LSND, Phys. Rev. Lett. 81
(1998), 17741777, preprint available as nuclex/9709006.
Results on nu_mu > nu_e oscillations from pion decay in flight,
Phys. Rev. C58 (1998), 24892511.
13) SuperKamiokande collaboration, Evidence for oscillation of
atmospheric neutrinos, Phys. Rev. Lett 81 (1998), 15621567,
preprint available as hepex/9807003.
14) MACRO collaboration, Measurement of the atmospheric neutrino
induced upgoing muon flux, Phys. Lett. B434 (1998), 451457,
preprint available as hepex/9807005.
15) IMB collaboration, A search for muonneutrino oscillations with
the IMB detector, Phys. Rev. Lett 69 (1992), 10101013.
For a fairly modelindependent attempt to figure out something about
neutrino masses from the latest crop of experiments, see:
16) V. Barger, T. J. Weiler, and K. Whisnant, Inferred 4.4 eV upper
limits on the muon and tauneutrino masses, preprint available as
hepph/9808367.
For a nice summary of the data, and an argument that it's evidence
for the existence of a sterile neutrino, see:
17) David O. Caldwell, The status of neutrino mass, preprint available
as hepph/9804367.
For a very readable argument that massive neutrinos are evidence for a
supersymmetric SO(10) grand unified theory, see
18) Frank Wilczek, Beyond the Standard Model: this time for real,
preprint available as hepph/9809509.
Finally, with all this talk of cracks in the Standard Model, it's nice
to think again about the rise of the Standard Model. The following book
is packed with the reminiscences of many theorists and experimentalists
involved in developing this wonderful theory of particles and forces,
including Bjorken, 't Hooft, Veltman, Susskind, Polyakov, Richter,
Iliopoulos, GellMann, Weinberg, Lederman, Goldhaber, Cronin, and
Kobayashi:
19) Lilian Hoddeson, Laurie Brown, Michael Riordan and Max Dresden,
eds., The Rise of the Standard Model: Particle Physics in the 1960s and
1970s.
It's a must for anyone with an interest in the history of physics!

Previous issues of "This Week's Finds" and other expository articles on
mathematics and physics, as well as some of my research papers, can be
obtained at
http://math.ucr.edu/home/baez/
For a table of contents of all the issues of This Week's Finds, try
http://math.ucr.edu/home/baez/twf.html
A simple jumpingoff point to the old issues is available at
http://math.ucr.edu/home/baez/twfshort.html
If you just want the latest issue, go to
http://math.ucr.edu/home/baez/this.week.html
get from the upper atmosphere to the
detector. Thus an obvious explanation for the experimental results is
that we're actually seeing the oscillation process AS IT Ttwf_ascii/week131000064400020410000157000000464231022134257100141260ustar00baezhttp00004600000001Also available at http://math.ucr.edu/home/baez/week131.html
March 7, 1999
This Week's Finds in Mathematical Physics (Week 131)
John Baez
I've been thinking more about neutrinos and their significance for grand
unified theories. The term "grand unified theory" sounds rather
pompous, but in its technical meaning it refers to something with
limited goals: a quantum field theory that attempts to unify all the
forces *except* gravity. This limitation lets you pretend spacetime
is flat.
The heyday of grand unified theories began in the mid1970s, shortly
after the triumph of the Standard Model. As you probably know, the
Standard Model is a quantum field theory describing all known particles
and all the known forces except gravity: the electromagnetic, weak and
strong forces. The Standard Model treats the electromagnetic and weak
forces in a unified way  so one speaks of the "electroweak" force 
but it treats the strong force seperately.
In 1975, Georgi and Glashow invented a theory which fit all the known
particles of each generation into two irreducible representations of
SU(5). Their theory had some very nice features: for example, it
unified the strong force with the electroweak force, and it explained
why quark charges come in multiples of 1/3. It also made some new
predictions, most notably that protons decay with a halflife of
something like 10^{29} or 10^{30} years. Of course, it's slightly
inelegant that one needs *two* irreducible representations of SU(5) to
account for all the particles of each generation. Luckily SU(5) fits
inside SO(10) in a nice way, and Georgi used this to concoct a slightly
bigger theory where all 15 particles of each generation, AND ONE MORE,
fit into a single irreducible representation of SO(10). I described the
mathematics of all this in "week119", so I won't do so again here.
What's the extra particle? Well, when you look at the math, one obvious
possibility is a righthanded neutrino. As I explained last week, the
existence of a righthanded neutrino would make it easier for neutrinos
to have mass. And this in turn would allow "oscillations" between
neutrinos of different generations  possibly explaining the mysterious
shortage of electron neutrinos that we see coming from the sun.
This "solar neutrino deficit" had already been seen by 1975, so everyone
got very excited about grand unified theories. The order of the day
was: look for neutrino oscillations and proton decay!
A nice illustration of the mood of the time can be found in a talk
Glashow gave in 1980:
1) Sheldon Lee Glashow, The new frontier, in First Workshop on Grand
Unification, eds. Paul H. Frampton, Sheldon L. Glashow and Asim Yildiz,
Math Sci Press, Brookline Massachusetts, 1980, pp. 38.
I'd like to quote some of his remarks because it's interesting to
reflect on what has happened in the intervening two decades:
Pions, muons, positrons, neutrons and strange particles were
found without the use of accelerators. More recently, most
developments in elementary particle physics depended upon these
expensive artificial aids. Science changes quickly. A time may
come when accelerators no longer dominate our field: not yet, but
perhaps sooner than some may think.
Important discoveries await the next generation of accelerators.
QCD and the electroweak theory need further confirmation. We need
know how b quarks decay. The weak interaction intermediaries must
be seen to be believed. The top quark (or perversions needed by
topless theories) lurks just out of range. Higgs may wait to be
found. There could well be a fourth family of quarks and leptons.
There may even be unanticipated surprises. We need the new machines.
Of course we now know how the b (or "bottom") quark decays, we've seen
the t (or "top") quark, we've seen the weak interaction intermediaries,
and we're quite sure there is not a fourth generation of quarks and
leptons. There have been no unanticipated surprises. Accelerators grew
ever more expensive until the U.S. Congress withdrew funding for the
Superconducting Supercollider in 1993. The Higgs is still waiting to be
found or proved nonexistent. Experiments at CERN should settle that
issue by 2003 or so.
On the other hand, we have for the first time an apparently
correct *theory* of elementary particle physics. It may be, in
a sense, phenomenologically complete. It suggests the possibility
that there are no more surprises at higher energies, at least for
energies that are remotely accessible. Indeed, PETRA and ISR have
produced no surprises. The same may be true for PEP, ISABELLE, and
the TEVATRON. Theorists do expect novel higherenergy phenomena,
but only at absurdly inacessible energies. Proton decay, if it is
found, will reinforce the belief in the great desert extending from
100 GeV to the unification mass of 10^{14} GeV. Perhaps the desert
is a blessing in disguise. Ever larger and more costly machines
conflict with dwindling finances and energy reserves. All frontiers
come to an end.
You may like this scenario or not; it may be true or false. But,
it is neither impossible, implausible, or unlikely. And, do not
despair nor prematurely lament the death of particle physics. We
have a ways to go to reach the desert, with exotic fauna along the
way, and even the desolation of a desert can be interesting. The
end of the highenergy frontier in no ways implies the end of
particle physics. There are many ways to skin a cat. In this talk
I will indicate several exciting lines of research that are well
away from the highenergy frontier. Important results, perhaps even
extraordinary surprises, await us. But, there is danger on the way.
The passive frontier of which I shall speak has suffered years of
benign neglect. It needs money and manpower, and it must compete
for this with the accelerator establishment. There is no labor
union of physicists who work at accelerators, but sometimes it seems
there is. It has been argued that plans for accelerator construction
must depend on the "needs" of the working force: several thousands
of dedicated highenergy experimenters. This is nonsense. Future
accelerators must be built in accordance with scientific, not
demographic, prioriries. The new machines are not laborintensive,
must not be forced to be so. Not all high energy physicsts can be
accomodated at the new machines. The highenergy physicist has no
guaranteed right to work at an accelerator, he has not that kind of
job security. He must respond to the challenge of the passive
frontier.
Of course, the collapse of the highenergy physics job market and the
death of the Superconducting Supercollider give these words a certain
poignancy. But what is this "passive frontier" Glashow mentions? It
means particle physics that doesn't depend on very high energy particle
accelerators. He lists a number of options:
A) CP phenomenology. The Standard Model is not symmetrical under
switching particles and their antiparticles  called "charge conjugation",
or "C". Nor is it symmetrical under switching left and right  called
"parity", or "P". It's almost, but not quite, symmetrical under the
combination of both operations, called "CP". Violation of CP symmetry
is evident in the behavior of the neutral kaon. Glashow suggests looking
for CP violation in the form of a nonzero magnetic dipole moment for the
neutron. As far as I know, this has still not been seen.
B) New kinds of stable matter. Glashow proposes the search for new
stable particles as "an ambitious and risky field of scientific
endeavor". People have looked and haven't found anything.
C) Neutrino masses and neutrino oscilllations. Glashow claims that
"neutrinos should have masses, and should mix". He now appears to
be right. It took almost 20 years for the trickle of experimental
results to become the lively stream we see today, but it happened.
He urges "Let us not miss the next nearby supernova!" Luckily we
did not.
D) Astrophysical neutrino physics. In addition to solar neutrinos and
neutrinos from supernovae, there are other interesting connections
between neutrinos and astrophysics. The background radiation from the
big bang should contain neutrinos as well as the easiertosee photons.
More precisely, there should be about 100 neutrinos of each generation
per cubic centimeter of space, thanks to this effect. These "relic
neutrinos" have not been seen, but that's okay: by now they would be too
low in energy to be easily detected. Glashow notes that if neutrinos
had a nonzero mass, relic neutrinos could contribute substantially to
the total density of the universe. The heaviest generation weighing 30
eV or so might be enough to make the universe eventually recollapse! On
the other hand, for neutrinos to be gravitationally bound to galaxies,
they'd need to be at least 20 eV or so.
E) Magnetic monopoles. Most grand unified theories predict the existence
of magnetic monopoles due to "topological defects" in the Higgs fields.
Glashow urges people to look for these. This has been done, and they
haven't been seen.
F) Proton decay. As Glashow notes, proton decay would be the "king of
the new frontier". Reflecting the optimism of 1980, he notes that "to
some, it is a foregone conclusion that proton decay is about to be seen
by experiments now abuilding". But alas, people looked very hard and
did not find it! This killed the SU(5) theory. Many people switched to
supersymmetric theories, which are more compatible with very slow proton
decay. But with the continuing lack of new experiments to explain,
enthusiasm for grand unified theories gradually evaporated, and
theoretical particle physics took refuge in the elegant abstractions of
string theory.
But now, 20 years later, interest in grand unified theories seems
to be coming back. We have a rich body of mysterious experimental
results about neutrino oscillations. Somebody should explain them!
On a slightly different note, one of my little side hobbies is to study
the octonions and dream about how they might be useful in physics. One
place they show up is in the E6 grand unified theory  the next theory
up from the SO(10) theory. I said a bit about this in "week119", but I
just bumped into another paper on it in the same conference proceedings
that Glashow's paper appears is:
2) Feza Gursey, Symmetry breaking patterns in E_6, in First Workshop on
Grand Unification, eds. Paul H. Frampton, Sheldon L. Glashow and Asim
Yildiz, Math Sci Press, Brookline Massachusetts, 1980, pp. 3955.
He says something interesting that I want to understand someday  maybe
someone can explain why it's true. He says that E6 is a "complex" group,
E7 is a "pseudoreal" group, and E8 is a "real" group. This use of
terminology may be nonstandard, but what he means is that E6 admits
complex representations that are not their own conjugates, E7 admits
complex reps that are their own conjugates, and that all complex reps of
E8 are complexifications of real ones (and hence their own conjugates).
This should have something to do with the symmetry of the Dynkin diagram
of E6.
Octonions are also prominent in string theory and in the grand unified
theories proposed by my friends Geoffrey Dixon and Tony Smith  see
"week59", "week91", and "week104". I'll probably say more about this
someday....
The reason I'm interested in neutrinos is that I want to learn what
evidence there is for laws of physics going beyond the Standard Model
and general relativity. This is also why I'm trying to learn a bit of
astrophysics. The new hints of evidence for a nonzero cosmological
constant, the missing mass problem, the largescale structure of the
universe, and even the puzzling gammaray bursters  they're all food
for thought along these lines.
The following book caught my eye since it looked like just what I
need  an easy tutorial in the latest developments in cosmology:
3) Greg Bothun, Modern Cosmological Observations and Problems, Taylor &
Francis, London, 1998.
On reading it, some of the remarks about particle physics made me
unhappy. For example, Bothun says "the observed entropy S of the
universe, as measured by the ratio of baryons to photons, is ~ 5 x
10^{10}." But as Ted Bunn explained to me, the entropy is actually
correlated to the ratio of photons to baryons  the reciprocal of this
number. Bothun also calls the kinetic energy density of the field
postulated in inflationary cosmology, "essentially an entropy field that
currently drives the uniform expansion and cooling of the universe".
This makes no sense to me. There are also a large number of typos, the
most embarrassing being "virilizing" for "virializing".
But there's a lot of good stuff in this book! The author's specialty is
largescale structure, and I learned a lot about that. Just to set the
stage, recall that the Milky Way has a disc about 30 kiloparsecs in
diameter and contains roughly 100 or 200 billion stars. But our galaxy
is one of a cluster of about 20 galaxies, called the Local Group. In
addition to our galaxy and the Large and Small Magellanic Clouds which
orbit it, this contains the Andromeda Galaxy (also known as M31),
another spiral galaxy called M33, and a bunch of dwarf irregular
galaxies. The Local Group is about a megaparsec in radius.
This is typical. Galaxies often lie in clusters which are a few
megaparsecs in radius, containing from a handful to hundreds of big
galaxies. Some famous nearby clusters include the Virgo cluster (about
20 megaparsecs away) and the Coma cluster (about 120 megaparsecs away).
Thousands of clusters have been cataloged by Abell and collaborators.
And then there are superclusters, each typically containing 310
clusters in an elongated "filament" about 50 megaparsecs in diameter. I
don't mean to make this sound more neat than it actually is, because
nobody is very sure about intergalactic distances, and the structures
are rather messy. But there are various discernible patterns. For
example, superclusters tend to occur at the edges of larger roundish
"voids" which have few galaxies in them. These voids are very large,
about 100 or 200 megaparsecs across. In general, galaxies tend to
be falling into denser regions and moving away from the voids. For
example, the Milky Way is falling towards the center of the Local
Supercluster at about 300 kilometers per second, and the Local
Supercluster is also falling towards the next nearest one  the
HydraCentaurus Supercluster  at about 300 kilometers per second.
Now, if the big bang theory is right, all this stuff was once very
small, and the universe was much more homogeneous. Obviously gravity
tends to amplify inhomogeneities. The problem is to understand in a
quantitative way how these inhomogeneities formed as the universe grew.
Here are a couple of other books that I'm finding useful  they're
a bit more mathematical than Bothun's. I'm trying to stick to new
books because this subject is evolving so rapidly:
4) Jayant V. Narlikar, Introduction to Cosmology, Cambridge U.
Press, Cambridge, 1993.
5) Peter Coles and Francesco Lucchin, Cosmology: The Origin and
Evolution of Cosmic Structure, Wiley, New York, 1995.
While I was looking around, I also bumped into the following book on
black holes:
6) Sandip K. Chakrabarti, ed., Observational Evidence for Black Holes
in the Universe, Kluwer, Dordrecht, 1998.
It mentioned some objects I'd never heard of before. I want to tell you
about them, just because they're so cool!
Xray novae: First, what's a nova? Well, remember that a white dwarf is
a small, dense, mostly burntout star. When one member of a binary star
is a white dwarf, and the other dumps some of its gas on this white
dwarf, the gas can undergo fusion and emit a huge burst of energy  as
much as 10,000 times what the sun emits in a year. To astronomers it
may look like a new star is suddenly born  hence the term "nova". But
not all novae emit mostly visible light  some emit Xrays or even gamma
rays. A "Xray nova" is an Xray source that suddenly appears in a few
days and then gradually fades away in a year or less. Many of these are
probably neutron stars rather than white dwarfs. But a bunch are
probably black holes!
Blazars: A "blazar" is a galactic nucleus that's shooting out a jet of
hot plasma almost directly towards us, exhibiting rapid variations in
power. Like quasars and other active galactic nuclei, these are
probably black holes sucking in stars and other stuff and forming big
accretion disks that shoot out jets of plasma from both poles.
Mega masers: A laser is a source of coherent light caused by stimulated
emission  a very quantummechanical gadget. A maser is the same sort
of thing but with microwaves. In fact, masers were invented before
lasers  they are easier to make because the wavelength is longer. In
galaxies, clouds of water vapor, hydroxyl, silicon monoxide, methanol
and other molecules can form enormous natural masers. In our galaxy the
most powerful such maser is W49N, which has a power equal to that of the
Sun. But recently, still more powerful masers have been bound in other
galaxies, usually associated with active galactic nuclei. These are
called "mega masers" and they are probably powered by black holes. The
first mega maser was discovered in 1982; it is a hydroxyl ion maser in
the galaxy IC4553, with a luminosity about 1000 times that of our sun.
Subsequently people have found a bunch of water mega masers. The most
powerful so far is in TXFS2226184  it has a luminosity of about 6100
times that of the Sun!

Addendum: Here is something from Allen Knutson in response to my remark
that E6 has complex representations that aren't their own conjugates.
I hoped that this is related to the symmetry of the Dynkin diagram
of E6, and Allen replied:
It does. The automorphism G>G that exchanges representations with their
duals, the Cartan involution, may or may not be an inner automorphism.
The group of outer automorphisms of G (G simple) is iso to the diagram
automorphism group. So no diagram auts, means the Cartan involution is inner,
means all reps are iso to their duals, i.e. possess invariant bilinear forms.
(Unfortunately it's not iff  the D_n's alternate between whether the
Cartan involution is inner, much as their centers alternate between
Z_4 and Z_2^2.)
Any rep either has complex trace sometimes, or a real, or a quaternionic
structure, morally because of ArtinWedderburn applied to the real
group algebra of G. Given a rep one can find out which by looking at
the integral over G of Tr(g^2), which comes out 0, 1, or 1 (respectively).
This is the "Schur indicator" and can be found in Serre's LinReps of
Finite Groups.
Allen K.

Previous issues of "This Week's Finds" and other expository articles on
mathematics and physics, as well as some of my research papers, can be
obtained at
http://math.ucr.edu/home/baez/
For a table of contents of all the issues of This Week's Finds, try
http://math.ucr.edu/home/baez/twf.html
A simple jumpingoff point to the old issues is available at
http://math.ucr.edu/home/baez/twfshort.html
If you just want the latest issue, go to
http://math.ucr.edu/home/baez/this.week.html
ting the optimism of 1980, he notes that "to
some, it is a foregone conclusion that proton decay is about to be seen
by experiments now abuilding". But alas, people looked very hard and
did not find it! This killed the SU(5) theory. Mtwf_ascii/week132000064400020410000157000000560160774011336400141360ustar00baezhttp00004600000001Also available at http://math.ucr.edu/home/baez/week132.html
April 2, 1999
This Week's Finds in Mathematical Physics  Week 132
John Baez
Today I want to talk about ncategories and quantum gravity again. For
starters let me quote from a paper of mine about this stuff:
1) John Baez, Higherdimensional algebra and Planckscale physics,
to appear in Physics Meets Philosophy at the Planck Scale, eds.
Craig Callender and Nick Huggett, Cambridge U. Press. Preprint
available as grqc/9902017.
By the way, this book should be pretty fun to read  it'll contain papers
by both philosophers and physicists, including a bunch who have already
graced the pages of This Week's Finds, like Barbour, Isham, Rovelli,
Unruh, and Witten. I'll say more about it when it comes out.
Okay, here are some snippets from this paper. It starts out talking
about the meaning of the Planck length, why it may be important in
quantum gravity, and what a theory of quantum gravity should be like:

Two constants appear throughout general relativity: the speed of light
c and Newton's gravitational constant G. This should be no
surprise, since Einstein created general relativity to reconcile the
success of Newton's theory of gravity, based on instantaneous action at
a distance, with his new theory of special relativity, in which no
influence travels faster than light. The constant c also appears in
quantum field theory, but paired with a different partner: Planck's
constant hbar. The reason is that quantum field theory takes
into account special relativity and quantum theory, in which hbar
sets the scale at which the uncertainty principle becomes important.
It is reasonable to suspect that any theory reconciling general
relativity and quantum theory will involve all three constants c, G,
and hbar. Planck noted that apart from numerical factors there
is a unique way to use these constants to define units of length, time,
and mass. For example, we can define the unit of length now
called the `Planck length' as follows:
L = sqrt(hbar G /c^3)
This is extremely small: about 1.6 x 10^{35} meters. Physicists have
long suspected that quantum gravity will become important for
understanding physics at about this scale. The reason is very simple:
any calculation that predicts a length using only the constants c, G and
hbar must give the Planck length, possibly multiplied by an unimportant
numerical factor like 2pi.
For example, quantum field theory says that associated to any mass m
there is a length called its Compton wavelength, L_C, such that
determining the position of a particle of mass m to within one Compton
wavelength requires enough energy to create another particle of that
mass. Particle creation is a quintessentially quantumfieldtheoretic
phenomenon. Thus we may say that the Compton wavelength sets the
distance scale at which quantum field theory becomes crucial for
understanding the behavior of a particle of a given mass. On the other
hand, general relativity says that associated to any mass m there is a
length called the Schwarzschild radius, L_S, such that compressing
an object of mass m to a size smaller than this results in the
formation of a black hole. The Schwarzschild radius is roughly the
distance scale at which general relativity becomes crucial for
understanding the behavior of an object of a given mass. Now, ignoring
some numerical factors, we have
L_C = hbar/mc
and
L_S = Gm/c^2.
These two lengths become equal when m is the Planck mass. And
when this happens, they both equal the Planck length!
At least naively, we thus expect that both general relativity and
quantum field theory would be needed to understand the behavior of an
object whose mass is about the Planck mass and whose radius is about the
Planck length. This not only explains some of the importance of the
Planck scale, but also some of the difficulties in obtaining
experimental evidence about physics at this scale. Most of our
information about general relativity comes from observing heavy objects
like planets and stars, for which L_S >> L_C. Most of our information
about quantum field theory comes from observing light objects like
electrons and protons, for which L_C >> L_S. The Planck mass is
intermediate between these: about the mass of a largish cell. But the
Planck length is about 10^{20} times the radius of a proton! To study
a situation where both general relativity and quantum field theory are
important, we could try to compress a cell to a size 10^{20} times that
of a proton. We know no reason why this is impossible in principle.
But we have no idea how to actually accomplish such a feat.
There are some wellknown loopholes in the above argument. The
`unimportant numerical factor' I mentioned above might actually be very
large, or very small. A theory of quantum gravity might make testable
predictions of dimensionless quantities like the ratio of the muon and
electron masses. For that matter, a theory of quantum gravity might
involve physical constants other than c, G, and hbar. The latter two
alternatives are especially plausible if we study quantum gravity as
part of a larger theory describing other forces and particles. However,
even though we cannot prove that the Planck length is significant for
quantum gravity, I think we can glean some wisdom from pondering the
constants c,G, and hbar  and more importantly, the physical insights
that lead us to regard these constants as important.
What is the importance of the constant c? In special relativity,
what matters is the appearance of this constant in the Minkowski
metric
ds^2 = c^2 dt^2  dx^2  dy^2  dz^2
which defines the geometry of spacetime, and in particular the lightcone
through each point. Stepping back from the specific formalism here, we
can see several ideas at work. First, space and time form a unified
whole which can be thought of geometrically. Second, the quantities
whose values we seek to predict are localized. That is, we can measure
them in small regions of spacetime (sometimes idealized as points).
Physicists call such quantities `local degrees of freedom'. And third,
to predict the value of a quantity that can be measured in some region
R, we only need to use values of quantities measured in regions that
stand in a certain geometrical relation to R. This relation is called
the `causal structure' of spacetime. For example, in a relativistic
field theory, to predict the value of the fields in some region R, it
suffices to use their values in any other region that intersects every
timelike path passing through R. The common way of summarizing this
idea is to say that nothing travels faster than light. I prefer to say
that a good theory of physics should have *local degrees of freedom
propagating causally*.
In Newtonian gravity, G is simply the strength of the gravitational
field. It takes on a deeper significance in general relativity, where
the gravitational field is described in terms of the curvature of the
spacetime metric. Unlike in special relativity, where the Minkowski
metric is a `background structure' given a priori, in general relativity
the metric is treated as a field which not only affects, but also is
affected by, the other fields present. In other words, the geometry of
spacetime becomes a local degree of freedom of the theory.
Quantitatively, the interaction of the metric and other fields is
described by Einstein's equation
G_{ab} = 8 pi G T_{ab}
where the Einstein tensor G_{ab} depends on the curvature of the
metric, while the stressenergy tensor T_{ab} describes the flow
of energy and momentum due to all the other fields. The role of the
constant G is thus simply to quantify how much the geometry of
spacetime is affected by other fields. Over the years, people have
realized that the great lesson of general relativity is that a good
theory of physics should contain no geometrical structures that affect
local degrees of freedom while remaining unaffected by them. Instead,
all geometrical structures  and in particular the causal structure 
should themselves be local degrees of freedom. For short, one says
that the theory should be backgroundfree.
The struggle to free ourselves from background structures began long
before Einstein developed general relativity, and is still not complete.
The conflict between Ptolemaic and Copernican cosmologies, the dispute
between Newton and Leibniz concerning absolute and relative motion, and
the modern arguments concerning the `problem of time' in quantum gravity
 all are but chapters in the story of this struggle. I do not have
room to sketch this story here, nor even to make more precise the
allimportant notion of `geometrical structure'. I can only point the
reader towards the literature, starting perhaps with the books by
Barbour and Earman, various papers by Rovelli, and the many references
therein.
Finally, what of hbar? In quantum theory, this appears most
prominently in the commutation relation between the momentum p and
position q of a particle:
pq  qp = i hbar,
together with similar commutation relations involving other pairs of
measurable quantities. Because our ability to measure two quantities
simultaneously with complete precision is limited by their failure to
commute, hbar quantifies our inability to simultaneously know
everything one might choose to know about the world. But there is far
more to quantum theory than the uncertainty principle. In practice,
hbar comes along with the whole formalism of complex Hilbert spaces
and linear operators.
There is a widespread sense that the principles behind quantum theory
are poorly understood compared to those of general relativity. This has
led to many discussions about interpretational issues. However, I do
not think that quantum theory will lose its mystery through such
discussions. I believe the real challenge is to better understand why
the mathematical formalism of quantum theory is precisely what it is.
Research in quantum logic has done a wonderful job of understanding the
field of candidates from which the particular formalism we use has been
chosen. But what is so special about this particular choice? Why, for
example, do we use complex Hilbert spaces rather than real or
quaternionic ones? Is this decision made solely to fit the experimental
data, or is there a deeper reason? Since questions like this do not yet
have clear answers, I shall summarize the physical insight behind hbar
by saying simply that a good theory of the physical universe should be a
*quantum theory*  leaving open the possibility of eventually saying
something more illuminating.
Having attempted to extract the ideas lying behind the constants c, G,
and hbar, we are in a better position to understand the task of
constructing a theory of quantum gravity. General relativity
acknowledges the importance of c and G but idealizes reality by treating
hbar as negligibly small. From our discussion above, we see that this
is because general relativity is a backgroundfree classical theory with
local degrees of freedom propagating causally. On the other hand,
quantum field theory as normally practiced acknowledges c and hbar but
treats G as negligible, because it is a backgrounddependent quantum
theory with local degrees of freedom propagating causally.
The most conservative approach to quantum gravity is to seek a theory
that combines the best features of general relativity and quantum field
theory. To do this, we must try to find a *backgroundfree
quantum theory with local degrees of freedom propagating causally*.
While this approach may not succeed, it is definitely worth pursuing.
Given the lack of experimental evidence that would point us towards
fundamentally new principles, we should do our best to understand
the full implications of the principles we already have!
From my description of the goal one can perhaps see some of the
difficulties. Since quantum gravity should be backgroundfree, the
geometrical structures defining the causal structure of spacetime should
themselves be local degrees of freedom propagating causally. This much
is already true in general relativity. But because quantum gravity
should be a quantum theory, these degrees of freedom should be treated
quantummechanically. So at the very least, we should develop a quantum
theory of some sort of geometrical structure that can define a causal
structure on spacetime.

Then I talk about topological quantum field theories, which are
backgroundfree quantum theories *without* local degrees of freedom, and
what we have learned from them. Basically what we've learned is that
there's a deep analogy between the mathematics of spacetime
(e.g. differential topology) and the mathematics of quantum theory.
This is interesting because in backgroundfree quantum theories we
expect that spacetime, instead of serving as a "stage" on which events
play out, actually becomes part of the play of events itself  and
must itself be described using quantum theory. So it's very interesting
to try to connect the concepts of spacetime and quantum theory. The
analogy goes like this:
DIFFERENTIAL TOPOLOGY QUANTUM THEORY
(n1)dimensional manifold Hilbert space
(space) (states)
cobordism between (n1)dimensional operator
manifolds (process)
(spacetime)
composition of cobordisms composition of operators
identity cobordism identity operator
And if you know a little category theory, you'll see what we have
here are two categories: the category of cobordisms and the category
of Hilbert spaces. A topological quantum field theory is a functor
from the first to the second....
Okay, now for some other papers:
2) Geraldine Brady and Todd H. Trimble. A string diagram calculus for
predicate logic, and C. S. Peirce's system Beta, available at
http://people.cs.uchicago.edu/~brady
Geraldine Brady and Todd H. Trimble, A categorical interpretation
of Peirce's propositional logic Alpha, Jour. Pure and Appl. Alg.
149 (2000), 213239.
Geraldine Brady and Todd H. Trimble, The topology of relational
calculus.
Charles Peirce is a famously underappreciated American philosopher who
worked in the late 1800s. Among other things, like being the father of
pragmatism, he is also one of the fathers of higherdimensional algebra.
As you surely know if you've read me often enough, part of the point of
higherdimensional algebra is to break out of "linear thinking". By
"linear thinking" I mean the tendency to do mathematics in ways that are
easily expressed in terms of 1dimensional strings of symbols. In his
work on logic, Peirce burst free into higher dimensions. He developed a
way of reasoning using diagrams that he called "existential graphs".
Unfortunately this work by Peirce was never published! One reason is
that existential graphs were difficult and expensive to print. As a
result, his ideas languished in obscurity.
By now it's clear that higherdimensional algebra is useful in physics:
examples include Feynman diagrams and the spin networks of Penrose.
The theory of ncategories is beginning to provide a systematic language
for all these techniques. So it's worth reevaluating Peirce's work and
seeing how it fits into the picture. And this is what the papers by
Brady and Trimble do!
3) J. Scott Carter, Louis H. Kauffman, and Masahico Saito, Structures
and diagrammatics of four dimensional topological lattice field theories,
to appear in Adv. Math., preprint available as math.GT/9806023.
We can get 3dimensional topological quantum field theories from certain
Hopf algebras. As I described in "week38", Crane and Frenkel made the
suggestion that by categorifying this construction we should get
4dimensional TQFTs from certain Hopf categories. This paper makes the
suggestion precise in a certain class of examples! Basically these are
categorified versions of the DijkgraafWitten theory.
4) J. Scott Carter, Daniel Jelsovsky, Selichi Kamada, Laurel Langford
and Masahico Saito, Quandle cohomology and statesum invariants of
knotted curves and surfaces, preprint available as math.GT/9903135.
Yet another attack on higher dimensions! This one gets invariants of
2links  surfaces embedded in R^4  from the cohomology groups of
"quandles". I don't really understand how this fits into the overall
scheme of higherdimensional algebra yet. They show their invariant
distinguishes between the 2twist spun trefoil (a certain sphere knotted
in R^4) and the same sphere with the reversed orientation.
5) Tom Leinster, Structures in higherdimensional category theory,
preprint available at http://www.dpmms.cam.ac.uk/~leinster
This is a nice tour of ideas in higherdimensional algebra. Right now
one big problem with the subject is that there are lots of approaches
and not a clear enough picture of how they fit together. Leinster's
paper is an attempt to start seeing how things fit together.
6) Claudio Hermida, Higherdimensional multicategories, slides of
a lecture given in 1997, available at http://www.math.mcgill.ca/~hermida
This talk presents some of the work by Makkai, Power and Hermida on
their definition of ncategories. For more on their work see "week107".
7) Carlos Simpson, On the BreenBaezDolan stabilization hypothesis for
Tamsamani's weak ncategories, preprint available as math.CT/9810058.
For quite a while now James Dolan and I have been talking about something
we call the "stabilization hypothesis". I gave an explanation of this in
"week121", but briefly, it says that the nth column of the following chart
(which extends to infinity in both directions) stabilizes after 2n+2 rows:
ktuply monoidal ncategories
n = 0 n = 1 n = 2
k = 0 sets categories 2categories
k = 1 monoids monoidal monoidal
categories 2categories
k = 2 commutative braided braided
monoids monoidal monoidal
categories 2categories
k = 3 " " symmetric weakly
monoidal involutory
categories monoidal
2categories
k = 4 " " " " strongly
involutory
monoidal
2categories
k = 5 " " " " " "
Carlos Simpson has now made this hypothesis precise and proved it using
Tamsamani's definition of ncategories! And he did it using the same
techniques that Graeme Segal used to study kfold loop spaces...
exploiting the relation between ncategories and homotopy theory. This
makes me really happy.
8) Mark Hovey, Model Categories, American Mathematical Society Mathematical
Surveys and Monographs, vol 63., Providence, Rhode Island, 1999. Preprint
available as http://www.math.uiuc.edu/Ktheory/0278/index.html
Speaking of that kind of thing, the technique of model categories is
really important for homotopy theory and ncategories, and this book
is a really great place to learn about it.
9) Frank Quinn, Groupcategories, preprint available as math.GT/9811047.
This one is about the algebra behind certain topological quantum field
theories. I'll just quote the abstract:
A groupcategory is an additively semisimple category with a
monoidal product structure in which the simple objects are
invertible. For example in the category of representations of a
group, 1dimensional representations are the invertible simple
objects. This paper gives a detailed exploration of "topological
quantum field theories" for groupcategories, in hopes of finding
clues to a better understanding of the general situation.
Groupcategories are classified in several ways extending results
of Frohlich and Kerler. Topological field theories based on homology
and cohomology are constructed, and these are shown to include
theories obtained from groupcategories by ReshetikhinTuraev
constructions. Braidedcommutative categories most naturally give
theories on 4manifold thickenings of 2complexes; the usual
3manifold theories are obtained from these by normalizing them
(using results of Kirby) to depend mostly on the boundary of the
thickening. This is worked out for groupcategories, and in
particular we determine when the normalization is possible and when
it is not.
10) Sjoerd Crans, A tensor product for Graycategories,
Theory and Applications of Categories, Vol. 5, 1999, No. 2, pp 1269,
available at http://www.tac.mta.ca/tac/volumes/1999/n2/n2.ps
A Graycategory is what some people call a semistrict 3category: not as
general as a weak 3category, but general enough. Technically,
Graycategories are defined as categories enriched over the category of
2categories equipped with a tensor product invented by John Gray. To
define semistrict 4categories one might similarly try to equip
Graycategories with a suitable tensor product. And this is what Crans
is studying. Let me quote the abstract:
In this paper I extend Gray's tensor product of 2categories to a
new tensor product of Graycategories. I give a description in
terms of generators and relations, one of the relations being an
``interchange'' relation, and a description similar to Gray's
description of his tensor product of 2categories. I show that this
tensor product of Graycategories satisfies a universal property
with respect to quasifunctors of two variables, which are defined
in terms of laxnatural transformations between
Graycategories. The main result is that this tensor product is
part of a monoidal structure on GrayCat, the proof requiring
interchange in an essential way. However, this does not give a
monoidal {(bi)closed} structure, precisely because of interchange
And although I define composition of laxnatural transformations,
this composite need not be a laxnatural transformation again,
making GrayCat only a partial GrayCatcategory.

Previous issues of "This Week's Finds" and other expository articles on
mathematics and physics, as well as some of my research papers, can be
obtained at
http://math.ucr.edu/home/baez/
For a table of contents of all the issues of This Week's Finds, try
http://math.ucr.edu/home/baez/twf.html
A simple jumpingoff point to the old issues is available at
http://math.ucr.edu/home/baez/twfshort.html
If you just want the latest issue, go to
http://math.ucr.edu/home/baez/this.week.html
elf  and
must itself be described using quantum theory. So it's very interesting
to try to connect the concepts of spacetime and quantum theory. The
analogy goes like this:
DIFFERENTIAL TOPOLOGY QUANTUM THEORY
(n1)dimensional manifold Hilbert space
(space) (states)
cobordism between (n1)dimensional operator
manifolds twf_ascii/week133000064400020410000157000000441200774011336400141300ustar00baezhttp00004600000001Also available at http://math.ucr.edu/home/baez/week133.html
April 23, 1999
This Week's Finds in Mathematical Physics  Week 133
John Baez
I'd like to start with a long quote from a paper by Ashtekar:
1) Abhay Ashtekar, Quantum Mechanics of Geometry, preprint available
as grqc/9901023.

During his Goettingen inaugural address in 1854, Riemann suggested that
the geometry of space may be more than just a fiducial, mathematical
entity serving as a passive stage for physical phenomena, and may in
fact have direct physical meaning in its own right. General relativity
provided a brilliant confirmation of this vision: curvature of space now
encodes the physical gravitational field. This shift is profound. To
bring out the contrast, let me recall the situation in Newtonian
physics. There, space forms an inert arena on which the dynamics of
physical systems  such as the solar system  unfolds. It is like a
stage, an unchanging backdrop for all of physics. In general
relativity, by contrast, the situation is very different. Einstein's
equations tell us that matter curves space. Geometry is no longer
immune to change. It reacts to matter. It is dynamical. It has
"physical degrees of freedom" in its own right. In general relativity,
the stage disappears and joins the troupe of actors! Geometry is a
physical entity, very much like matter.
Now, the physics of this century has shown us that matter has
constituents and the 3dimensional objects we perceive as solids are in
fact made of atoms. The continuum description of matter is an
approximation which succeeds brilliantly in the macroscopic regime but
fails hopelessly at the atomic scale. It is therefore natural to ask:
Is the same true of geometry? If so, what is the analog of the `atomic
scale?' We know that a quantum theory of geometry should contain three
fundamental constants of Nature, c, G, hbar, the speed of light,
Newton's gravitational constant and Planck's constant. Now, as Planck
pointed out in his celebrated paper that marks the beginning of quantum
mechanics, there is a unique combination,
L_P = sqrt(hbar G/c^3),
of these constants which has dimension of length. (L_P ~ 10^{33} cm.)
It is now called the Planck length. Experience has taught us that the
presence of a distinguished scale in a physical theory often marks a
potential transition; physics below the scale can be very different from
that above the scale. Now, all of our welltested physics occurs at
length scales much bigger than L_P. In this regime, the continuum
picture works well. A key question then is: Will it break down at the
Planck length? Does geometry have constituents at this scale? If so,
what are its atoms? Its elementary excitations? Is the spacetime
continuum only a `coarsegrained' approximation? Is geometry quantized?
If so, what is the nature of its quanta?
To probe such issues, it is natural to look for hints in the
procedures that have been successful in describing matter. Let us
begin by asking what we mean by quantization of physical quantities.
Take a simple example  the hydrogen atom. In this case, the answer is
clear: while the basic observables  energy and angular momentum 
take on a continuous range of values classically, in quantum mechanics
their eigenvalues are discrete; they are quantized. So, we can ask if
the same is true of geometry. Classical geometrical quantities such as
lengths, areas and volumes can take on continuous values on the phase
space of general relativity. Are the eigenvalues of corresponding
quantum operators discrete? If so, we would say that geometry is
quantized and the precise eigenvalues and eigenvectors of geometric
operators would reveal its detailed microscopic properties.
Thus, it is rather easy to pose the basic questions in a precise
fashion. Indeed, they could have been formulated soon after the advent
of quantum mechanics. Answering them, on the other hand, has proved to
be surprisingly difficult. The main reason, I believe, is the
inadequacy of standard techniques. More precisely, to examine the
microscopic structure of geometry, we must treat Einstein gravity
quantum mechanically, i.e., construct at least the basics of a quantum
theory of the gravitational field. Now, in the traditional approaches
to quantum field theory, one *begins* with a continuum, background
geometry. To probe the nature of quantum geometry, on the other hand,
we should *not* begin by assuming the validity of this picture. We must
let quantum gravity decide whether this picture is adequate; the theory
itself should lead us to the correct microscopic model of geometry.
With this general philosophy, in this article I will summarize the
picture of quantum geometry that has emerged from a specific approach to
quantum gravity. This approach is nonperturbative. In perturbative
approaches, one generally begins by assuming that spacetime geometry is
flat and incorporates gravity  and hence curvature  step by step by
adding up small corrections. Discreteness is then hard to unravel.
[Footnote: The situation can be illustrated by a harmonic oscillator:
While the exact energy levels of the oscillator are discrete, it would
be very difficult to "see" this discreteness if one began with a free
particle whose energy levels are continuous and then tried to
incorporate the effects of the oscillator potential step by step via
perturbation theory.]
In the nonperturbative approach, by contrast, there is no background
metric at all. All we have is a bare manifold to start with. All
fields  matter as well as gravity/geometry  are treated as dynamical
from the beginning. Consequently, the description can not refer to a
background metric. Technically this means that the full diffeomorphism
group of the manifold is respected; the theory is generally covariant.
As we will see, this fact leads one to Hilbert spaces of quantum states
which are quite different from the familiar Fock spaces of particle
physics. Now gravitons  the three dimensional wavy undulations on a
flat metric  do not represent fundamental excitations. Rather, the
fundamental excitations are *one* dimensional. Microscopically, geometry
is rather like a polymer. Recall that, although polymers are
intrinsically one dimensional, when densely packed in suitable
configurations they can exhibit properties of a three dimensional
system. Similarly, the familiar continuum picture of geometry arises as
an approximation: one can regard the fundamental excitations as `quantum
threads' with which one can `weave' continuum geometries. That is, the
continuum picture arises upon coarsegraining of the semiclassical
`weave states'. Gravitons are no longer the fundamental mediators of the
gravitational interaction. They now arise only as approximate notions.
They represent perturbations of weave states and mediate the
gravitational force only in the semiclassical approximation. Because
the nonperturbative states are polymerlike, geometrical observables
turn out to have discrete spectra. They provide a rather detailed
picture of quantum geometry from which physical predictions can be made.
The article is divided into two parts. In the first, I will indicate
how one can reformulate general relativity so that it resembles gauge
theories. This formulation provides the starting point for the quantum
theory. In particular, the onedimensional excitations of geometry
arise as the analogs of "Wilson loops" which are themselves analogs of
the line integrals exp(i integral A.dl) of electromagnetism. In the
second part, I will indicate how this description leads us to a quantum
theory of geometry. I will focus on area operators and show how the
detailed information about the eigenvalues of these operators has
interesting physical consequences, e.g., to the process of Hawking
evaporation of black holes.

I feel like quoting more, but I'll resist. It's a nice semitechnical
introduction to loop quantum gravity  a very good place to start if you
know some math and physics but are just getting started on the quantum
gravity business.
Next, here are some papers by younger folks working on loop quantum
gravity:
2) Fotini Markopoulou, The internal description of a causal set: What
the universe looks like from the inside, preprint available as
grqc/9811053.
Fotini Markopoulou, Quantum causal histories, preprint available as
hepth/9904009.
Fotini Markopoulou is perhaps the first person to take the issue of
causality really seriously in loop quantum gravity. In her earlier work
with Lee Smolin (see "week99" and "week114") she proposed a way to equip an
evolving spin network (or what I'd call a spin foam) with a partial order on
its vertices, representing a causal structure. In these papers she is
further developing these ideas. The first one uses topos theory! It's
good to see brave young physicists who aren't scared of using a little
category theory here and there to make their ideas precise. Personally
I feel confused about causality in loop quantum gravity  I think we'll
have to muck around and try different things before we find out what
works. But Markopoulou's work is the main reason I'm even *daring* to
think about these issues....
3) Seth A. Major, Embedded graph invariants in ChernSimons theory,
preprint available as hepth/9810071.
In This Week's Finds I've already mentioned Seth Major has worked with
Lee Smolin on qdeformed spin networks in quantum gravity (see "week72").
There is a fair amount of evidence, though as yet no firm proof, that
qdeforming your spin networks corresponds to introducing a nonzero
cosmological constant. The main technical problem with qdeformed spin
networks is that they require a "framing" of the underlying graph.
Here Major tackles that problem....
And now for something completely different, arising from a thread on
sci.physics.research started by Garrett Lisi. What's the gauge group
of the Standard Model? Everyone will tell you it's U(1) x SU(2) x
SU(3), but as Marc Bellon pointed out, this is perhaps not the most
accurate answer. Let me explain why and figure out a better answer.
Every particle in the Standard Model transforms according to some
representation of U(1) x SU(2) x SU(3), but some elements of this
group act trivially on all these representations. Thus we can find
a smaller group which can equally well be used as the gauge group
of the Standard Model: the quotient of U(1) x SU(2) x SU(3)
by the subgroup of elements that act trivially.
Let's figure out this subgroup! To do so we need to go through all
the particles and figure out which elements of U(1) x SU(2) x SU(3)
act trivially on all of them.
Start with the gauge bosons. In any gauge theory, the gauge bosons
transform in the adjoint representation, so the elements of the gauge
group that act trivially are precisely those in the *center* of the
group. U(1) is abelian so its center is all of U(1). Elements of SU(n)
that lie in the center must be diagonal. The n x n diagonal unitary
matrices with determinant 1 are all of the form exp(2 pi i / n),
and these form a subgroup isomorphic to Z/n. It follows that the
center of U(1) x SU(2) x SU(3) is U(1) x Z/2 x Z/3.
Next let's look at the other particles. If you forget how these work,
see "week119". For the fermions, it suffices to look at those of the
first generation, since the other two generations transform exactly
the same way. First of all, we have the lefthanded electron and
neutrino:
(nu_L, e_L)
These form a 2dimensional representation. This representation is the
tensor product of the irreducible rep of U(1) with hypercharge 1, the
isospin1/2 rep of SU(2), and the trivial rep of SU(3).
A word about notation! People usually describe irreducible reps of U(1)
by integers. For historical reasons, hypercharge comes in integral
multiples of 1/3. Thus to get the appropriate integer we need to
multiply the hypercharge by 3. Also, the group SU(2) here is
associated, not to spin in the sense of angular momentum, but to
something called "weak isospin". That's why we say "isopin1/2 rep"
above. Mathematically, though, this is just the usual spin1/2
representation of SU(2).
Next we have the lefthanded up and down quarks, which come in 3
colors each:
(u_L, u_L, u_L, d_L, d_L, d_L)
This 6dimensional representation is the tensor product of the
irreducible rep of U(1) with hypercharge 1/3, the isospin1/2
rep of SU(2), and the fundamental rep of SU(3).
That's all the lefthanded fermions. Note that they all transform
transform according to the isospin1/2 rep of SU(2)  we call them
"isospin doublets". The righthanded fermions all transform according
to the isospin0 rep of SU(2)  they're "isospin singlets". First we
have the righthanded electron:
e_R
This is the tensor product of the irreducible rep of U(1) with
hypercharge 2, the isospin0 rep of SU(2), and the trivial rep of
SU(3). Then there are the righthanded up quarks:
(u_R, u_R, u_R)
which form the tensor product of the irreducible rep of U(1) with
hypercharge 4/3, the isospin0 rep of SU(2), and the fundamental rep of
SU(3). And then there are the righthanded down quarks:
(d_R, d_R, d_R)
which form the tensor product of the irreducible rep of U(1)
with hypercharge 2/3, the isospin0 rep of SU(2), and the
3dimensional fundamental rep of SU(3).
Finally, besides the fermions, there is the  so far unseen  Higgs
boson:
(H_+, H_0)
This transforms according to the tensor product of the irreducible
rep of U(1) with hypercharge 1, the isospin1/2 rep of SU(2), and
the 1dimensional trivial rep of SU(3).
Okay, let's see which elements of U(1) x Z/2 x Z/3 act trivially on all
these representations! Note first that the generator of Z/2 acts as
multiplication by 1 on the isospin singlets and 1 on the isospin
doublets. Similarly, the generator of Z/3 acts as multiplication by
1 on the leptons and exp(2 pi i / 3) on the quarks. Thus everything
in Z/2 x Z/3 acts as multiplication by some sixth root of unity. So
to find elements of U(1) x Z/2 x Z/3 that act trivially, we only need
to consider guys in U(1) that are sixth roots of unity.
To see what's going on, we make a little table using the information
I've described:
ACTION OF ACTION OF ACTION OF
exp(pi i / 3) 1 exp(2 pi i / 3)
IN U(1) IN SU(2) IN SU(3)
e_L 1 1 1
nu_L 1 1 1
u_L exp(pi i / 3) 1 exp(2 pi i / 3)
d_L exp(pi i / 3) 1 exp(2 pi i / 3)
e_R 1 1 1
u_R exp(4 pi i / 3) 1 exp(2 pi i / 3)
d_R exp(4 pi i / 3) 1 exp(2 pi i / 3)
H 1 1 1
And we look for patterns!
See any?
The most important one for our purposes is that if we multiply all three
numbers in each row, we get 1.
This means that the element (exp(pi i / 3), 1, exp(2 pi i / 3)) in U(1)
x SU(2) x SU(3) acts trivially on all particles. This element generates
a subgroup isomorphic to Z/6. If you think a bit harder you'll see
there are no *other* patterns that would make any *more* elements of
U(1) x SU(2) x SU(3) act trivially. And if you think about the relation
between charge and hypercharge, you'll see this pattern has a lot to do
with the fact that quark charges in multiples of 1/3, while leptons have
integral charge. There's more to it than that, though....
Anyway, the "true" gauge group of the Standard Model  i.e., the
smallest possible one  is not U(1) x SU(2) x SU(3), but the quotient of
this by the particular Z/6 subgroup we've just found. Let's call
this group G.
There are two reasons why this might be important. First, Marc Bellon
pointed out a nice way to think about G: it's the subgroup of U(2) x U(3)
consisting of elements (g,h) with
(det g)(det h) = 1.
If we embed U(2) x U(3) in U(5) in the obvious way, then this subgroup G
actually lies in SU(5), thanks to the above equation. And this is what
people do in the SU(5) grand unified theory. They don't actually stuff
all of U(1) x SU(2) x SU(3) into SU(5), just the group G! For more
details, see "week119". Better yet, try this book that Brett McInnes
recommended to me:
4) Lochlainn O'Raifeartaigh, Group structure of gauge theories,
Cambridge University Press, Cambridge, 1986.
Second, this magical group G has a nice action on a 7dimensional
manifold which we can use as the fiber for a 11dimensional KaluzaKlein
theory that mimics the Standard Model in the lowenergy limit. The way
to get this manifold is to take S^3 x S^5 sitting inside C^2 x C^3 and
mod out by the action of U(1) as multiplication by phases. The group
G acts on C^2 x C^3 in an obvious way, and using this it's easy to see
that it acts on (C^2 x C^3)/U(1).
I'm not sure where to read more about this, but you might try:
5) Edward Witten, Search for a realistic KaluzaKlein theory,
Nucl. Phys. B186 (1981), 412428.
Edward Witten, Fermion quantum numbers in KaluzaKlein theory,
Shelter Island II, Proceedings: Quantum Field Theory and the
Fundamental Problems of Physics, ed. T. Appelquist et al,
MIT Press, 1985, pp. 227277.
6) Thomas Appelquist, Alan Chodos and Peter G.O. Freund, editors,
Modern KaluzaKlein Theories, AddisonWesley, Menlo Park, California,
1987.

Previous issues of "This Week's Finds" and other expository articles on
mathematics and physics, as well as some of my research papers, can be
obtained at
http://math.ucr.edu/home/baez/
For a table of contents of all the issues of This Week's Finds, try
http://math.ucr.edu/home/baez/twf.html
A simple jumpingoff point to the old issues is available at
http://math.ucr.edu/home/baez/twfshort.html
If you just want the latest issue, go to
http://math.ucr.edu/home/baez/this.week.html
tum
gravity business.
Next, here are some papers by younger folks working on loop quantum
gravity:
2) Fotini Markopoulou, The internal description of a causal set: What
the universe looks like from the inside, preprint available as
grqc/9811053.
Fotini Markopoulou, Quantum causal histories, preprint available as
hepth/9904009.
Fotini Markopoulou is perhaps the first person to take the issue of
causality really serioustwf_ascii/week134000064400020410000157000000425440774011336500141420ustar00baezhttp00004600000001Also available at http://math.ucr.edu/home/baez/week134.html
June 8, 1999
This Week's Finds in Mathematical Physics (Week 134)
John Baez
My production of "This Week's Finds" has slowed to a trickle as I've
been struggling to write up a bunch of papers. Deadlines, deadlines!
I hate deadlines, but when you write things for other people, or with
other people, that's what you get into. I'll do my best to avoid them
in the future. Now I'm done with my chores and I want to have some fun.
I spent last weekend with a bunch of people talking about quantum
gravity in a hunting lodge by a lake in Minnowbrook, New York:
1) Minnowbrook Symposium on SpaceTime Structure, program and
transparencies of talks available at
http://www.phy.syr.edu/research/he_theory/minnowbrook/#PROGRAM
The idea of this gettogether, organized by Kameshwar Wali and some other
physicists at Syracuse University, was to bring together people working
on string theory, loop quantum gravity, noncommutative geometry, and
various discrete approaches to spacetime. People from these different
schools of thought don't talk to each other as much as they should, so
this was a good idea. People gave lots of talks, asked lots of tough
questions, argued, and learned what each other were doing. But I came
away with a sense that we're quite far from understanding quantum
gravity: every approach has obvious flaws.
One big problem with string theory is that people only know how to study
it on a spacetime with a fixed background metric. Even worse, things
are poorly understood except when the metric is static  that is, roughly
speaking, when geometry of space does not change with the passage of time.
For example, people understand a lot about string theory on spacetimes
that are the product of Minkowski spacetime and a fixed CalabiYau
manifold. There are lots of CalabiYau manifolds, organized in
continuous multiparameter families called moduli spaces. This suggests
the idea that the geometry of the CalabiYau manifold could change with
time. This idea is lurking behind a lot of interesting work. For
example, Brian Greene gave a nice talk on "mirror symmetry". Different
CalabiYau manifolds sometimes give the same physics; these are called
"mirror manifolds". Because of this, a curve in one moduli space of
CalabiYau manifolds can be physically equivalent to a curve in some
other moduli space, which sometimes lets you continue the curve beyond a
singularity in the first moduli space. Physicists like to think of
these curves as representing spacetime geometries where the CalabiYau
manifold changes with time. The problem is, there's no fully worked out
version of string theory that allows for a timedependent CalabiYau
manifold!
There's a good reason for this: one shouldn't expect anything so simple
to make sense, except in the "adiabatic approximation" where things
change very slowly with time. The product of Minkowski spacetime with a
fixed CalabiYau manifold is a solution of the 10dimensional Einstein
equations, and this is part of why this kind of spacetime serves as a
good background for string theory. But we do not get a solution if
the geometry of the CalabiYau manifold varies from point to point in
Minkowski spacetime  except in the adiabatic approximation.
There are also problems with "unitarity" in string theory when the
geometry of space changes with time. This is already familiar from
ordinary quantum field theory on curved spacetime. In quantum field
theory, people usually like to describe time evolution using unitary
operators on a Hilbert space of states. But this approach breaks down
when the geometry of space changes with time. People have studied this
problem in detail, and there seems to be no completely satisfactory way
to get around it. No way, that is, except the radical step of ceasing
to treat the geometry of spacetime as a fixed "background". In other
words: stop doing quantum field theory on spacetime with a preestablished
metric, and invent a backgroundfree theory of quantum gravity! But this
is not so easy  see "week132" for more on what it would entail.
Apparently this issue is coming to the attention of string theorists now
that they are trying to study their theory on nonstatic background
metrics, such as antide Sitter spacetime. Indeed, someone at the
conference said that a bunch of top string theorists recently got
together to hammer out a strategy for where string theory should go
next, but they got completely stuck due to this problem. I think
this is good: it means string theorists are starting to take the
foundational issues of quantum gravity more seriously. These issues
are deep and difficult.
However, lest I seem to be picking on string theory unduly, I should
immediately add that all the other approaches have equally serious
flaws. For example, loop quantum gravity is wonderfully backgroundfree,
but so far it is almost solely a theory of kinematics, rather than
dynamics. In other words, it provides a description of the geometry of
*space* at the quantum level, but says little about *spacetime*.
Recently people have begun to study dynamics with the help of "spin
foams", but we still can't compute anything well enough to be sure we're
on the right track. So, pessimistically speaking, it's possible that
the backgroundfree quality of loop quantum gravity has only been
achieved by simplifying assumptions that will later prevent us from
understanding dynamics.
Alain Connes expressed this worry during Abhay Ashtekar's talk, as did
Arthur Jaffe afterwards. Technically speaking, the main issue is that
loop quantum gravity assumes that unsmeared Wilson loops are sensible
observables at the kinematical level, while in other theories, like
YangMills theory, one always needs to smear the Wilson loops. Of
course these other theories aren't backgroundfree, so loop quantum
gravity probably *should* be different. But until we know that loop
quantum gravity really gives gravity (or some fancier theory like
supergravity) in the largescale limit, we can't be sure it should be
different in this particular way. It's a legitimate worry... but only
time will tell!
I could continue listing approaches and their flaws, including Connes'
own approach using noncommutative geometry, but let me stop here. The
only really good news is that different approaches have *different*
flaws. Thus, by comparing them, one might learn something!
Some more papers have come out recently which delve into the
philosophical aspects of this muddle:
2) Carlo Rovelli, Quantum spacetime: what do we know?, to appear in
Physics Meets Philosophy at the Planck Scale, eds. Craig Callender
and Nick Huggett, Cambridge U. Press. Preprint available as grqc/9903045.
3) J. Butterfield and C. J. Isham, Spacetime and the philosophical
challenge of quantum gravity, to appear in Physics Meets Philosophy
at the Planck Scale, eds. Craig Callender and Nick Huggett, Cambridge
U. Press. Preprint available as grqc/9903072.
Rovelli's paper is a bit sketchy, but it outlines ideas which I find
very appealing  I always find him to be very clearheaded about the
conceptual issues of quantum gravity. I found the latter paper a bit
frustrating, because it lays out a wide variety of possible positions
regarding quantum gravity, but doesn't make a commitment to any one of
them. However, this is probably good when one is writing to an audience
of philosophers: one should explain the problems instead of trying to
sell them on a particular claimed solution, because the proposed
solutions come and go rather rapidly, while the problems remain. Let me
quote the abstract:
"We survey some philosophical aspects of the search for a quantum theory
of gravity, emphasising how quantum gravity throws into doubt the
treatment of spacetime common to the two `ingredient theories' (quantum
theory and general relativity), as a 4dimensional manifold equipped
with a Lorentzian metric. After an introduction, we briefly review the
conceptual problems of the ingredient theories and introduce the
enterprise of quantum gravity. We then describe how three main research
programmes in quantum gravity treat four topics of particular
importance: the scope of standard quantum theory; the nature of
spacetime; spacetime diffeomorphisms, and the socalled problem of time.
By and large, these programmes accept most of the ingredient theories'
treatment of spacetime, albeit with a metric with some type of quantum
nature; but they also suggest that the treatment has fundamental
limitations. This prompts the idea of going further: either by
quantizing structures other than the metric, such as the topology; or by
regarding such structures as phenomenological. We discuss this in
Section 5."
Now let me mention a few more technical papers that have come out in
the last few months:
4) John Baez and John Barrett, The quantum tetrahedron in 3
and 4 dimensions, preprint available as grqc/9903060.
The idea here is to form a classical phase whose points represent
geometries of a tetrahedron in 3 or 4 dimensions, and then apply
geometric quantization to obtain a Hilbert space of states. These
Hilbert spaces play an important role in spin foam models of quantum
gravity. The main goal of the paper is to explain why the quantum
tetrahedron has fewer degrees of freedom in 4 dimensions than in 3
dimensions. Let me quote from the introduction:
"State sum models for quantum field theories are constructed by giving
amplitudes for the simplexes in a triangulated manifold. The simplexes
are labelled with data from some discrete set, and the amplitudes depend
on this labelling. The amplitudes are then summed over this set of
labellings, to give a discrete version of a path integral. When the
discrete set is a finite set, then the sum always exists, so this
procedure provides a bona fide definition of the path integral.
State sum models for quantum gravity have been proposed based on the Lie
algebra so(3) and its qdeformation. Part of the labelling scheme
is then to assign irreducible representations of this Lie algebra to
simplexes of the appropriate dimension. Using the qdeformation, the
set of irreducible representations becomes finite. However, we will
consider the undeformed case here as the geometry is more elementary.
Irreducible representations of so(3) are indexed by a nonnegative
halfintegers j called spins. The spins have different interpretations
in different models. In the PonzanoRegge model of 3dimensional
quantum gravity, spins label the edges of a triangulated 3manifold, and
are interpreted as the quantized lengths of these edges. In the
OoguriCraneYetter state sum model, spins label triangles of a
triangulated 4manifold, and the spin is interpreted as the norm of a
component of the Bfield in a BF Lagrangian. There is also a state sum
model of 4dimensional quantum gravity in which spins label triangles.
Here the spins are interpreted as areas.
Many of these constructions have a topologically dual formulation. The
dual 1skeleton of a triangulated surface is a trivalent graph, each of
whose edges intersect exactly one edge in the original triangulation.
The spin labels can be thought of as labelling the edges of this graph,
thus defining a spin network. In the PonzanoRegge model, transition
amplitudes between spin networks can be computed as a sum over
labellings of faces of the dual 2skeleton of a triangulated 3manifold.
Formulated this way, we call the theory a `spin foam model'.
A similar dual picture exists for 4dimensional quantum gravity. The
dual 1skeleton of a triangulated 3manifold is a 4valent graph each of
whose edges intersect one triangle in the original triangulation. The
labels on the triangles in the 3manifold can thus be thought of as
labelling the edges of this graph. The graph is then called a
`relativistic spin network'. Transition amplitudes between relativistic
spin networks can be computed using a spin foam model. The path
integral is then a sum over labellings of faces of a 2complex
interpolating between two relativistic spin networks.
In this paper we consider the nature of the quantized geometry of a
tetrahedron which occurs in some of these models, and its relation to
the phase space of geometries of a classical tetrahedron in 3 or 4
dimensions. Our main goal is to solve the following puzzle: why does
the quantum tetrahedron have fewer degrees of freedom in 4 dimensions
than in 3 dimensions? This seeming paradox turns out to have a simple
explanation in terms of geometric quantization. The picture we develop
is that the four face areas of a quantum tetrahedron in four dimensions
can be freely specified, but that the remaining parameters cannot, due
to the uncertainty principle."
Naively one would expect the quantum tetrahedron to have the same number
of degrees of freedom in 3 and 4 dimensions (since one is considering
tetrahedra mod rotations). However, quantum mechanics is funny about
these things! For example, the Hilbert space of two spin1/2 particles
whose angular momenta point in opposite directions is smaller than the
Hilbert space of a single spin1/2 particle, even though classically you
might think both systems have the same number of degrees of freedom.
In fact a very similar thing happens for the quantum tetrahedron in 3
and 4 dimensions.
5) Abhay Ashtekar, Alejandro Corichi and Kirill Krasnov, Isolated
horizons: the classical phase space, preprint available as grqc/9905089.
This paper explains in more detail the classical aspects of the
calculation of the entropy of a black hole in loop quantum gravity
(see "week112" for a description of this calculation). Let me
quote the abstract:
"A Hamiltonian framework is introduced to encompass nonrotating (but
possibly charged) black holes that are "isolated" near future
timelike infinity or for a finite time interval. The underlying
spacetimes need not admit a stationary Killing field even in a
neighborhood of the horizon; rather, the physical assumption is that
neither matter fields nor gravitational radiation fall across the
portion of the horizon under consideration. A precise notion of
nonrotating isolated horizons is formulated to capture these ideas.
With these boundary conditions, the gravitational action fails to be
differentiable unless a boundary term is added at the horizon. The
required term turns out to be precisely the ChernSimons action for the
selfdual connection. The resulting symplectic structure also acquires,
in addition to the usual volume piece, a surface term which is the
ChernSimons symplectic structure. We show that these modifications
affect in subtle but important ways the standard discussion of
constraints, gauge and dynamics. In companion papers, this framework
serves as the point of departure for quantization, a statistical
mechanical calculation of black hole entropy and a derivation of laws of
black hole mechanics, generalized to isolated horizons. It may also have
applications in classical general relativity, particularly in the
investigation of analytic issues that arise in the numerical studies
of black hole collisions."
The following are some review articles on spin networks, spin
foams and the like:
6) Roberto De Pietri, Canonical "loop" quantum gravity and spin foam
models, to appear in the proceedings of the XXIIIth Congress of the
Italian Society for General Relativity and Gravitational Physics
(SIGRAV), 1998, preprint available as grqc/9903076.
7) Seth Major, A spin network primer, to appear in Amer. Jour. Phys.,
preprint available as grqc/9905020.
8) Seth Major, Operators for quantized directions, preprint available
as grqc/9905019.
9) John Baez, An introduction to spin foam models of BF theory
and quantum gravity, to appear in Geometry and Quantum Physics,
eds. Helmut Gausterer and Harald Grosse, Lecture Notes in Physics,
SpringerVerlag, Berlin. Preprint available as grqc/9905087.
By the way, Barrett and Crane have come out with a paper sketching a
spin foam model for Lorentzian (as opposed to Riemannian) quantum
gravity:
10) John Barrett and Louis Crane, A Lorentzian signature model for quantum
general relativity, preprint available as grqc/9904025.
However, this model is so far purely formal, because it involves
infinite sums that probably diverge. We need to keep working on this!
Now that I'm getting a bit of free time, I want to tackle this issue.
Meanwhile, Iwasaki has come out with an alternative spin foam model of
Riemannian quantum gravity:
11) Junichi Iwasaki, A surface theoretic model of quantum gravity,
preprint available as grqc/9903112.
Alas, I don't really understand this model yet. Finally, to wrap
things up, something completely different:
12) Richard E. Borcherds, Quantum vertex algebras, preprint available
as math.QA/9903038.
I like how the abstract of this paper starts: "The purpose of this paper
is to make the theory of vertex algebras trivial". Good! Trivial is
not bad, it's good. Anything one understands is automatically trivial.

Previous issues of "This Week's Finds" and other expository articles on
mathematics and physics, as well as some of my research papers, can be
obtained at
http://math.ucr.edu/home/baez/
For a table of contents of all the issues of This Week's Finds, try
http://math.ucr.edu/home/baez/twf.html
A simple jumpingoff point to the old issues is available at
http://math.ucr.edu/home/baez/twfshort.html
If you just want the latest issue, go to
http://math.ucr.edu/home/baez/this.week.html
xplain the problems instead of trying to
sell them on a particular claimed solution, because the proposed
solutions come and go rather rapidly, while the prtwf_ascii/week135000064400020410000157000000423730774011336500141430ustar00baezhttp00004600000001Also available at http://math.ucr.edu/home/baez/week135.html
July 31, 1999
This Week's Finds in Mathematical Physics (Week 135)
John Baez
Well, darn it, now I'm too busy running around to conferences to write
This Week's Finds! First I went to Vancouver, then to Santa Barbara,
and for almost a month now I've been in Portugal, bouncing between
Lisbon and Coimbra. But let me try to catch up....
From June 16th to 19th, Steve Savitt and Steve Weinstein of the
University of British Columbia held a workshop designed to get
philosophers and physicists talking about the conceptual problems
of quantum gravity:
1) Toward a New Understanding of Space, Time and Matter, workshop
home page at http://axion.physics.ubc.ca/Workshop/
After a day of lectures by Chris Isham, John Earman, Lee Smolin and
myself, we spent the rest of the workshop sitting around in a big
room with a beautiful view of Vancouver Bay, discussing various issues
in a fairly organized way. For example, Chris Isham led a discussion
on "What is a quantum theory?" in which he got people to question the
assumptions underlying quantum physics, and Simon Saunders led one on
"Quantum gravity: physics, metaphysics or mathematics?" in which we
pondered the scientific and sociological implications of the fact that
work on quantum gravity is motivated more by desire for consistency,
clarity and mathematical elegance than the need to fit new experimental
data.
It's pretty clear that understanding quantum gravity will make us rethink
some fundamental concepts  the question is, which ones? By the end of
the conference, almost every basic belief or concept relevant to physics
had been held up for careful scrutiny and found questionable. Space,
time, causality, the real numbers, set theory  you name it! It was a
bit unnerving  but it's good to do this sort of thing now and then, to
prevent hardening of the mental arteries, and it's especially fun to do
it with a big bunch of physicists and philosophers. However, I must
admit that I left wanting nothing more than to do lots of grungy
calculations in order to bring myself back down to earth  relatively
speaking, of course.
I particularly enjoyed Chris Isham's talk about topos theory because
it helped me understand one way that topos theory could be applied to
quantum theory. I've tended to regard topoi as "too classical" for
quantum theory, because while the internal logic of a topos is
intuitionistic (the principle of exclude middle may fail), it's
still not very quantum. For example, in a topos the operation
"and" still distributes over "or", and vice versa, while failure
of this sort of distributivity is a hallmark of quantum logic. If
you don't know what I mean, try these books, in rough order of
increasing difficulty:
2) David W. Cohen, An Introduction to Hilbert Space and Quantum
Logic, SpringerVerlag, New York, 1989.
3) C. Piron, Foundations of Quantum Physics, W. A. Benjamin,
Reading, Massachusetts, 1976.
4) C. A. Hooker, editor, The Logicoalgebraic Approach to Quantum
Mechanics, two volumes, D. Reidel, Boston, 19751979.
Perhaps even more importantly, topoi are Cartesian! What does this
mean? Well, it means that we can define a "product" of any two
objects in a topos. That is, given objects a and b, there's an
object a x b equipped with morphisms
p: a x b > a
and
q: a x b > b
called "projections", satisfying the following property: given
morphisms from some object c into a and b, say
f: c > a
and
g: c > b
there's a unique morphism f x g: c > a x b such that if we
follow it by p we get f, and if we follow it by q we get g. This
is just an abstraction of the properties of the usual Cartesian
product of sets, which is why we call a category "Cartesian" if
any pair of objects has a product.
Now, it's a fun exercise to check that in a category with
products, every object has a morphism
Delta: a > a x a
called the "diagonal", which when composed with either of the two
projections from a x a to a gives the identity. For example, in
the topos of sets, the diagonal morphism is given by
Delta(x) = (x,x)
We can think of the diagonal morphism as allowing "duplication
of information". This is not generally possible in quantum
mechanics:
5) William Wooters and Wocjciech Zurek, A single quantum cannot
be cloned, Nature 299 (1982), 802803.
The reason is that in the category of Hilbert spaces, the
tensor product is not a product in the above sense! In
particular, given a Hilbert space H, there isn't a natural
diagonal operator
Delta: H > H tensor H
and there aren't even natural projection operators from H tensor H
to H. As pointed out to me by James Dolan, this nonCartesianness
of the tensor product gives quantum theory much of its special flavor.
Besides making it impossible to "clone a quantum", it's closely
related to how quantum theory violates Bell's inequality, because
it means we can't think of an arbitrary state of a twopart quantum
system as built by tensoring states of both parts.
Anyway, this has made me feel for a while that topos theory isn't
sufficiently "quantum" to be useful in understanding the peculiar
special features of quantum physics. However, after Isham and I
gave our talks, someone pointed out to me that one can think of a
topological quantum field theory as a presheaf of Hilbert spaces
over the category nCob whose morphisms are ndimensional cobordisms.
Now, presheaves over any category form a topos, so this means we
should be able to think of a topological quantum field theory as a
"Hilbert space object" in the topos of presheaves over nCob. From
this point of view, the peculiar "quantumness" of topological quantum
field theory comes from it being a Hilbert space object, while its
peculiar "variability"  i.e., the fact that it assigns a different
Hilbert space to each (n1)dimensional manifold representing space 
comes from the fact that it's an object in a topos. (Topoi are
known for being very good at handling things like "variable sets".)
I'm not sure how useful this is, but it's worth keeping in mind.
While I'm talking about quantum logic, let me raise a puzzle
concerning the KochenSpecker theorem. Remember what this says:
if you have a Hilbert space H with dimension but more than 2,
there´s no map F from selfadjoint operators on H to real numbers
with the following properties:
a) For any selfadjoint operator A, F(A) lies in the spectrum of A,
and
b) For any continuous f: R > R, f(F(A)) = F(f(A)).
This means there's no sensible consistent way of thinking of all
observables as simultaneously having values in a quantum system!
Okay, the puzzle is: what happens if the dimension of H equals 2?
I don't actually know the answer, so I'd be glad to hear it if
someone can figure it out!
By the way, I once wanted to do an undergraduate research project
on mathematical physics with Kochen. He asked me if I knew the
spectral theorem, I said "no", and he said that in that case there
was no point in me trying to work with him. I spent the next summer
reading Reed and Simon's book on Functional Analysis and learning
lots of different versions of the spectral theorem. I shudder to
think that perhaps this is why I spent years studying analysis
before eventually drifting towards topology and algebra. But no:
now that I think about it, I was already interested in analysis at
the time, since I'd had a wonderful real analysis class with Robin
Graham.
Okay, now let me say a bit about the next conference I went to. From
June 22nd to 26th there was a conference on "Strong Gravitational
Fields" at the Institute for Theoretical Physics at U. C. Santa Barbara.
This finished up a wonderful semesterlong program by Abhay Ashtekar,
Gary Horowitz and Jim Isenberg:
6) Classical and Quantum Physics of Strong Gravitational Fields,
program homepage with transparencies and audio files of talks at
http://www.itp.ucsb.edu/~patrick/gravity99/
Like the whole program, the conference covered a wide range of
topics related to gravity: string theory and loop quantum gravity,
observational and computational black hole physics, and gamma ray
bursters. I can't summarize all this stuff; since I usually spend
a lot of talking about quantum gravity here, let me say a bit about
other things instead.
John Friedman gave an interesting talk on gravitational waves from
unstable neutron stars. When a pulsar is young, like about 5000
years old, it typically rotates about its axis once every 16
milliseconds or so. A good example is N157B, a pulsar in the Large
Magellanic Cloud. Using the current spindown rate one can extrapolate
and guess that pulsars have about a 5millisecond period at birth.
It's interesting to think about what makes a newly formed neutron
star slow down. Theorists have recently come up with a new possible
mechanism: namely, a new sort of gravitationalwavedriven instability
of relativistic stars that could force newly formed slow down to a
10millisecond period. It's very clever: the basic idea is that if
a star is rotating very fast, a rotational mode that rotates slower
than the star will gravitationally radiate *positive* angular momentum,
but such modes carry *negative* angular momentum, since they rotate
slower than the star. If you think about it carefully, you'll see
this means that gravitational radiation should tend to amplify such
modes! I asked for a lowbrow analog of this mechanism and it turns
out that a similar sort of thing is at work in the formation of water
waves by the wind  with linear momentum taking the place of angular
momentum. Anyway, it's not clear that this process really ever has
a chance to happen, because it only works when the neutron star is
not too hot and not too cold, but it's pretty cool.
Richard Price gave a nice talk on computer simulation of black
hole collisions. Quantitatively understanding the gravitational
radiation emitted in black hole and neutron star collisions is a
big business these days  it's one of the NSF's "grand challenge"
problems. The reason is that folks are spending a lot of money
building gravitational wave detectors like LIGO:
7) LIGO project home page, http://www.ligo.caltech.edu/
8) Other gravitational wave detection projects,
http://www.ligo.caltech.edu/LIGO_web/other_gw/gw_projects.html
and they need to know exactly what to look for. Now, headon
collisions are the easiest to understand, since one can simplify
the calculation using axial symmetry. Unfortunately, it's not
very likely that two black holes are going to crash into each
other headon. One really wants to understand what happens when
two black holes spiral into each other. There are two extreme
cases: the case of black holes of equal mass, and the case of
a very light black hole of mass falling into a heavy one.
The latter case is 95% understood, since we can think of the
light black hole as a "test particle"  ignoring its effect
on the heavy one. The light one slowly spirals into the
heavy one until it reaches the innermost stable orbit, and then
falls in. We can use the theory of a relativistic test particle
falling into a black hole to understand the early stages of this
process, and use black hole perturbation theory to study the
"ringdown" of the resulting single black hole in the late stages
of the process. (By "ringdown" I mean the process whereby an
oscillating black hole settles down while emitting gravitational
radiation.) Even the intermediate stages are manageable, because
the radiation of the small black hole doesn't have much effect on
the big one.
By contrast, the case of two black holes of equal mass is less
well understood. We can treat the early stages, where relativistic
effects are small, using a postNewtonian approximation, and
again we can treat the late stages using black hole perturbation
theory. But things get complicated in the intermediate stage,
because the radiation of each hole greatly effects the other,
and there is no real concept of "innermost stable orbit" in this
case. To make matters worse, the intermediate stage of the process
is exactly the one we really want to understand, because this is
probably when most of the gravitational waves are emitted!
People have spent a lot of work trying to understand black hole
collisions through numbercrunching computer calculation, but
it's not easy: when you get down to brass tacks, general relativity
consists of some truly scary nonlinear partial differential equations.
Current work is bedeviled by numerical instability and also the
problem of simulating enough of a region of spacetime to understand
the gravitational radiation being emitted. Fans of mathematical
physics will also realize that gaugefixing is a major problem.
There is a lot of interest in simplifying the calculations through
"black hole excision": anything going on inside the event horizon
can't affect what happens outside, so if one can get the computer
to *find* the horizon, one can forget about simulating what's going
on inside! But nobody is very good at doing this yet... even using
the simpler concept of "apparent horizon", which can be defined
locally. So there is some serious work left to be done!
(For more details on both these talks, go to the conference website
and look at the transparencies.)
I also had some interesting talks with people about black hole entropy,
some of which concerned a new paper by Steve Carlip. I'm not really
able to do justice to the details, but it seems important....
9) Steve Carlip, Entropy from conformal field theory at Killing
horizons, preprint available at grqc/9906126.
Let me just quote the abstract:
On a manifold with boundary, the constraint algebra of general
relativity may acquire a central extension, which can be computed
using covariant phase space techniques. When the boundary is a
(local) Killing horizon, a natural set of boundary conditions
leads to a Virasoro subalgebra with a calculable central charge.
Conformal field theory methods may then be used to determine the
density of states at the boundary. I consider a number of cases 
black holes, Rindler space, de Sitter space, TaubNUT and TaubBolt
spaces, and dilaton gravity  and show that the resulting density
of states yields the expected BekensteinHawking entropy. The
statistical mechanics of black hole entropy may thus be fixed
by symmetry arguments, independent of details of quantum gravity.
There was also a lot of talk about "isolated horizons", a concept
that plays a fundamental role in certain treatments of black holes
in loop quantum gravity:
10) Abhay Ashtekar, Christopher Beetle, and Stephen Fairhurst,
Mechanics of isolated horizons, preprint available as grqc/9907068.
11) Jerzy Lewandowski, Spacetimes admitting isolated horizons,
preprint available as grqc/9907058.
For more on isolated horizons try the references in "week128".
Finally, on a completely different note, I've recently seen some
new papers related to the McKay correspondence  see "week65" if
you don't know what *that* is! I haven't read them yet, but I
just want to remind myself that I should, so I'll list them here:
12) John McKay, Semiaffine CoxeterDynkin graphs and $G \subseteq
SU_2(C)$, preprint available as math.QA/9907089.
13) Igor Frenkel, Naihuan Jing and Weiqiang Wang, Vertex
representations via finite groups and the McKay correspondence,
preprint available as math.QA/9907166.
Igor Frenkel, Naihuan Jing and Weiqiang Wang, Quantum vertex
representations via finite groups and the McKay correspondence,
preprint available as math.QA/9907175.
Next time I want to talk about the big category theory conference
in honor of MacLane's 90th birthday! Then I'll be pretty much
caught up on the conferences....

Robert Israel's answer to my puzzle about the KochenSpecker theorem:
It's not true in dimension 2. Note that for a selfadjoint 2x2 matrix
A, any f(A) is of the form a A + b I for some real scalars a and b
(this is easy to see if you diagonalize A). The selfadjoint matrices
that are not multiples of I split into equivalence classes, where
A and B are equivalent if B = a A + b I for some scalars a, b (a <> 0).
Pick a representative A from each equivalence class, choose
F(A) as one of the eigenvalues of A, and then F(a A + b I) = a F(A) + b.
Of course, F(b I) = b. Then F satisfies the two conditions.
The reason this doesn't work in higher dimensions is that in higher
dimensions you can have two selfadjoint matrices A and B which don't
commute, F(A) = G(B) for some functions F and G, and F(A) is not a
multiple of I.
Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia
Vancouver, BC, Canada V6T 1Z2

Previous issues of "This Week's Finds" and other expository articles on
mathematics and physics, as well as some of my research papers, can be
obtained at
http://math.ucr.edu/home/baez/
For a table of contents of all the issues of This Week's Finds, try
http://math.ucr.edu/home/baez/twf.html
A simple jumpingoff point to the old issues is available at
http://math.ucr.edu/home/baez/twfshort.html
If you just want the latest issue, go to
http://math.ucr.edu/home/baez/this.week.html
it, I was already interested in analysis at
the time, since I'd had a wonderful real analysis class with Robin
Graham.
Okay, now let me say a bit about the next conference I went to. From
June 22nd to 26th there was a conference on "Strong Gravitationaltwf_ascii/week136000064400020410000157000000351411110035727200141270ustar00baezhttp00004600000001Also available at http://math.ucr.edu/home/baez/week136.html
August 21, 1999
This Week's Finds in Mathematical Physics (Week 136)
John Baez
I spent most of last month in Portugal, spending time with Roger Picken
at the Instituto Superior Tecnico in Lisbon and attending the category
theory school and conference in Coimbra, which was organized by Manuela
Sobral:
1) Category Theory 99 website, with abstracts of talks,
http://www.mat.uc.pt/~ct99/
The conference was a big deal this year, because it celebrated the
90th birthday of Saunders Mac Lane, who with Samuel Eilenberg invented
category theory in 1945. Mac Lane was there and in fine fettle. He
gave a nice talk about working with Eilenberg, and after the banquet
in his honor, he even sang a song about Riemann while wrapped in a
black cloak!
(In case you're wondering, the cloak was contributed by some musicians.
A few days ago we'd seen them serenade a tearful old man and then wrap
him in a cloak, so one of our number suggested that they try this trick
on Mac Lane. Far from breaking into tears, he burst into song.)
The conference was exquisitely wellorganized, packed with top
category theorists, and stuffed with so many cool talks I scarcely
know where to begin describing them... I'll probably say a bit
about a random sampling of them next time, and the proceedings
will appear in a special issue of the Journal of Pure and Applied
Algebra honoring Mac Lane's 90th birthday, so keep your eye out for
that if you're interested. The school featured courses by Cristina
Pedicchio, Vaughan Pratt, and some crazy mathematical physicist who
thinks the laws of physics are based on ncategories. The notes can
be found in the following book:
2) School on Category Theory and Applications, Coimbra, July
1317, 199, Textos de Matematica Serie B No. 21, Departamento De
Matematica da Universideade de Coimbra. Contains: "nCategories"
by John Baez, "Algebraic theories" by M. Cristina Pedicchio, and
"Chu Spaces: duality as a common foundation for computation and
mathematics" by Vaughan Pratt.
Pedicchio's course covered various generalizations of Lawvere's
wonderful concept of an algebraic theory. Recall from "week53"
that we can think of a category C with extra properties or structure
as a kind of "theory", and functors F: C > Set preserving this
structure as "models" of the theory. For example, a "finite
products theory" C is just a category with finite products. In
this case, a model is a functor F: C > Set preserving finite
products, and a morphism of models is a natural transformation
between such functors. This gives us a category Mod(C) of models
of C.
To understand what this really means, let's restrict attention
the simplest case, when all the objects in C are products of a
given object x. In this case Pedicchio calls C an "algebraic
theory". A model F is then really just a set F(x) together
with a bunch of nary operations coming from the morphisms in C,
satisfying equational laws coming from the equations between
morphisms in C. Any sort of algebraic gadget that's just
a set with a bunch of nary operations satisfying equations can
be described using a theory of this sort. For example: monoids,
groups, abelian groups, rings... and so on. We can describe
any of these using a suitable algebraic theory, and in each case,
the category Mod(C) will be the category of these algebraic gadgets.
Now, what I didn't explain last time I discussed this was the
notion of theorymodel duality. Fans of "duality" in all its
forms are sure to like this! There's a functor
R: Mod(C) > Set
which carries each model F to the set F(x). We can think of this
as a functor which forgets all the operations of our algebraic
gadget and remembers only the underlying set. Now, if you know
about adjoint functors (see "week77""week79"), this should
immediately make you want to find a left adjoint for R, namely
a functor
L: Set > Mod(C)
sending each set to the "free" algebraic gadget on this set.
Indeed, such a left adjoint exists!
Given this pair of adjoint functors we can do all sorts of fun
stuff. In particular, we can talk about the category of "finitely
generated free models" of our theory. The objects here are objects
of Mod(C) of the form L(S) where S is a finite set, and the morphisms
are the usual morphisms in Mod(C). Let me call this category
fgFreeMod(C).
Now for the marvelous duality theorem: fgFreeMod(C) is equivalent
to the opposite of the category C. In other words, you can
reconstruct an algebraic theory from its category of finitely
generated free algebras in the simplest manner imaginable: just
reversing the direction of all the morphisms! This is so nice
I won't explain why it's true... I don't want to deprive you of
the pleasure of looking at it in some simple examples and seeing
for yourself how it works. For example, take the theory of groups,
and figure out how every operation appearing in the definition of
"group" corresponds to a homomorphism between finitely generated
free groups.
There are lots of other interesting questions related to
theorymodel duality. For example: what kinds of categories
arise as categories of models of an algebraic theory? Pedicchio
calls these "algebraic categories", and she told us some nice
theorems characterizing them. Or: given the category of free
models of an algebraic theory, can you fatten it up to get
the category of *all* models? Pedicchio mentioned a process
called "exact completion" that does the job. Or: starting
from just the category of models of a theory, can you tell
which are the free models? Alas, I don't know the answer
to this... but I'm sure people do.
Even better, all of this can be generalized immensely, to
theories of a more flexible sort than the "algebraic theories"
I've been talking about so far. For example, we can study
"essentially algebraic theories", which are just categories
with finite limits. Given one of these, say C, we define a
model to be a functor F: C > Set preserving finite limits.
This allows one to study algebraic structures with
partially defined operations. I already gave an example in
"week53"  there's a category with finite limits called
"the theory of categories", whose models are categories!
One can work out theorymodel duality in this bigger context,
where it's called GabrielUlmer duality:
3) P. Gabriel and F. Ulmer, Lokal praesentierbare Kategorien,
Springer Lecture Notes in Mathematics, Berlin, 1971.
But this stuff goes far beyond that, and Pedicchio led us at a
rapid pace all the way up to the latest work. A lot of the basic
ideas here came from Lawvere's famous thesis on algebraic
semantics, so it was nice to see him attending these lectures, and
even nicer to hear that 26 years after he wrote it, his thesis
is about to be published:
4) William Lawvere, Functorial Semantics of Algebraic Theories,
Ph.D. Dissertation, University of Columbia, 1963. Summary
appears under same title in: Proceedings of the National
Academy of Sciences of the USA 50 (1963), 869872.
(Unfortunately I forget who is publishing it!) It was also nice
to find out that Lawvere and Schanuel are writing a book on
"objective number theory"... which will presumably be more
difficult, but hopefully not less delightful, than their
wonderful introduction to category theory for people who
know *nothing* about fancy mathematics:
5) William Lawvere and Steve Schanuel, Conceptual Mathematics:
A First Introduction to Categories, Cambridge U. Press, Cambridge 1997.
This is the book to give to all your friends who are wondering
what category theory is about and want to learn a bit without
too much pain. If you've read this far and understood what I
was talking about, you must have such friends! If you *didn't*
understand what I was talking about, read this book!
By the way, Lawvere told me that he started out wanting to do
physics, and wound up doing his thesis on algebraic semantics
when he started to trying to formalize what a physical theory
was. It's interesting that the modern notion of "topological
quantum field theory" is very much modelled after Lawvere's ideas,
but with symmetric monoidal categories with duals replacing the
categories with finite products which Lawvere considered! I guess
he was just ahead of his time. In fact, he has returned to physics
in more recent years  but that's another story.
Okay, let me change gears now....
Some ncategory gossip. Ross Street has a student who has defined a
notion of semistrict ncategory up to n = 5, and Sjoerd Crans has
defined semistrict ncategories (which he calls "teisi") for n up to
6. However, the notion still seems to resist definition for general
n, which prompted my pal Lisa Raphals to compose the following
limerick:
A theoretician of "n"
Considered conditions on when
Some mathematicians
Could find definitions
For n even greater than 10.
Interestingly, work on weak ncategories seems to be proceeding at
a slightly faster clip  they've gotten to n = infinity already.
In fact, during the conference Michael Batanin came up to me and said
that a fellow named Penon had published a really terse definition of
weak omegacategories that seems equivalent to Batanin's own
(see "week103")  at least after some minor tweaking. Batanin was quite
enthusiastic and said he plans to write a paper about this stuff.
Later, when I went to Cambridge England, Tom Leinster gave a talk
summarizing Penon's definition:
2) Jacques Penon, Approache polygraphique des $\infty$categories
non strictes, in Cahiers Top. Geom. Diff. 40 (1999), 3179.
It seems pretty cool, so I'd like to tell you what Leinster said 
using his terminology rather than Penon's (which of course is in
French). To keep this short I'm going to assume you know a
reasonable amount of category theory.
First of all, a "reflexive globular set" is a collection of sets and
functions like this:
<s <s <s
X_0 i> X_1 i> X_2 i> .....
<t <t <t
going on to infinity, satisfying these equations:
s(s(x)) = s(t(x))
t(s(x)) = t(t(x))
s(i(x)) = t(i(x)) = x.
We call the elements of X_n are "ncells", and call s(x) and t(x) the
"source" and "target" of the ncell x, respectively. If s(x) = a and
t(x) = b, we think of x as going from a to b, and write x: a > b.
If we left out all the stuff about the maps i we would simply have a
"globular set". These are important in ncategory theory because
strict omegacategories, and also Batanin's weak omegacategories, are
globular sets with extra structure. This also true of Penon's
definition, but he starts right away with "reflexive" globular sets,
which have these maps i that are a bit like the degeneracies in the
definition of a simplicial set (see "week115"). In Penon's definition
i(x) plays the role of an "identity nmorphism", so we also write i(x)
as 1_x: x > x.
Let RGlob be the category of reflexive globular sets, where morphisms
are defined in the obvious way. (In other words, RGlob is a presheaf
category  see "week115" for an explanation of this notion.)
In this setup, the usual sort of strict omegacategory may be
defined as a reflexive globular set X together with various
"composition" operations that allow us to compose ncells x and y
whenever t^j(x) = s^j(x), obtaining an ncell
x o_j y
We get one such composition operation for each n and each j such
that 1 <= j <= n. We impose some obvious axioms of two
sorts:
A: axioms determining the source and target of a composite
B: strict associativity, unit and interchange laws
I'll assume you know these axioms or can fake it. (If you
read the definition of strict 2category in "week80", perhaps
you can get an idea for what kinds of axioms I'm talking about.)
Now, strict omegacategories are great, but we need to weaken this
notion. So, first Penon defines an "omegamagma" to be something
exactly like a strict omegacategory but without the axioms of type B.
You may recall that a "magma" is defined by Bourbaki to be a set with
a binary operation satisyfing no laws whatsoever  the primeval
algebraic object! An omegamagma is just as lawless, and a lot bigger
and meaner.
Strict omegacategories are too strict: all laws hold as equations.
Omegamagmas are too weak: no laws hold at all! How do we get what
we want?
We define a category Q whose objects are quadruples (M,p,C,[.,.])
where:
M is an omegamagma
C is a strict omegacategory
p: M > C is a morphism of omegamagmas (i.e., a morphism of
reflexive globular sets strictly preserving all the omegamagma
operations)
[.,.] is a way of lifting equations between nmorphisms in the
image of the projection p to (n+1)morphisms in M. More precisely:
given ncells
f,g: a > b
in M such that p(f) = p(g), we have an (n+1)cell
[f,g]: f > g
in M such that p([f,g]) = 1_{p(f)} = 1_{p(g)}. We require that
[f,f] = 1_f.
A morphism in Q is defined to be the obvious thing: a morphism
f: M > M' of omegamagmas and a morphism f: C > C' of strictomega
categories, strictly preserving all the structure in sight.
Okay, now we define a functor
U: Q > RGlob
by
U(M,p,C,[.,.]) = M
where we think of M as just a reflexive globular set. Penon
proves that U has a left adjoint
F: RGlob > Q
This adjunction defines a monad
T: RGlob > RGlob
and Penon defines a "weak omegacategory" to be an algebra of this
monad.
(See "week92" and "week118" for how you get monads from adjunctions.
Alas, I think I haven't gotten around to explaining the concept of an
algebra of a monad! So much to explain, so little time!)
Now, if you know some category theory and think a while about this,
you will see that in a weak omegacategory defined this way, all
the laws like associativity hold *up to equivalence*, with the
equivalences satisfying the necessary coherence laws *up to
equivalence*, and so ad infinitum. Crudely speaking, the
lifting [.,.] is what turns equations into nmorphisms. To get
a feeling for how this work, you have to figure out what the left
adjoint F looks like. Penon works this out in detail in the second
half of his paper.

Previous issues of "This Week's Finds" and other expository articles on
mathematics and physics, as well as some of my research papers, can be
obtained at
http://math.ucr.edu/home/baez/
For a table of contents of all the issues of This Week's Finds, try
http://math.ucr.edu/home/baez/twf.html
A simple jumpingoff point to the old issues is available at
http://math.ucr.edu/home/baez/twfshort.html
If you just want the latest issue, go to
http://math.ucr.edu/home/baez/this.week.html
ntics, so it was nice to see him attending these lectures, and
even nicer to hear that 26 years after he wrote it, his thesis
is about to be published:
4) William Lawvere, Functorial Semantics of Algebraic Theories,
Ph.D. Dissertation, University of Columbia, 1963. Summary
appears under same title in: Proceedings of the National
Academy of Sciences of the USA 50 (1963), 869872.
(Unfortunately I forget wtwf_ascii/week137000064400020410000157000000347650774011336500141530ustar00baezhttp00004600000001Also available at http://math.ucr.edu/home/baez/week137.html
September 4, 1999
This Week's Finds in Mathematical Physics (Week 137)
John Baez
Now I'm in Cambridge England, chilling out with the category theorists,
so it makes sense for me to keep talking about category theory. I'll
start with some things people discussed at the conference in Coimbra
(see last week).
1) Michael Mueger, Galois theory for braided tensor categories
and the modular closure, preprint available as math.CT/9812040.
A braided monoidal category is simple algebraic gadget that captures
a bit of the essence of 3dimensionality in its rawest form. It
has a bunch of "objects" which we can draw a labelled dots like
this:
x
.
So far this is just 0dimensional. Next, given a bunch of objects
we get a new object, their "tensor product", which we can draw by setting
the dots side by side. So, for example, we can draw x tensor y like this:
x y
. .
This is 1dimensional. But in addition we have, for any pair of
objects x and y, a bunch of "morphisms" f: x > y. We can draw a
morphism from a tensor product of objects to some other tensor
product of objects as a picture like this:
x y z
  
  
  

 f 

 
 
 
u v
This picture is 2dimensional. In addition, we require that for any
pair of objects x and y there is a "braiding", a special morphism
from x tensor y to y tensor x. We draw it like this:
x y
 
\ /
\ /
/
/ \
/ \
 
y x
With this crossing of strands, the picture has become 3dimensional!
We also require that we can "compose" a morphism f: x > y and a morphism
g: y > z and get a morphism fg: x > z. We draw this by sticking one
picture on top of each other like this... I'll draw a fancy example where
all the objects in question are themselves tensor products of other objects:
x y z
  
  
  

 f 

 
 
 

 g 

   
   
   
a b c d
Finally, we require that the tensor product, braiding and composition
satisfy a bunch of axioms. I won't write these down because I already
did so in "week121", but the point is that they all make geometrical
sense  or more precisely, topological sense  in terms of the above
pictures.
The pictures I've drawn should make you think about knots and tangles
and circuit diagrams and Feynman diagrams and all sorts of things
like that  and it's true, you can understand all these things very
elegantly in terms of braided monoidal categories! Sometimes it's
nice to throw in another rule:
x y x y
   
\ / \ /
\ / \ /
/ = \
/ \ / \
/ \ / \
   
y x y x
where we cook up the second picture using the inverse of the braiding.
This rule is good when you don't care about the difference between
overcrossings and undercrossings. If this rule holds we say our
braided monoidal category is "symmetric". Topologically, this rule
makes sense when we study 4dimensional or higherdimensional situations,
where we have enough room to untie all knots. For example, the
traditional theory of Feynman diagrams is based on symmetric monoidal
categories (like the category of representations of the Poincare
group), and it works very smoothly in 4dimensional spacetime.
But 3dimensional spacetime is a bit different. For example,
when we interchange two identical particles, it really makes
a difference whether we do it like this:
x y
 
\ /
\ /
/
/ \
/ \
 
y x
or like this:
x y
 
\ /
\ /
\
/ \
/ \
 
y x
Thus in 3d spacetime, besides bosons and fermions, we have other sorts
of particles that act differently when we interchange them  sometimes
people call them "anyons", and sometimes people talk about "exotic
statistics".
Now let me dig into some more technical aspects of the picture.
Starting with Reshetikhin and Turaev, people have figured out
how to use braided monoidal categories to construct topological
quantum field theories in 3dimensional spacetime. But they can't
do it starting from any old braided monoidal category, because
quantum field theory has a lot to do with Hilbert spaces. So usually
they start from a special sort called a "modular tensor category".
This is a kind of hybrid of a braided monoidal category and a Hilbert
space.
In fact, apart from one technical condition  which is at the
heart of Mueger's work  we can get the definition of a modular
tensor category by taking the definition of "Hilbert space",
categorifying it once to get the definition of "2Hilbert
space", and then throwing in a tensor product and braiding
that are compatible with this structure.
It's amazing that by such abstract conceptual methods we come
up with almost precisely what's needed to construct topological
quantum field theories in 3 dimensions! It's a great illustration
of the power of category theory. It's almost like getting something
for nothing! But I'll resist the temptation to tell you the details,
since "week99" explains a bunch of it, and the rest is in here:
2) John Baez, Higherdimensional algebra II: 2Hilbert spaces,
Adv. Math. 127 (1997), 125189. Also available as qalg/9609018.
In this paper I call a 2Hilbert space with a compatible tensor
product a "2H*algebra", and if it also has a compatible braiding,
I call it a "braided 2H*algebra". This terminology is bit
clunky, but for consistency I'll use it again here.
Okay, great: we *almost* get the definition of modular tensor
category by elegant conceptual methods. But there is one
niggling but crucial technical condition that remains! There
are lots of different ways to state this condition, but Mueger
proves they're equivalent to the following very elegant one.
Let's define the "center" of a braided monoidal category to
be the category consisting of all objects x such that
x y x y
   
\ / \ /
\ / \ /
/ = \
/ \ / \
/ \ / \
   
y x y x
for all y, and all morphisms between such objects. The center
of a braided monoidal category is obviously a symmetric monoidal
category. The term "center" is supposed to remind you of the
usual center of a monoid  the elements that commute with all
the others. And indeed, both kinds of center are special cases
of a general construction that pushes you down the columns of
the "periodic table":
ktuply monoidal ncategories
n = 0 n = 1 n = 2
k = 0 sets categories 2categories
k = 1 monoids monoidal monoidal
categories 2categories
k = 2 commutative braided braided
monoids monoidal monoidal
categories 2categories
k = 3 " " symmetric weakly
monoidal involutory
categories monoidal
2categories
k = 4 " " " " strongly
involutory
monoidal
2categories
k = 5 " " " " " "
I described this in "week74" and "week121", so I won't do so
again. My point here is really just that lots of this
3dimensional stuff is part of a bigger picture that applies
to all different dimensions. For more details, including a
description of the center construction, try:
3) John Baez and James Dolan, Categorification, in Higher
Category Theory, eds. Ezra Getzler and Mikhail Kapranov,
Contemporary Mathematics vol. 230, AMS, Providence, 1998,
pp. 136. Also available at math.QA/9802029.
Anyway, Mueger's elegant characterization of a modular tensor
category amounts to this: it's a braided 2H*algebra whose
center is "trivial". This means that every object in the center
is a direct sum of copies of the object 1  the unit for the
tensor product.
Mueger does a lot more in his paper that I won't describe here,
and he also said a lot of interesting things in his talk about
the general concept of center. For example, the center of a
monoidal category is a braided monoidal category. In particular,
if you take the center of a 2H*algebra you get a braided
2H*algebra. But what if you then take this braided 2H*algebra
and look at *its* center? Well, it turns out to be "trivial" in
the above sense!
There's a bit of overlap between Mueger's paper and this one:
4) A. Bruguieres, Categories premodulaires, modularisations et
invariants des varietes de dimension 3, preprint.
One especially important issue they both touch upon is this:
if you have a braided 2H*algebra, is there any way to mess
with it slightly to get a modular tensor category? The answer
is yes. Thus we can really get a topological quantum field theory
from any braided 2H*algebra. But this raises another question:
can we describe this topological quantum field theory directly,
without using the modular tensor category? The answer is again
yes! For details see:
5) Stephen Sawin, JonesWitten invariants for nonsimplyconnected
Lie groups and the geometry of the Weyl alcove, preprint available
as math.QA/9905010.
This paper uses this machinery to get topological quantum field
theories related to ChernSimons theory. People have thought about
this a lot, ever since Reshetikhin and Turaev, but the really
great thing about this paper is that it handles the case when
the gauge group isn't simplyconnected. This introduces a lot
of subtleties which previous papers touched upon only superficially.
Sawin works it out much more thoroughly by an analysis of subsets
of the Weyl alcove that are closed under tensor product. It's
very pretty, and reading it is very good exercise if you want to
learn more about representations of quantum groups.
Now, I said that a lot of this is part of a bigger picture that
works in higher dimensions. However, a lot of this higherdimensional
stuff remains very mysterious. Here are two cool papers that make
some progress in unlocking these mysteries:
6) Marco Mackaay, Finite groups, spherical 2categories, and 4manifold
invariants, preprint available as math.QA/9903003.
7) Mikhail Khovanov, A categorification of the Jones polynomial,
preprint available as math.QA/9908171.
Marco Mackaay spoke about his work in Coimbra, and I had grilled
him about it in Lisbon beforehand, so I think I understand it
pretty well. Basically what he's doing is categorifying the
3dimensional topological quantum field theories studied by Dijkgraaf
and Witten to get 4dimensional theories. It fits in very nicely
with his earlier work described in "week121".
People have been trying to categorify the magic of quantum groups
for quite some time now, and Khovanov appears to have made a good
step in that direction by describing the Jones polynomial of a
link as the "graded Euler characteristic" of a chain complex of
graded vector spaces. Since graded Euler characteristic is a
generalization of the dimension of a vector space, and taking the
dimension is a process of decategorification (i.e., vector spaces
are isomorphic iff they have the same dimension), Khovanov's
chain complex can be thought of as a categorified version of the
Jones polynomial.
I would like to understand better the relation between Khovanov's
work and the work of Crane and Frenkel on categorifying quantum
groups (see "week58"). For this, I guess I should read the
following papers:
8) J. Bernstein, I. Frenkel and M. Khovanov, A categorification
of the TemperleyLieb algebra and Schur quotients of U(sl_2)
by projective and Zuckerman functors, to appear in Selecta
Mathematica.
9) Mikhail Khovanov, Graphical calculus, canonical bases and
KazhdanLusztig theory, Ph.D. thesis, Yale, 1997.

Previous issues of "This Week's Finds" and other expository articles on
mathematics and physics, as well as some of my research papers, can be
obtained at
http://math.ucr.edu/home/baez/
For a table of contents of all the issues of This Week's Finds, try
http://math.ucr.edu/home/baez/twf.html
A simple jumpingoff point to the old issues is available at
http://math.ucr.edu/home/baez/twfshort.html
If you just want the latest issue, go to
http://math.ucr.edu/home/baez/this.week.html
group), andtwf_ascii/week138000064400020410000157000000221140774011336500141350ustar00baezhttp00004600000001Also available at http://math.ucr.edu/home/baez/week138.html
September 12, 1998
This Week's Finds in Mathematical Physics (Week 138)
John Baez
I haven't been going to the Newton Institute much during my stay
in Cambridge, even though it's right around the corner from where
I live. There's always interesting math and physics going on
at the Newton Institute, and this summer they had some conferences
on cosmology, but I've been trying to get away from it all for a
while. Still, I couldn't couldn't resist the opportunity to go to
James Hartle's 60th birthday party, which was held there on September
2nd.
Hartle is famous for his work on quantum gravity and the foundations
of quantum mechanics, so some physics bigshots came and gave talks.
First Chris Isham spoke on applications of topos theory to quantum
mechanics, particularly in relation to Hartle's work on the socalled
"decoherent histories" approach to quantum mechanics, which he developed
with Murray GellMann. Then Roger Penrose spoke on his ideas of
gravitationally induced collapse of the wavefunction. Then Gary Gibbons
spoke on the geometry of quantum mechanics. All very nice talks!
Finally, Stephen Hawking spoke on his work with Hartle. This talk was
the most personal in nature: Hawking interspersed technical descriptions
of the papers they wrote together with humorous reminiscences of their
gettogethers in Santa Barbara and elsewhere, including a long trip in a
Volkswagen beetle with Hawking's wheelchair crammed into the back seat.
I think Hawking said he wrote 4 papers with Hartle. The first really
important one was this:
1) James Hartle and Stephen Hawking, Path integral derivation of
black hole radiance, Phys. Rev. D13 (1976), 2188.
As I explained in "week111", Hawking wrote a paper in 1975 establishing
a remarkable link between black hole physics and thermodynamics. He
showed that a black hole emits radiation just as if it had a temperature
inversely proportional to its mass. However, this paper was regarded
with some suspicion at the time, not only because the result was so
amazing, but because the calculation involved modes of the electromagnetic
field of extremely short wavelengths near the event horizon  much shorter
than the Planck length.
For this reason, Hartle and Hawking decided to redo the calculation
using path integrals  a widely accepted technique in particle physics.
Hawking's background was in general relativity, so he wasn't too good
at path integrals; Hartle had more experience with particle physics
and knew how to do that kind of stuff.
(By now, of course, Hawking can do path integrals quicker than most
folks can balance their checkbook. This was a while ago.)
This wasn't straightforward. In particle physics people usually do
calculations assuming spacetime is flat, so Hartle and Hawking needed
to adapt the usual pathintegral techniques to the case when spacetime
contains a black hole. The usual trick in path integrals is to replace
the time variable t by an imaginary number, then do the calculation, and
then analytically continue the answer back to real times. This isn't so
easy when there's a black hole around!
For starters, you have to analytically continue the Schwarzschild solution
(the usual metric for a nonrotating black hole) to imaginary values of the
time variable. When you do this, something curious happens: you find that
the Schwarzschild solution is periodic in the imaginary time direction.
And the period is proportional to the black hole's mass.
Now, if you are good at physics, you know that doing quantum field
theory calculations where imaginary time is periodic with period 1/T
is the same as doing statistical mechanics calculations where the
temperature is T. So right away, you see that a black hole acts like
it has a temperature inversely proportional to its mass!
(In case you're worried, I'm using units where hbar, c, G, and k are
equal to 1.)
Anyway, that's how people think about the HartleHawking paper these
days. I haven't actually read it, so my description may be a bit
anachronistic. Things usually look simpler and clearer in retrospect.
The other really important paper by Hartle and Hawking is this one:
2) James Hartle and Stephen Hawking, Wavefunction of the universe,
Phys. Rev. D28 (1983), 2960.
In quantum mechanics, we often describe the state of a physical
system by a wavefunction  a complexvalued function on the
classical configuration space. If quantum mechanics applies to
the whole universe, this naturally leads to the question: what's
the wavefunction of the universe? In the above paper, Hartle
and Hawking propose an answer.
Now, it might seem a bit overly ambitious to guess the wavefunction
of the entire universe, since we haven't even seen the whole thing yet.
And indeed, if someone claims to know the wavefunction of the whole
universe, you might think they were claiming to know everything
that has happened or will happen. Which naturally led GellMann to
ask Hartle: "If you know the wavefunction of the universe, why aren't
you rich yet?"
But the funny thing about quantum theory is that, thanks to the
uncertainty principle, you can know the wavefunction of the universe,
and still be completely clueless as to which horse will win at the
races tomorrow, or even how many planets orbit the sun.
That will either make sense to you, or it won't, and I'm not sure
anything *short* I might write will help you understand it if you
don't already. A full explanation of this business would lead me
down paths I don't want to tread just now  right into that morass
they call "the interpretation of quantum mechanics".
So instead of worrying too much about what it would *mean* to know the
wavefunction of the universe, let me just explain Hartle and Hawking's
formula for it. Mind you, this formula may or may not be correct, or
even rigorously welldefined  there's been a lot of argument about it
in the physics literature. However, it's pretty cool, and definitely
worth knowing.
Here things get a wee bit more technical. Suppose that space is a
3sphere, say X. The classical configuration space of general relativity
is the space of metrics on X. The wavefunction of the universe should
be some complexvalued function on this classical configuration space.
And here's Hartle and Hawking's formula for it:
psi(q) = integral exp(S(g)/hbar) dg
gX = q
Now you can wow your friends by writing down this formula and
saying "Here's the wavefunction of the universe!"
But, what does it mean?
Well, the integral is taken over Riemannian metrics g on a 4ball
whose boundary is X, but we only integrate over metrics that
restrict to a given metric q on X  that's what I mean by writing
gX = q. The quantity S(g) is the EinsteinHilbert action of the
metric g  in other words, the integral of the Ricci scalar curvature
of g over the 4ball. Finally, of course, hbar is Planck's constant.
The idea is that, formally at least, this wavefunction is a solution of
the WheelerDeWitt equation, which is the basic equation of quantum
gravity (see "week43").
The measure "dg" is, unfortunately, illdefined! In other words, one
needs to use lots of clever tricks to extract physics from this formula,
as usual for path integrals. But one can do it, and Hawking and others
have spent a lot of time ever since 1983 doing exactly this. This led
to a subject called "quantum cosmology".
I should add that there are lots of ways to soup up the basic
HartleHawking formula. If we have other fields around besides
gravity, we just throw them into the action in the action in the
obvious way and integrate over them too. If our manifold X representing
space is not a 3sphere, we can pick some other 4manifold having it
as boundary. If we can't make up our mind which 4manifold to use,
we can try a "sum over topologies", summing over all 4manifolds
with X as boundary. We can do this even when X is a 3sphere,
actually  but it's a bit controversial whether we should, and
also whether the sum converges.
Well, there's a lot more to say, like what the physical interpretation
of the HartleHawking formula is, and what predicts. It's actually quite
cool  in a sense, it says that the universe tunnelled into being out of
nothingness! But that sounds like a bunch of nonsense  the sort of fluff
they write on the front of science magazines to sell copies. To really
explain it takes quite a bit more work. And unfortunately, it's just
about dinnertime, so I want to stop now.
Anyway, it was an interesting birthday party.

Previous issues of "This Week's Finds" and other expository articles on
mathematics and physics, as well as some of my research papers, can be
obtained at
http://math.ucr.edu/home/baez/
For a table of contents of all the issues of This Week's Finds, try
http://math.ucr.edu/home/baez/twf.html
A simple jumpingoff point to the old issues is available at
http://math.ucr.edu/home/baez/twfshort.html
If you just want the latest issue, go to
http://math.ucr.edu/home/baez/this.week.html
eally
important one was this:
1) James Hartle and Stephen Hawking, Path integral derivation of
black hole radiance, Phys. Rev. D13 (1976), 2188.
As I explained in "week111", Hawking wrote a paper in 1975 establishing
a remarkable link between black hole physics and thermodynamics. He
showed that a black hole emits radiation just as if it had a temperature
inversely proportional to its mass. However, this paper was regarded twf_ascii/week139000064400020410000157000000441050774011336500141420ustar00baezhttp00004600000001Also available at http://math.ucr.edu/home/baez/week139.html
September 19, 1999
This Week's Finds in Mathematical Physics (Week 139)
John Baez
Last time I described some of the talks at James Hartle's 60th birthday
celebration at the Newton Institute. But I also met some people at that
party that I'd been wanting to talk to. There's a long story behind this,
so if you don't mind, I'll start at the beginning....
A while ago, Phillip Helbig, one of the two moderators of
sci.physics.research who do astrophysics, drew my attention to an
interesting paper:
1) Vipul Periwal, Cosmological and astrophysical tests of
quantum gravity, preprint available at astroph/9906253.
The basic idea behind this is that quantum gravity effects could cause
deviations from Newton's inverse square law at large distance scales, and
that these deviations might explain various puzzles in astrophysics, like
the "missing mass problem" and the possibly accelerating expansion of the
universe.
This would be great, because it might not only help us understand these
astrophysics puzzles, but also help solve the big problem with quantum
gravity, namely the shortage of relevant experimental data.
But of course one needs to read the fine print before getting too excited
about ideas like this!
Following the argument in Periwal's paper requires some familiarity
with the renormalization group, since that's what people use to study
how "constants" like the charge of the electron or Newton's gravitational
constant depend on the distance scale at which you measure them  due
to quantum effects. Reading the paper, I immediately became frustrated
with my poor understanding of the renormalization group. It's really
important, so I decided to read more about it and explain it in the
simplest possible terms on sci.physics.research  since to understand
stuff, I like to try to explain it.
In the process, I found this book very helpful:
1) Michael E. Peskin and Daniel V. Schroeder, An Introduction
to Quantum Field Theory, AddisonWesley, Reading, Massachusetts
1995.
The books I'd originally learned quantum field theory from didn't
incorporate the modern attitude towards renormalization, due to
Kenneth Wilson  the idea that quantum field theory may not
ultimately be true at very short distance scales, but that's
okay, because if we assume it's a good approximation at pretty
short distance scales, it becomes a *better* approximation at
*larger* distance scales. This is especially important when you're
thinking about quantum gravity, where godawful strange stuff may be
happening at the Planck length. Peskin and Schroeder explain this
idea quite well. For my own sketchy summary, try this:
2) John Baez, Renormalization made easy,
http://math.ucr.edu/home/baez/renormalization.html
I deliberately left out as much math as possible, to concentrate
on the basic intuition.
Thus fortified, I returned to Periwal's paper, and it made a bit
more sense. Let me describe the main idea: how we might expect
Newton's gravitational constant to change with distance.
So, suppose we have any old quantum field theory with a coupling constant
G in it. In fact, G will depend on the length scale at which we
measure it. But using Planck's constant and the speed of light we
can translate length into 1/momentum. This allows us to think of G
as a function of momentum. Roughly speaking, when you shoot particles
at each other at higher momenta, they come closer together before
bouncing off, so measuring a coupling constant at a higher momentum
amounts to measuring at a shorter distance scale.
The equation describing how G depends on the momentum p is called
the "CallanSymanzik equation". In general it looks like this:
dG
 = beta(G)
d(ln p)
But all the fun starts when we use our quantum field theory to calculate
the right hand side, which is called  surprise!  the "beta function"
of our theory. Typically we get something like this:
dG
 = (n  d)G + aG^2 + bG^3 + ....
d(ln p)
Here n is the dimension of spacetime and d is a number called the
"upper critical dimension". You see, it's fun when possible to think
of our quantum field theory as defined in a spacetime of arbitrary
dimension, and then specialize to the case at hand. I'll show you
how work out d in a minute. It's harder to work out the numbers
a, b, and so on  for this, you need to do some computations using the
quantum field theory in question.
What does the CallanSymanzik equation really mean? Well, for starters
let's neglect the higherorder terms and suppose that
dG(p)
 = (n  d)G
d(ln p)
This says G is proportional to p^{nd}. There are 3 cases:
A) When n < d, our coupling constant gets *smaller* at higher momentum
scales, and we say our theory is "superrenormalizable". Roughly, this
means that at larger and larger momentum scales, our theory looks more
and more like a "free field theory"  one where particles don't interact
at all. This makes superrenormalizable theories easy to study by
treating them as a free field theory plus a small perturbation.
B) When n > d, our coupling constant gets *larger* at higher momentum
scales, and we say our theory is "nonrenormalizable". Such theories
are hard to study using perturbative calculations in free field theory.
C) When n = d, we are right on the brink between the two cases above.
We say our theory is "renormalizable", but we really have to work out
the next term in the beta function to see if the coupling constant
grows or shrinks with increasing momentum.
Consider the example of general relativity. We can figure out
the upper critical dimension using a bit of dimensional analysis
and handwaving. Let's work in units where Planck's constant and the
speed of light are 1. The Lagrangian is the Ricci scalar curvature
divided by 8 pi G, where G is Newton's gravitational constant. We
need to get something dimensionless when we integrate the Lagrangian
over spacetime to get the action, since we exponentiate the action
when doing path integrals in quantum field theory. Curvature has
dimensions of 1/length^2, so when spacetime has dimension n, G must
have dimensions of length^{n2}.
This means that if you are a tiny little person with a ruler X
times smaller than mine, Newton's constant will seem X^{n2} times
bigger to you. But measuring Newton's constant at a length scale
that's X times smaller is the same as measuring it at a momentum scale
that's X times bigger. We already solved the CallanSymanzik equation
and saw that when we measure G at the momentum scale p, we get an
answer proportional to p^{nd}. We thus conclude that d = 2.
(If you're a physicist, you might enjoy finding the holes in the
above argument, and then plugging them.)
This means that quantum gravity is nonrenormalizable in 4 dimensions.
Apparently gravity just keeps looking stronger and stronger at
shorter and shorter distance scales. That's why quantum gravity has
traditionally been regarded as hard  verging on hopeless.
However, there is a subtlety. We've been ignoring the higherorder
terms in the beta function, and we really shouldn't!
This is obvious for renormalizable theories, since when n = d, the
beta function looks like
dG
 = aG^2 + bG^3 + ....
d(ln p)
so if we ignore the higherorder terms, we are ignoring the whole
righthand side! To see the effect of these higherorder terms let's
just consider the simple case where
dG
 = aG^2
d(ln p)
If you solve this you get
c
G = 
1  ac ln p
where c is a positive constant. What does this mean? Well, if a < 0,
it's obvious even before solving the equation that G slowly *decreases*
with increasing momentum. In this case we say our theory is
"asymptotically free". For example, this is true for the strong
force in the Standard Model, so in collisions at high momentum quarks
and gluons act a lot like free particles. (For more on this, try "week94".)
On the other hand, if a > 0, the coupling constant G *increases* with
increasing momentum. To make matters worse, it becomes INFINITE
at a sufficiently high momentum! In this case we say our theory has
a "Landau pole", and we cluck our tongues disapprovingly, because it's
not a good thing. For example, this is what happens in quantum
electrodynamics when we don't include the weak force. Of course,
one should really consider the effect of even higherorder terms in
the beta function before jumping to conclusions. However, particle
physicists generally feel that among renormalizable field theories,
the ones with a < 0 are good, and the ones with a > 0 are bad.
Okay, now for the really fun part. Perturbative quantum gravity
in 2 dimensions is not only renormalizable (because this is the
upper critical dimension), it's also asympotically free! Thus
in n dimensions, we have
dG
 = (n  2)G + aG^2 + ....
d(ln p)
where a < 0. If we ignore the higherorder terms which I have
written as "....", this implies something very interesting for
quantum gravity in 4 dimensions. Suppose that at low momenta
G is small. Then the righthand side is dominated by the first
term, which is positive. This means that as we crank up the
momentum scale, G keeps getting bigger. This is what we already
saw about nonrenormalizable theories. But after a while, when G
gets big, the second term starts mattering more  and it's negative.
So the growth of G starts slowing!
In fact, it's easy to see that as we keep cranking up the momentum,
G will approach the value for which
dG
 = 0
d(ln p)
We call this value an "ultraviolet stable fixed point" for the
gravitational constant. Mathematically, what we've got is a flow
in the space of coupling constants, and an ultraviolet stable fixed
point is one that attracts nearby points as we flow in the direction
of higher momenta. This particular kind of ultraviolet stable fixed
point  coming from an asymptotically free theory in dimensions above
its upper critical dimension  is called a "WilsonFisher fixed point".
So: perhaps quantum gravity is saved from an evergrowing Newton's
constant at small distance scales by a WilsonFisher fixed point!
But before we break out the champagne, note that we neglected the
higherorder terms in the beta function in our last bit of reasoning.
They can still screw things up. For example, if
dG
 = (n  2)G + aG^2 + bG^3
d(ln p)
and b is positive, there will not be a WilsonFisher fixed point
when the dimension n gets too far above 2. Is 4 too far above 2?
Nobody knows for sure. We can't really work out the beta function
exactly. So, as usual in quantum gravity, things are a bit iffy.
However, Periwal cites the following paper as giving numerical
evidence for a WilsonFisher fixed point in quantum gravity:
4) Herbert W. Hamber and Ruth M. Williams, Newtonian potential in
quantum Regge gravity, Nucl. Phys. B435 (1995), 361397.
And he draws some startling conclusions from the existence of
this fixed point. He says it should have consequences for the
missing mass problem and the value of the cosmological constant!
However, I found it hard to follow his reasoning, so I decided
to track down some of the references  starting with the above
paper.
Now, Ruth Williams works at Cambridge University, so I was not
surprised to find her at Hartle's party. She was busy talking
to John Barrett, who also does quantum gravity, up at Nottingham
University. I arranged to stop by her office, get a copy of her
paper, and have her explain it to me. I also arranged to visit
John in Nottingham and have him explain his work with Louis Crane
on Lorentzian spin foam models  but more about that next week!
Anyway, here's how the HamberWilliams paper goes, very roughly.
They simulate quantum gravity by chopping up a 4dimensional torus
into 16 x 16 x 16 x 16 hypercubes, chopping each hypercube into 24
4simplices in the obvious way, and then doing a Monte Carlo calculation
of the path integral using the Regge calculus, which is a discretized
version of general relativity suited to triangulated manifolds (see
"week119" for details). Their goal was to work out how Newton's
constant varies with distance. They did it by calculating correlations
between Wilson loops that wrap around the torus. They explain how
you can deduce Newton's constant from this information, but I don't
have the energy to describe that here. Anyway, they claim that Newton's
constant varies with distance as one would expect if there was a
WilsonFisher fixed point!
(It's actually more complicated that this because besides Newton's
constant, there is also another coupling constant in their theory:
the cosmological constant. And of course this is very important
for potential applications to astrophysics.)
Unfortunately, I'm still mystified about a large number of things.
Let me just mention two. First, Hamber and Williams consider values
of G which are *greater* than the WilsonFisher fixed point. Since
this is an ultraviolet stable fixed point, such values of G flow *down*
to the fixed point as we crank up the momentum scale. Or in other
words, in this regime Newton's constant gets *bigger* with increasing
distances. At least to my naive brain, this sounds nice for explaining
the missing mass problem. But the funny thing is, this regime is
utterly different from the regime where G is close to zero  namely,
*less* than the WilsonFisher fixed point. I thought all the usual
perturbative quantum gravity calculations were based on the assumption
that at macroscopic distance scales G is small, and flows *up* to
the fixed point as we crank up the momentum scale! Are these folks
claiming this picture is completely wrong? I'm confused.
Another puzzle is that Periwal thinks Newton's constant will start
to grow at distance scales roughly comparable to the radius of the
universe (or more precisely, the Hubble length). But it looks like
Hamber and Williams say their formula for G as a function of momentum
holds at *short* distance scales.
I guess I need to read more stuff, starting perhaps with Weinberg's
old paper on quantum gravity and the renormalization group:
5) Steven Weinberg, Ultraviolet divergences in quantum theories of
gravitation, in General Relativity: an Einstein Centenary Survey,
eds. Stephen Hawking and Werner Israel, Cambridge U. Press, Cambridge
(1979).
and then perhaps turning to his paper on the cosmological constant:
6) Steven Weinberg, The cosmological constant problem, Rev. Mod. Phys.
61 (1989), 1.
and some books on the renormalization group and quantum gravity:
7) Claude Itzykson and JeanMichel Drouffe, Statistical Field Theory,
2 volumes, Cambridge U. Press, 1989.
8) Jean ZinnJustin, Quantum Field Theory and Critical Phenomena,
Oxford U. Press, Oxford, 1993.
9) Jan Ambjorn, Bergfinnur Durhuus, and Thordur Jonsson, Quantum
Geometry: A Statistical Field Theory Approach, Cambridge U. Press,
Cambridge, 1997.
I should also think more about this recent paper, which claims to
find a phase transition in a toy model of quantum gravity where
one does the path integral over a special class of metrics  namely
those with 2 Killing vector fields.
10) Viqar Husain and Sebastian Jaimungal, Phase transition in
quantum gravity, preprint available as grqc/9908056.
But if anyone can help me clear up these issues, please let me know!
Okay, enough of that. Another person I met at the party was Roger
Penrose! Later I visited him in Oxford. Though recently retired,
he still holds monthly meetings at his house in the country, attended
by a bunch of young mathematicians and physicists. At the one I went
to, the discussion centered around Penrose's forthcoming book. The goal
of this book is to explain modern physics to people who know only a
little math, but are willing to learn more. A nice thing about it is
that it treats various modern physics fads without the uncritical
adulation that mars many popularizations. In particular, when I
visited, he was busy writing a chapter on inflationary cosmology,
so he talked about a bunch of problems with that theory, and cosmology
in general.
I've never been sold on inflation, since it relies on fairly speculative
aspects of grand unified theories (or GUTs), so most of these problems merely
amused me. Theorists take a certain wicked glee in seeing someone else's
theory in trouble. However, one of these problems concerned the Standard
Model, and this hit closer to home. Penrose made the standard observation
that the most distant visible galaxies in opposite directions have not had
time to exchange information  at least not since the time of recombination,
when the initial hot fireball cooled down enough to become transparent.
But if the symmetry between the electromagnetic and weak forces is
spontaneously broken only when the Higgs field cools down enough to line
up, as the Standard Model suggests, this raises the danger that the Higgs
field could wind up pointing in different directions in different patches
of the visible universe!  since these different "domains" would not yet
have had time to expand to the point where a single one fills the whole
visible universe. But we don't see such domains  or more precisely, we
don't see the "domain walls" one would expect at their boundaries.
Of course, inflation is an attempt to deal with similar problems, but
inflation is posited to happen at GUT scale energies, too soon (it seems)
to solve *this* problem, which happens when things cool down to the
point where the electroweak symmetry breaks.
Again, if anyone knows anything about this, I'd love to hear about it.

Previous issues of "This Week's Finds" and other expository articles on
mathematics and physics, as well as some of my research papers, can be
obtained at
http://math.ucr.edu/home/baez/
For a table of contents of all the issues of This Week's Finds, try
http://math.ucr.edu/home/baez/twf.html
A simple jumpingoff point to the old issues is available at
http://math.ucr.edu/home/baez/twfshort.html
If you just want the latest issue, go to
http://math.ucr.edu/home/baez/this.week.html
a sufficiently high momentum! In this case we say our theory has
a "Landau pole", and we cluck our tongues disapprovingly, because it's
not a good thing. For example, this is what happens in quantum
electrodynamics when we don't include the weak force. Of course,
one should really consider the effect of even higherorder terms in
the beta function before jumping to conclusions. However, particle
physicists generally feel that amontwf_ascii/week14000064400020410000157000000564160774011336500140620ustar00baezhttp00004600000001Newsgroups: sci.physics.research,sci.math
From: jbaez@ucrmath.ucr.edu (John Baez)
Subject: This Week's Finds in Mathematical Physics (Week 14)
Keywords: Source for Graph Paper
Date: 8 May 93
Organization: Math. Dept., UC Riverside
Week 14
Things are moving very fast in the quantum gravity/4d topology game, so
I feel I should break my vow not to continue this series until after
next weekend's conference on Knots and Quantum Gravity.
Maybe I should recall where things were when I left off. The physics
problem motivating a lot of work in theoretic physics today is
reconciling general relativity and quantum theory. The key feature of
general relativity is that time and space do not appear as a "background
structure," but rather are dynamical variables. In mathematical terms,
this just means that there is not a fixed metric; instead gravity *is*
the metric, and the metric evolves with time like any other physical
field, satisfying some field equations called the Einstein equations.
But it is worth stepping back from the mathematics and trying to put
into simple words why this makes general relativity so special. Of
course, it's very hard to put this sort of thing into words. But
roughly, we can say this: in Newtonian mechanics, there is a universal
notion of time, the "t" coordinate that appears in all the equations of
physics, and we assume that anyone with a decent watch will be able to
keep in synch with everyone else, so there is no confusion about what
this "t" is (apart from choosing when to call t = 0, which is a small
sort of arbitrariness one has to live with). In special relativity
this is no longer true; watches moving relative to each other will no
longer stay in synch, so we need to pick an "inertial frame," a notion
of rest, in order to have a "t" coordinate to play with. Once we pick
this inertial frame, we can write the laws of physics as equations
involving "t". This is not too bad, because there is only a
finiteparameter family of inertial frames, and simple recipes to
translate between them, and also because nothing going on will screw up
the functioning of our (idealized) clocks: that is, the "t" coordinate
doesn't give a damn about the *state* of the universe. That's what is
meant by saying a "background structure"  it's some aspect of the
universe that is unaffected by everything else that's going on.
In general relativity, things get much more interesting: there is no
such thing as an inertial frame that defines coordinates on spacetime,
because there is no way you can get a lot of things at different places
to remain at rest with each other  this is what is meant by saying that
spacetime is curved. You can measure time with your watch, socalled
"proper time," but this applies only near you. More interestingly
still, to compare what your watch is doing to what someone else's is
doing, you actually need to know a lot about the state of the universe,
e.g., whether there are any heavy masses around that are curving
spacetime. The "metric," whereby one measures distances and proper
time, depends on the state of the universe  or more properly, it is
part of the state of the universe.
Trying to do *quantum* theory in this context has always been too hard
for people. Part of the reason why is that built into the heart of
traditional quantum theory is the "Hamiltonian," which describes the
evolution of the state of the system relative to a Godgiven
"background" notion of "t". Anyone who has taken quantum mechanics will
know that the star of the show is the Schrodinger equation:
i dPsi/dt = H Psi
saying how the wavefunction Psi changes with time in a way depending on
the Hamiltonian H. No "t," no "H"  this is one basic problem with
trying to reconcile quantum theory with general relativity.
Actually, it turns out that the analog to Schrodinger's equation for
quantum gravity is the WheelerDeWitt equation. The Hamiltonian is
replaced by an operator called the "Hamiltonian constraint" and we have
H Psi = 0.
Note how this cleverly avoids mentioning "t"! The problem is, people
still aren't quite sure what to do with the solutions to this equation 
we're so used to working with Schrodinger's equation.
Now in 1988 Witten wrote a paper in which he coined the term
"topological quantum field theory," or TQFT, for short. This was meant
to capture in a rigorous way what field theories like quantum gravity
should be like. Actually, Witten was working on a different theory
called Donaldson theory, which also has the property of having no
background structures. Shortly thereafter the mathematician Atiyah came
up with a formal definition of a TQFT. To get an idea of this
definition, try the files "symmetries" and (if you don't know what
categories are, and you'll need to) "categories" available
electronically from the ftp site listed at the end of this article.
For a serious tour of TQFTs and the like, try his book:
The Geometry and Physics of Knots, by Michael Atiyah, Cambridge U.
Press, 1990.
One can think of a TQFT as a framework in which a WheelerDeWittlike
equation governs the dynamics of a quantum field theory. Experts may
snicker here, but it is true, if not as enlightening as other things one
can say.
I won't bother to define TQFTs here, but I think Smolin put it very well
when he said the idea of TQFTs really helped us break out of our
traditional idea of fields as being something defined at every point of
spacetime, wiggling around, and allowed us to see field theory from many
new angles. For example, TQFTs let us wiggle out of the old conundrum
of whether spacetime is continuous or discrete, because many TQFTs can
be *equivalently* described in either of two ways: via a continuum model
of spacetime, or via a discrete one in which spacetime is given a
"simplicial structure," like a big tetrahedral tinkertoy lattice kind of
thing. The latter idea appears to be due to Turaev and Viro, although
certainly physicists have had similar ideas for years, going back to
Ponzano and Regge, who worked on simplicial quantum gravity.
Now the odd thing is that while interesting 3d TQFTs have been found,
the most notable being ChernSimons theory, nobody has quite been able
to make 4d TQFTs rigorous. Witten's original work on Donaldson theory
has led to many interesting things, but not yet a fullfledged TQFT in
the rigorous sense of Atiyah. And quantum gravity still resists being
formulated as a TQFT.
A while back I noted that Crane and Yetter had invented a 4d TQFT using
the simplicial approach. There has been a lot of argument over whether
this TQFT is interesting or "trivial." Of course, trivial is not a
precise concept. For a while Ocneanu claimed that the partition
function of every compact 4manifold equalled 1 in this TQFT, which
counts as very trivial. But this appears not to be the case. Broda
invented another 4d TQFT and here on "This Week's Finds" Ruberman showed
it was trivial in the sense that the partition function of any compact
4manifold was a function of the "signature" of the 4manifold. This is
trivial because the signature is a wellunderstood invariant and if we
are trying to do something new and interesting that just isn't good enough.
In the following paper:
1) Skein theory and TuraevViro invariants, by Justin Roberts, Pembroke
College preprint, April 14, 1993 (Roberts is at J.D.Roberts@pmms.cam.ac.uk)
Roberts *almost* claims to show that the CraneYetter invariant is
trivial in the same sense, namely that the partition function of any
compact 4manifold is an exponential of the signature. Now if Crane and
Yetter's own computations are correct, this cannot be the case, but it
*could* be an exponential of a linear combination of the signature and
the Euler characteristic, as far as I know. The catch is that Roberts
does not normalize his version of the CraneYetter invariant in the same
way that Crane and Yetter do, so it is hard to compare results. But
Roberts says: "The normalisations here do not agree with those in Crane
and Yetter, and I have not checked the relationship. However, when
dealing with the [3d TQFT] invariants, different normalisations of the
initial data change the invariants by factors depending on standard
topological invariants (for example Betti numbers), so there is every
reason to belive that these [4d TQFT] invariants are trivial (that is,
they differ from 1 only by standard invariant factors) in all
normalisations."
This is a bit of a disappointment, because Crane at least had hoped that
their TQFT might actually turn out to *be* quantum gravity. This was
not idle dreaming; it was because the CraneYetter construction was a
rigorous analog of some work by Ooguri on simplicial quantum gravity.
Then, about a week ago, Rovelli put a paper onto the net:
2) The basis of the PonzanoReggeTuraevViroOoguri model is the loop
representation basis, 16 pages in LaTeX, Friday April 30, available as
hepth/9304164.
(To get stuff from hepth see the end of this article.)
This is a remarkable paper that I have not been able to absorb yet.
First it goes over 3d quantum gravity  which *has* been made into a
rigorous TQFT. It works with the simplicial formulation of the theory.
That is, we consider our (3dimensional) spacetime as being chopped up
into tetrahedra, and assign to each edge a length, which is required to
be 0,1/2,1,3/2,.... This idea of quantized edgelengths goes back to
4d work of Ponzano and Regge, but recently Ooguri showed that in 3d
this assumption gives the same answers as Witten's continuum approach to
3d quantum gravity. The "halfintegers" 0,1/2,1,3/2, etc. should remind
physicists of spin, which is quantized in the same way, and
mathematically this is exactly what is going on: we are really labelling
edges with representations of the group SU(2), that is, spins. What
Rovelli shows is that if one starts with the loop representation of 3d
quantum gravity (yet another approach), one can prove it equivalent to
Ooguri's approach, and what's more, using the loop representation one
can *calculate* the lengths of edges of triangles in a given state of
space (space here is a 2dimensional triangulated surface) and *show*
that lengths are quantized in units of the Planck length over 2. (Here
the Planck length L is the fundamental length scale in quantum gravity,
about 1.6 times 10^{33} meters.)
And, most tantalizing of all, he sketches a generalization of the above
to 4d. In 4d it is known that in the loop representation of quantum
gravity it is areas of surfaces that are quantized in units of L^2/2,
rather than lengths. Rovelli considers an approach where one chops
4dimensional spacetime up into simplices and assigns to each
2dimensional face a halfinteger area. He uses this to write down a
formula for the inner product in the Hilbert space of quantum gravity 
thus, at least formally, answering the longstanding "inner product
problem" in quantum gravity. The problem is that, unlike in 3d quantum
gravity, here one must sum over ways of dividing spacetime into
simplices, so the formula for the inner product involves a sum that does
not obviously converge. This is however sort of what one might expect,
since in 4d quantum gravity, unlike 3d, there are "local excitations" 
local wigglings of the metric, if you will  and this makes the Hilbert
space be infinitedimensional, whereas the 3d TQFTs have
finitedimensional Hilbert spaces.
I think I'll quote him here. It's a bit technical in patches, but worth
it...

We conclude with a consideration on the formal structure of 4d quantum
gravity, which is important to understand the above construction. Standard
quantum field theories, as QED and QCD, as well as their generalizations like
quantum field theories on curved spaces and perturbative string theory, are
defined on metric spaces. Witten's introduction of the topological quantum
field theories has shown that one can construct quantum field theories
defined on a manifold which has only its differential structure, and no fixed
metric structure. The theories introduced by Witten and axiomatized by
Atiyah have the following peculiar feature: they have a finite number of
degrees of freedom, or, equivalently, their quantum mechanical Hilbert
spaces are finite dimensional; classically this follows from the fact that
the number of fields is equal to the number of gauge transformations. However,
not any diffinvariant field theory on a manifold has a finite number of
degrees of freedom. Witten's gravity in 3d is given by the action
$S[A,E] = \int F \wedge E$,
which has finite number of degrees of freedom. Consider the action
$S[A,E] = \int F \wedge E \wedge E$,
in 3+1 dimensions, for a (self dual) SO(3,1) connection $A$ and a (real)
one form $E$ with values in the vector representation of
SO(3,1). This theory has a strong resemblance with its 2+1 dimensional
analog: it is still defined on a differential manifold without any fixed
metric structure, and is diffeomorphism invariant. We expect that a
consistent quantization of such a theory should be found along lines
which are more similar to the quantization of the $\int F \wedge E$,
theory than to the quantization of theories on flat space, based on the
Wightman axioms namely on npoints functions and related objects.
Still, the theory $\int F \wedge E \wedge E$ has genuine field
degrees of freedom: its physical phase space is infinite dimensional, and we
expect that its Hilbert state space will also be infinite dimensional. There
is a popular belief that a theory defined on a differential manifold
without metric and diffeomorphism invariant has necessarily a finite
number of degrees of freedom ("because thanks to general covariance
we can gauge away any local excitation"). This belief is of course wrong. A
theory as the one defined by the action $\int F \wedge E \wedge E$
is a theory that shares many features with the topological theories, in
particular, no quantity defined "in a specific point" is gauge
invariant; but at the same time it has genuinely infinite degrees of
freedom. Indeed, this theory is of course nothing but (Ashtekar's
form of) standard general relativity.
The fact that "local" quantities like the npoint functions are not
appropriate to describe quantum gravity nonperturbatively has been
repeatedly noted in the literature. As a consequence, the issue of
what are the quantities in terms of which a quantum theory of gravity can be
constructed is a much debated issue. The above discussion indicates
a way to face the problem: The topological quantum field theories studied by
Witten and Atiyah provide a framework in terms of which quantum gravity
itself may be framed, in spite of the infinite degrees of freedom. In
particular, Atiyah's axiomatization of the topological field theories
provides us with a clean way of formulating the problem. Of course, we
have to relax the requirement that the theory has a finite number of
degrees of freedom. These considerations leads us to propose that the
correct general axiomatic scheme for a physical quantum theory of
gravity is simply Atiyah's set of axioms up to finite dimensionality
of the Hilbert state space. We denote a structure that satisfies all
Atiyah's axioms, except the finite dimensionality of the state space, as
a *generalized topological theory*.
The theory we have sketched is an example of such a generalized topological
theory. We associate to the connected
components $\partial M_i$ of the boundary of M the infinite
dimensional state space of the Loop Representation of quantum
gravity. Eq. 5 [the magic formula I alluded to  jb], then, provides a map,
in Atiyah's sense, between the state spaces constructed on two of these
boundary components. Equivalently, it provides the definition of the Hilbert
product in the state space.
One could argue that the framework we have described cannot be
consistent, because it cannot allow us to recover the "broken phase
of gravity" in which we have a nondegenerate background metric: in
the proposed framework one has only nonlocal observables on the
boundaries, while in the broken phase a *local* field in M has
nonvanishing vacuum expectation value. We think that this argument is
weak because it disregards the diffeomorphism invariance of the theory:
in classical general relativity no experiment can distinguish a
Minkowskian spacetime metric from a nonMinkowkian flat metric. The two
are physically equivalent, as two gaugerelated Maxwell potentials. For
the same reason, no experiment could detect the absolute *position* of,
say, a gravitational wave, (while of course the position of an e.m. wave
is observable in Maxwell theory [after fixing an inertial frame  jb]).
Physical locality in general relativity is only defined as coincidence
of some physical variable with some other physical variable, while in
non general relativistic physics locality is defined with respect to a
fixed metric structure. In classical general relativity, there is no
gaugeinvariant obervable which is local in the coordinates. Thus,
any observation can be described by means of the value of the fields
on arbitrary boundaries of spacetime. This is the correct consequence
of the fact that "thanks to general covariance we can gauge away any
local excitation", and this is the reason for which one can have the ADM
"frozen time" formalism. The spacetime manifold of general relativity
is, in a sense, a much weaker physical object than the spacetime metric
manifold of ordinary theories. All the general relativistic physics can
be read from the boundaries of this manifold. At the same time, however,
these boundaries still carry an infinite dimensional number of degrees
of freedom.

Next, let me take the liberty of describing some work of my own,
3) Diffeomorphisminvariant generalized measures on the space of
connections modulo gauge transformations, by John Baez, to appear in the
proceedings of the Conference on Quantum Topology, Manhattan, Kansas,
May 8, 1993, soon available as state.tex from the ftp site described below
(and later in hepth).
This is an extremely interesting paper by a very good mathematician.
Whoops! Let's be objective here. In the loop representation of
quantum gravity, states of quantum gravity are given naively by certain
"measures" on a space A/G of connections modulo gauge transformations.
The ChernSimons path integral is also such a "measure". In both cases,
the "measure" in question is invariant under diffeomorphisms of space.
And in both cases, the loop transform allows one to think of these
measures as instead being functions of multiloops (collections of loops
in space). This is the origin of the relationship to knot theory.
The problem, as always in quantum field theory, is that the notion of
"measure" must be taken with a big grain of salt  it's not the sort of
measure they taught you about in real analysis. Instead, these measures
are a kind of "generalized measure" that allows you to integrate not all
continuous functions on A/G but only those lying in an algebra called
the "holonomy algebra," defined by Ashtekar, Isham and Lewandowski.
To be precise and technical, this is the closure in the L^infty norm of
the algebra of functions on A/G generated by "Wilson loops," or traced
holonomies around loops. So what we are really interested in is not
diffeomorphisminvariant measures on A/G, but diffeomorphism invariant
elements of the dual of the holonomy algebra. I begin with a review of
generalized measures, introduce the holonomy algebra, and then do a
bunch of new work in which I show how to rigorously construct lots
of diffeomorphisminvariant elements of the dual of the holonomy algebra
by doing lattice gauge theory on graphs embedded in space. Again, as
with the work discussed above, we see that the discrete and continuum
approaches to space go handinhand! And we see that there are some
interesting connections between singularity theory and group
representation theory showing up when we try to understand "measures" on
the space A/G.
The following is a part of a paper discussed in "week5", now available
from grqc:
4) Completeness of Wilson loop functionals on the moduli space of
SL(2,C) and SU(1,1)connections, Abhay Ashtekar and Jerzy Lewandowski,
Plain TeX, 7 pages, available as grqc/9304044.
I didn't discuss this aspect of the paper, so let me quote the abstract:

The structure of the moduli spaces M := A/G of (all, not just
flat) SL(2,C) and SU(1,1) connections on a nmanifold is analysed.
For any topology on the corresponding spaces A of all connections
which satisfies the weak requirement of compatibility with the affine
structure of A, the moduli space M is shown to be nonHausdorff.
It is then shown that the Wilson loop functionals i.e., the traces
of holonomies of connections around closed loops are complete in the
sense that they suffice to separate all separable points of M. The
methods are general enough to allow the underlying nmanifold to be
topologically nontrivial and for connections to be defined on
nontrivial bundles. The results have implications for canonical
quantum general relativity in 4 and 3 dimensions.

By the way, someone should extend this result to more general noncompact
semisimple Lie groups, and also show that for all compact semisimple Lie
groups the Wilson loop functionals in any faithful representation *do*
separate points (this is known for the fundamental representation of
SU(n)). If I had a bunch of grad students I would get one to do so.
The following was discussed in an earlier edition of this series,
"week11," but is now available from grqc:
5) An algebraic approach to the quantization of constrained systems: finite
dimensional examples, by Ranjeet S. Tate, (Ph.D. Dissertation, Syracuse
University), 124 pages, LaTeX (run thrice before printing), available as
grqc/9304043.
I haven't read the following one but it seems like an interesting
application of loop variables to more downtoearth physics; Gambini was
one of the originators of the loop representation, and intended it for
use in QCD:
6) SU(2) QCD in the path representation, by Rodolfo Gambini and Leonardo
Setaro, LaTeX 37 pages (7 figures included), available as
heplat/9305001.
In case "heplat" is not familiar, it is the computational and lattice
physics preprint list, at heplat@ftp.scri.fsu.edu. (Don't send email
there unless you know what you're doing! Read the end of this article
first!)
Let me quote the abstract:

We introduce a pathdependent hamiltonian representation (the path
representation) for SU(2) with fermions in 3 + 1 dimensions. The
gaugeinvariant operators and hamiltonian are realized in a Hilbert
space of open path and loop functionals. We obtain a new type of
relation, analogous to the Mandelstam identity of second kind, that
connects open path operators with loop operators. Also, we describe the
cluster approximation that permits to accomplish explicit calculations
of the vacuum energy density and the mass gap.

Previous editions of "This Week's Finds," and other expository posts
on mathematical physics, are available by anonymous ftp from
math.princeton.edu, thanks to Francis Fung. They are in the directory
/pub/fycfung/baezpapers. The README file contains lists of the
papers discussed in each week of "This Week's Finds."
Please don't ask me about hepth, grqc or heplat; instead, read the
sci.physics FAQ or the file "preprint.info" in /pub/fycfung/baezpapers.
dimensional, and we
expect that its Hilbert state space will also be infinite dimensional. There
is a popular belief that a theory defined on a differential manifold
without metric and diffeomorphism invariant has necessarily a finite
numbertwf_ascii/week140000064400020410000157000000352411002316231500141160ustar00baezhttp00004600000001Also available at http://math.ucr.edu/home/baez/week140.html
October 15, 1999
This Week's Finds in Mathematical Physics (Week 140)
John Baez
Let's start with something fun: biographies!
1) Norman Macrae, John von Neumann: The Scientific Genius Who Pioneered
the Modern Computer, Game Theory, Nuclear Deterrence and Much More,
American Mathematical Society, Providence, Rhode Island, 1999.
2) Steve Batterson, Stephen Smale: The Mathematician Who Broke the
Dimension Barrier, American Mathematical Society, Providence, Rhode
Island, 2000.
Von Neumann might be my candidate for the best mathematical physicist of
the 20th century. His work ranged from the ultrapure to the
ultraapplied. At one end: his work on axiomatic set theory. At the
other: designing and building some of the first computers to help design
the hydrogen bomb  which was so applied, it got him in trouble at the
Institute for Advanced Studies! But there's so much stuff in between:
the mathematical foundations of quantum mechanics (von Neumann algebras,
the StoneVon Neumann theorem and so on), ergodic theory, his work on
Hilbert's fifth problem, the Manhattan project, game theory, the theory
of selfreproducing cellular automata.... You may or may not like him,
but you can't help being awed. Hans Bethe, no dope himself, said of von
Neumann that "I always thought his brain indicated that he belonged to a
new species, an evolution beyond man". The mathematician Polya said
"Johnny was the only student I was ever afraid of." Definitely an
interesting guy.
While von Neumann is one of those titans that dominated the first
half of the 20th century, Smale is more typical of the latter half 
he protested the Vietnam war, and now he even has his own web page!
3) Stephen Smale's web page, http://www.math.berkeley.edu/~smale/
He won the Fields medal in 1966 for his work on differential topology.
Some of his work is what you might call "pure": figuring out how to
turn a sphere inside out without any crinkles, proving the Poincare
conjecture in dimensions 5 and above, stuff like that. But a lot of it
concerns dynamical systems: cooking up strange attactors using the
horseshoe map, proving that structural stability is not generic, and so
on  long before the recent hype about chaos theory began! More
recently he's also been working on economics, game theory, and the
relation of computational complexity to algebraic geometry.
Now for some papers on spin networks and spin foams:
4) Roberto De Pietri, Laurent Freidel, Kirill Krasnov, and Carlo Rovelli,
BarrettCrane model from a BoulatovOoguri field theory over a
homogeneous space, preprint available as hepth/9907154.
The BarrettCrane model is a very interesting theory of quantum
gravity. I've described it already in "week113", "week120" and
"week128", so I won't go into much detail about it  I'll just
plunge in....
The original BarrettCrane model involved a fixed triangulation of
spacetime. One can also cook up versions where you sum over
triangulations. In some ways the most natural is to sum over all ways
of taking a bunch of 4simplices and gluing them facetoface until no
faces are left free. Some of these ways give you manifolds; others
don't. In this paper, the authors show that this "sum over
triangulations" version of the BarrettCrane model can be thought of
as a quantum field theory on a product of 4 copies of the 3sphere.
Weird, eh?
But it's actually not so weird. The space of complex functions on the
(n1)sphere is naturally a representation of SO(n). But there's
another way to think of this representation. Consider an triangle in
R^n. We can associate vectors to two of its edges, say v and w, and
form the wedge product of these vectors to get a bivector v ^ w. This
bivector describes the area element associated to the triangle. If we
pick an orientation for the triangle, this bivector is uniquely
determined. Now, a bivector of the form v ^ w is called "simple". The
space of simple bivectors is naturally a Poisson manifold  i.e., we can
define Poisson brackets of functions on this space  so we can think of
it as a "classical phase space". Using geometric quantization, we can
quantize this classical phase space and get a Hilbert space. Since
rotations act on the classical phase space, they act on this Hilbert
space, so it becomes a representation of SO(n). And this representation
is isomorphic to the space of complex functions on the (n1)sphere!
Thus, we can think of a complex function on the (n1)sphere as a
"quantum triangle" in R^n, as long as we really just care about the
area element associated to the triangle. One can develop this analogy
in detail and make it really precise. In particular, one can describe a
"quantum tetrahedron" in R^n as a collection of 4 quantum triangles
satisfying some constraints that say the fit together into a
tetrahedron. These quantum tetrahedra act almost like ordinary
tetrahedra when they are large, but when the areas of their faces
becomes small compared to the square of the Planck length, the
uncertainty principle becomes important: you can't simultaneously know
everything about their geometry with perfect precision.
Let me digress for a minute and sketch the history of this stuff. The
quantum tetrahedron in 3 dimensions was invented by Barbieri  see
"week110". It quickly became important in the study of spin foam
models. Then Barrett and I systematically worked out how to construct
the quantum tetrahedron in 3 and 4 dimensions using geometric
quantization  see "week134". Subsequently, Freidel and Krasnov figured
out how to generalize this stuff to higher dimensions:
5) Laurent Freidel, Kirill Krasnov and Raymond Puzio, BF description of
higherdimensional gravity, preprint available as hepth/9901069.
6) Laurent Freidel and Kirill Krasnov, Simple spin networks as Feynman
graphs, preprint available as hepth/9903192.
So much for history  back to business. So far I've told you that
the state of a "quantum triangle" in 4 dimensions is given by a complex
function on the 3sphere. And I've told you that a "quantum
tetrahedron" is a collection of 4 quantum triangles satisfying some
constraints. More precisely, let H = L^2(S^3) be the Hilbert space for
a quantum triangle in 4 dimensions. Then the Hilbert space for a
quantum tetrahedron is a certain subspace T of H x H x H x H, where "x"
denotes the tensor product of Hilbert spaces. Concretely, we can think
of states in T as complex functions on the product of 4 copies of S^3.
These complex functions need to satisfy some constraints, but let's not
worry about that here....
Now let's "second quantize" the Hilbert space T. This is physics jargon
for making a Hilbert space out of the algebra of polynomials on T 
usually called the "Fock space" on T. As usual, there are two pictures
of states in this Fock space: the "field" picture and the "particle"
picture. On the one hand, they are states of a quantum field theory on
the product of 4 copies of S^3. But on the other hand, they are states
of an arbitrary collection of quantum tetrahedra in 4 dimensions. In
other words, we've got ourselves a quantum field theory whose
"elementary particles" are quantum tetrahedra!
The idea of the de PietriFreidelKrasnovRovelli paper is to play these
two pictures off each other and develop a new way of thinking about the
BarrettCrane model. The main thing these guys do is write down a
Lagrangian with some nice properties. Throughout quantum field theory,
one of the big ideas is to start with a Lagrangian and use it to compute
the amplitudes of Feynman diagrams. A Feynman diagram is a graph with
edges corresponding to "particles" and vertices corresponding to
"interactions" where a bunch of particles turns into another bunch of
particles.
But in the present context, the socalled "particles" are really quantum
tetrahedra! Thus the trick is to write down a Lagrangian giving Feynman
diagrams with 5valent vertices. If you do it right, these 5valent
vertices correspond exactly to ways that 5 quantum tetrahedra can fit
together as the 5 faces of a 4simplex! Let's call such a thing a
"quantum 4simplex". Then your Feynman diagrams correspond exactly to
ways of gluing together a bunch of quantum 4simplices facetoface.
Better yet, if you set things up right, the amplitude for such a Feynman
diagram exactly matches the amplitude that you'd compute for a
triangulated manifold using the BarrettCrane model!
In short, what we've got here is a quantum field theory whose Feynman
diagrams describe "quantum geometries of spacetime"  where spacetime
is not just a fixed triangulated manifold, but any possible way of
gluing together a bunch of 4simplices facetoface.
Sounds great, eh? So what are the problems? I'll just list three.
First, we don't know that the "sum over Feynman diagrams" converges in
this theory. In fact, it probably does not  but there are probably
ways to deal with this. Second, the model is Riemannian rather than
Lorentzian: we are using the rotation group when we should be using the
Lorentz group. Luckily this is addressed in a new paper by Barrett and
Crane. Third, we aren't very good at computing things with this sort of
model  short of massive computer simulations, it's tough to see what it
actually says about physics. In my opinion this is the most serious
problem: we should either start doing computer simulations of spin foam
models, or develop new analytical techniques for handling them  or
both!
Now, this "new paper by Barrett and Crane" is actually not brand new.
It's a revised version of something they'd already put on the net:
7) John Barrett and Louis Crane, A Lorentzian signature model for
quantum general relativity, preprint available as grqc/9904025.
However, it's much improved. When I went up to Nottingham at the end of
the summer, I had Barrett explain it to me. I learned all sorts of cool
stuff about representations of the Lorentz group. Unfortunately, I
don't now have the energy to explain all that stuff. I'll just say
this: everything I said above generalizes to the Lorentzian case. The
main difference is that we use the 3dimensional hyperboloid
H^3 = {t^2  x^2  y^2  z^2 = 1}
wherever we'd been using the 3sphere
S^3 = {t^2 + x^2 + y^2 + z^2 = 1}
It's sort of obvious in retrospect, but it's nice that it works out so
neatly!
Okay, here are some more papers on spin networks and spin foams.
Since I'm getting lazy, I'll just quote the abstracts:
8) Sameer Gupta, Causality in spin foam models, preprint available as
grqc/9908018.
We compute Teitelboim's causal propagator in the context of canonical
loop quantum gravity. For the Lorentzian signature, we find that the
resultant power series can be expressed as a sum over branched, colored
twosurfaces with an intrinsic causal structure. This leads us to
define a general structure which we call a "causal spin foam". We also
demonstrate that the causal evolution models for spin networks fall in
the general class of causal spin foams.
9) Matthias Arnsdorf and Sameer Gupta, Loop quantum gravity on
noncompact spaces, preprint available as grqc/9909053.
We present a general procedure for constructing new Hilbert spaces for
loop quantum gravity on noncompact spatial manifolds. Given any fixed
background state representing a noncompact spatial geometry, we use the
Gel'fandNaimarkSegal construction to obtain a representation of the
algebra of observables. The resulting Hilbert space can be interpreted
as describing fluctuation of compact support around this background
state. We also give an example of a state which approximates classical
flat space and can be used as a background state for our construction.
10) Seth A. Major, Quasilocal energy for spinnet gravity, preprint
available as grqc/9906052.
The Hamiltonian of a gravitational system defined in a region with
boundary is quantized. The classical Hamiltonian, and starting point for
the regularization, is required by functional differentiablity of the
Hamiltonian constraint. The boundary term is the quasilocal energy of
the system and becomes the ADM mass in asymptopia. The quantization is
carried out within the framework of canonical quantization using spin
networks. The result is a gauge invariant, welldefined operator on the
Hilbert space induced from the state space on the whole spatial
manifold. The spectrum is computed. An alternate form of the operator,
with the correct naive classical limit, but requiring a restriction on
the Hilbert space, is also defined. Comparison with earlier work and
several consequences are briefly explored.
11) C. Di Bartolo, R. Gambini, J. Griego, J. Pullin, Consistent
canonical quantization of general relativity in the space of
Vassiliev knot invariants, preprint available as grqc/9909063.
We present a quantization of the Hamiltonian and diffeomorphism
constraint of canonical quantum gravity in the spin network
representation. The novelty consists in considering a space of
wavefunctions based on the Vassiliev knot invariants. The constraints
are finite, well defined, and reproduce at the level of quantum
commutators the Poisson algebra of constraints of the classical theory.
A similar construction can be carried out in 2+1 dimensions leading to
the correct quantum theory.
12) John Baez, Spin foam perturbation theory, preprint available as
grqc/9910050 or at http://math.ucr.edu/home/baez/foam3.ps
We study perturbation theory for spin foam models on triangulated
manifolds. Starting with any model of this sort, we consider an
arbitrary perturbation of the vertex amplitudes, and write the
evolution operators of the perturbed model as convergent power series in
the coupling constant governing the perturbation. The terms in the
power series can be efficiently computed when the unperturbed model is a
topological quantum field theory. Moreover, in this case we can
explicitly sum the whole power series in the limit where the number of
topdimensional simplices goes to infinity while the coupling constant
is suitably renormalized. This `dilute gas limit' gives spin foam
models that are triangulationindependent but not topological quantum
field theories. However, we show that models of this sort are rather
trivial except in dimension 2.

Previous issues of "This Week's Finds" and other expository articles on
mathematics and physics, as well as some of my research papers, can be
obtained at
http://math.ucr.edu/home/baez/
For a table of contents of all the issues of This Week's Finds, try
http://math.ucr.edu/home/baez/twf.html
A simple jumpingoff point to the old issues is available at
http://math.ucr.edu/home/baez/twfshort.html
If you just want the latest issue, go to
http://math.ucr.edu/home/baez/this.week.html
one can describe a
"quantum tetrahedron" in R^n as a collection of 4 quantum triangles
satisfying some constraints that say the fit together into a
tetrahedron. These quantum tetrahedra act almost like ordinary
tetrahedra when they are large, but when the areas of their faces
becomes small compared to the square of the Planck length, the
uncertaintwf_ascii/week141000064400020410000157000000623750774011336500141440ustar00baezhttp00004600000001Keywords: Also available at http://math.ucr.edu/home/baez/week141.html
October 26, 1999
This Week's Finds in Mathematical Physics (Week 141)
John Baez
How can you resist a book with a title like "Inconsistent Mathematics"?
1) Chris Mortensen, Inconsistent Mathematics, Kluwer, Dordrecht, 1995.
Ever since Goedel showed that all sufficiently strong systems formulated
using the predicate calculus must either be inconsistent or incomplete,
most people have chosen what they perceive as the lesser of two evils:
accepting incompleteness to save mathematics from inconsistency. But
what about the other option?
This book begins with the startling sentence: "The following idea has
recently been gaining support: that the world is or might be inconsistent."
As we reel in shock, Mortensen continues:
Let us consider set theory first. The most natural set theory to
adopt is undoubtedly one which has unrestricted set abstraction
(also known as naive comprehension). This is the natural principle
which declares that to every property there is a unique set of
things having the property. But, as Russell showed, this leads
rapidly to the contradiction that the the Russell set [the set of
all sets that do not contain themselves as a member] both is and is
not a member of itself. The overwhelming majority of logicians
took the view that this contradiction required a weakening of
unrestricted abstraction in order to ensure a consistent set
theory, which was in turn seen as necessary to provide a consistent
foundation for mathematics. But all ensuing attempts at weakening
set abstraction proved to be in various ways ad hoc. Da Costa and
Routley both suggested instead that the Russell set might be dealt
with more naturally in an inconsistent but nontrivial set theory
(where triviality means that every sentence is provable).
An inconsistent but nontrivial logical system is called *paraconsistent*.
But it's not so easy to create such systems. To keep an inconsistency
from infecting the whole system and making it trivial, we need to drop
the rule of classical logic which says that "A and not(A) implies B"
for all propositions A and B. Unfortunately, this rule is built into the
propositional calculus from the very start!
So, we need to revise the propositional calculus.
One way to do it is to abandon "material implication"  the form of
implication where you can calculate the truth value of "P implies Q"
from those of P and Q using the following truth table:
P  Q  P implies Q

T  T  T
T  F  F
F  T  T
F  F  T
With material implication, a false statement implies *every* statement,
so any inconsistency is fatal. But in real life, if we discover we have
one inconsistent belief, we don't conclude we can fly and go jump off a
building! Material implication is really just our best attempt to define
implication using truth tables with 2 truth values: true and false. So
it's not surprising that logicians have investigated other forms of
implication.
One obvious approach is to use more truth values, like "true", "false",
and "don't know". There's a long history of work on such multivalued
logics.
Another approach, initiated by Anderson and Belnap, is called "relevance
logic". In relevance logic, "P implies Q" can only be true if there
is a conceptual connection between P and Q. So if B has nothing to do
with A, we don't get "A and not(A) implies B".
This book describes a logical system called "RQ"  relevance logic with
quantifiers. It also describes a system called "R#", which is a version
of the Peano axioms of arithmetic based on RQ instead of the usual
predicate calculus. Following the work of Robert Meyer, it proves
that R# is nontrivial in the sense described above. Moreover, this
proof can be carried out R# itself! However, you can carry out the
proof of Goedel's 2nd incompleteness theorem in R#, so R# cannot prove
itself consistent.
To paraphrase Mortensen: "But this is not really a puzzle. The
explanation is that relevant and other paraconsistent logics turn on
making a distinction between inconsistency and triviality, the former
being weaker than the latter; whereas classical logical cannot make this
distinction. For what the present author's intuitions are worth, these
do seem to be different concpets. Thus for R#, consistency cannot be
proved by finitistic means by Goedel's second theorem, whereas
nontriviality can be shown. Since Peano arithmetic collapses this
distinction, both kinds of consistency are infected by the same
unprovability."
Mortensen also mentions another approach to get rid of "A and not(A)
implies B" without getting rid of material implication. This is to get
rid of the rule that "A and not(A)" is false! He calls this "Brazilian
logic". Presumably this is not because your average Brazilian thinks
this way, but because the inventor of this approach, Da Costa, is Brazilian.
Brazilian logic sounds very bizarre at first, but in fact it's just the
dual of intuitionistic logic, where you drop the rule that "A or not(A)"
is true. Intuitionistic logic is nicely modeled by open sets in a
topological space: "and" is intersection, "or" is union, and "not" is
the interior of the complement. Similarly, Brazilian logic is modeled
by closed sets. In intuitionistic logic we allow a slight gap between A
and not(A); in Brazilian logic we allow a slight overlap.
In short, this book is full of fascinating stuff. Lots of passages are
downright amusing at first, like this:
[...] there have been calls recently for inconsistent calculus,
appealing to the history of calculus in which inconsistent claims
abound, especially about infinitesimals (Newton, Leibniz,
Bernoulli, l'Hospital, even Cauchy). However, inconsistent
calculus has resisted development.
But you always have to remember that the author is interested in
theories which, though inconsistent, are still paraconsistent. And I
think he really makes a good case for his claim that inconsistent
mathematics is worth studying  even if our ultimate goal is to *avoid*
inconsistency!
Okay, now let me switch gears drastically and say a bit about "exotic
spheres"  smooth manifolds that are homeomorphic but not diffeomorphic
to the nsphere with its usual smooth structure. People on
sci.physics.research have been talking about this stuff lately, so it
seems like a good time for a miniessay on the subject. Also, my
colleague Fred Wilhelm works on the geometry of exotic spheres, and he
just gave a talk on it here at U. C. Riverside, so I should pass along
some of his wisdom while I still remember it.
First, recall the "Hopf bundle". It's easy to describe starting with
the complex numbers. The unit vectors in C^2 form the sphere S^3. The
unit complex numbers form a group under multiplication. As a manifold
this is just the circle S^1, but as a group it's better known as U(1).
You can multiply a unit vector by a unit complex number and get a new
unit vector, so S^1 acts on S^3. The quotient space is the complex
projective space CP^1, which is just the sphere S^2. So what we've got
here is fiber bundle:
S^1 > S^3 > S^2 = CP^1
with fiber S^1, total space S^3 and base space S^2. This is the Hopf
bundle. It's famous because the map from the total space to the base
was the first example of a topologically nontrivial map from a sphere to
a sphere of lower dimension. In the lingo of homotopy theory, we say
it's the generator of the group pi_3(S^2).
Now in "week106" I talked about how we can mimic this construction by
replacing the complex numbers with any other division algebra. If we
use the real numbers we get a fiber bundle
S^0 > S^1 > RP^1 = S^1
where S^0 is the group of unit real numbers, better known as Z/2. This
bundle looks like the edge of a Moebius strip. If we use the quaternions
we get a more interesting fiber bundle:
S^3 > S^7 > HP^1 = S^4
where S^3 is the group of unit quaternions, better known as SU(2). We
can even do something like this with the octonions, and we get a fiber
bundle
S^7 > S^{15} > OP^1 = S^8
but now S^7, the unit octonions, doesn't form a group  because the
octonions aren't associative.
Anyway, it's the quaternionic version of the Hopf bundle that serves as
the inspiration for Milnor's construction of exotic 7spheres. These
exotic 7spheres are actually total spaces of *other* bundles with fiber
S^3 and base space S^4. The easiest way to get your hands on these
bundles is to take S^4, chop it in half along the equator, put a trivial
S^3bundle over each hemisphere, and then glue these together. To glue
these bundles together we need a way to attach the fibers over each
point x of the equator. In other words, for each point x in the equator
of S^4 we need a map
f_x: S^3 > S^3
which should vary smoothly with x. But the equator of S^4 is just S^3, and
S^3 is a group  the unit quaternions  so we can take
f_x(y) = x^n y x^m
for any pair of integers (n,m).
This gives us a bunch of S^3bundles over S^4. The total space X(n,m)
of any one of these bundles is obviously a smooth 7dimensional manifold.
But when is it homeomorphic to the 7sphere? And when is it *diffeomorphic*
to the 7sphere with its usual smooth structure?
Well, first we use some Morse theory. You can learn a lot about the
topology of a smooth manifold if you have a "Morse function" on the
manifold: a smooth realvalued function all of whose critical points
are nondegenerate. If you don't believe me, read this book:
2) John Milnor, Morse Theory, Princeton U. Press, Princeton, 1960.
When n + m = 1 there's a Morse function on X(n,m) with only two critical
points  a maximum and a minimum. This implies that X(n,m) is
homeomorphic to a sphere!
Once we know that X(n,m) is homeomorphic to S^7, we have to decide
when it's diffeomorphic to S^7 with its usual smooth structure.
This is the hard part. Notice that X(n,m) is the unit sphere bundle of
a vector bundle over S^4 whose fiber is the quaternions. We can
understand a bunch about X(n,m) using the characteristic classes
of this vector bundle. In particular, we can compute the Euler
number and the Pontrjagin number of this vector bundle. Using the
Euler number we can show that X(n,m) is homeomorphic to a sphere
*only* if n + m = 1  you can't really do this using Morse theory.
But more importantly, using the Pontrjagin number, we can show that
in this case X(n,m) is diffeomorphic to S^7 with its usual smooth
structure if and only if (n  m)^2 = 1 mod 7. Otherwise it's "exotic".
For the details of the above argument you can try the following book:
3) B. A. Dubrovin, A. T. Fomenko and S. P. Novikov, Modern Geometry 
Methods and Applications, Part III: Introduction to Homology Theory,
SpringerVerlag Graduate Texts, number 125, Springer, New York, 1990.
or the original paper:
4) John Milnor, On manifolds homeomorphic to the 7sphere, Ann.
Math 64 (1956), 399405.
Now, with quite a bit more work, you can show that smooth structures on
the nsphere form an group under connected sum  the operation of chopping
out a small hole in two spheres and gluing them together  and you can
show that this group is Z/28 for n = 7. This means that if we consider
two smooth structures on the 7sphere the same when they're related by
an *orientationpreserving* diffeomorphism, we get exactly 28 kinds.
Unfortunately we don't get all of them by the above explicit construction.
For more details, see:
5) M. Kervaire and J. Milnor, Groups of homotopy spheres I, Ann. Math.
77 (1963), 504537.
By the way, part II of the above paper doesn't exist! Instead, you
should read this:
6) J. Levine, Lectures on groups of homotopy spheres, in Algebraic and
Geometric Topology, Springer Lecture Notes in Mathematics number
1126, Springer, Berlin, 1985, pp. 6295.
Anyway, if you're wondering why I'm talking so much about exotic 7spheres,
instead of lowerdimensional examples that are easier to visualize, check
out this table of groups of smooth structures on the nsphere:
n group of smooth structures on the nsphere
0 1
1 1
2 1
3 1
4 ?
5 1
6 1
7 Z/28
8 Z/2
9 Z/2 x Z/2 x Z/2
10 Z/6
11 Z/992
12 1
13 Z/3
14 Z/2
15 Z/8128 x Z/2
16 Z/2
17 Z/2 x Z/2 x Z/2 x Z/2
18 Z/8 x Z/2
Dimension 7 is the simplest interesting case  except perhaps for
dimension 4, where the answer is unknown! The "smooth Poincare
conjecture" says that there's only one smooth structure on the
4sphere, but this remains a conjecture....
As you can see, there are lots of exotic 11spheres. In fact, this is
relevant to string theory! Yo