Quantum Gravity Seminar  Fall 2006
John Baez and Derek Wise
This fall the Quantum Gravity
Seminar will cover two subjects:
As usual, John Baez gave lectures and
Derek Wise took
beautiful notes which you can read here.
If you discover any errors in these notes,
please email me, and we'll try to correct them.
We'll keep a list of errors that
haven't been fixed yet.
These courses are also available via my blog at the
nCategory
Café, so you can follow along and ask questions there!
This is a new experiment, and I hope you try it.
Both these courses are continuing in the Winter
of 2007.
There are some LaTeX, encapsulated
PostScript and xfig files to download
if for some bizarre reason you want them.
However, we reserve all rights to this work.
In these lectures I spoke about:

the Lagrangian approach to classical mechanics,

the pathintegral approach to quantum mechanics,

symplectic geometry,

geometric quantization,

how going from point particles to strings makes us "categorify" all the above.
My colleague
Apoorva Khare produced
notes in LaTeX, and
Christine Dantas
drew pictures to go along with them:

John Baez and Apoorva Khare, with figures by Christine Dantas,
Course Notes on Quantization and Cohomology,
Fall 2006,
in PDF and
Postscript.
You can also see Derek's handwritten notes, week by week.
Each week's notes come with a blog
entry where you can ask questions and make comments.
There are also homework problems, and answers:

Week 1 (Oct. 3) 
How the dynamics of pbranes resembles the statics of (p+1)branes.
Blog entry.

Week 2 (Oct. 10) 
The Lagrangian approach to classical mechanics.
Action as the integral of a 1form (prelude).
Blog entry.

Week 3 (Oct. 17) 
From Lagrangian to Hamiltonian dynamics. Momentum as a cotangent vector.
The Legendre transform. The Hamiltonian. Hamilton's equations.
Blog entry.

Week 4 (Oct. 24) 
Hamiltonian dynamics and symplectic geometry. Hamiltonian
vector fields. Getting Hamiltonian vector field from a symplectic structure.
The canonical 1form on a cotangent bundle, and how this gives
a symplectic structure.
Blog entry.

Week 5 (Oct. 31)  The canonical 1form
α on T^{*}X. Symplectic structures. Why a symplectic
structure should be a nondegenerate 2form (so we get time evolution
from a Hamiltonian) and closed (so time evolution preserves this 2form).
The action expressed in terms of the canonical 1form.
Blog entry.

Homework: show that if α is the canonical 1form on the
cotangent bundle of a manifold, then
ω = dα
is a nondegenerate 2form.

Answers by Alex Hoffnung.

Week 6 (Nov. 7) 
The canonical 1form. The symplectic structure and the action of a loop
in phase space. Extended phase space: the cotangent bundle of
(configuration space) × time. The action as an integral of the
canonical 1form over a path in the extended phase space. Rovelli's
covariant formulation of classical mechanics, as a warmup for generalizing
classical mechanics from particles to strings.
Blog entry.

Week 7 (Nov. 14)  From particles
to strings and membranes. Generalizing everything we've done
so far from particles (p = 1) to strings (p = 2) and membranes
that trace out pdimensional surfaces in spacetime (p ≥ 0).
The concept of "pvelocity". The canonical pform on
the extended phase space Λ^{p} T*M, where M is spacetime.
Blog entry.

Week 8 (Nov. 28)  From particles to membranes,
continued. A coordinatefree
definition of pvelocity. The action for a charged
point particle in general relativity, versus the action for a charged
membrane. The electromagnetic field versus its pform generalization.
Blog entry.

Week 9 (Dec. 5) 
A glimpse of what's to come. Geometric quantization:
finding a connection on a U(1) bundle whose curvature is the
symplectic 2form ω on phase space. Why doing this
is only possible if ω defines an integral cohomology class 
hence the term "quantization".
Blog entry.
When you're done with these, try the
continuation of this seminar in
Winter 2007.
Also try these related materials:
In these lectures I spoke about:

the lambda calculus and its role in classical computation,

how quantum computation differs from classical computation,

the quantum lambda calculus and its role in quantum computation,

cartesian closed categories and symmetric monoidal closed categories,

how treating computation as a process makes us "categorify" all the above.
Some — but not all! — of these lectures are summarized
and further developed in this long paper:

John Baez and Mike Stay, Physics, Topology, Logic and Computation: a
Rosetta Stone.
Available in PDF and
Postscript
Here are the notes.
Each week's notes comes with a blog
entry where you can ask questions and make comments:

Week 0 (Sept. 28) 
Computation, the lambda calculus and cartesian closed
categories: an overview.
Blog entry.

Week 1 (Oct. 5) 
Types and operations. Categories as theories.
Monoidal categories versus categories with finite products  also
known as "cartesian categories".
Blog entry.

Week 2 (Oct. 12) 
Monoidal categories versus cartesian categories.
Closed monoidal categories.
Blog entry.

Week 3 (Oct. 19) 
Guest lecture by James Dolan: Holodeck strategies
and cartesian closed categories.
Blog entry.

Week 4 (Oct. 26) 
Currying and uncurrying, evaluation and coevaluation.
Basic aspects of the "quantum lambdacalculus": so far, the
fragment of the lambdacalculus that works in any closed monoidal
category. The "name" of a morphism. Compact categories.
Blog entry.

Week 5 (Nov. 2) 
Theorem: evaluating the "name" of a morphism gives that morphism!
The naturality of currying. A new "bubble"
notation for currying and uncurrying.
Popping bubbles to reveal the quantum world.
Blog entry.

Week 6 (Nov. 9) 
Classical versus quantum lambdacalculus. From lambdaterms to
string diagrams. Internalizing composition. The "untyped"
lambdacalculus. Church numerals and booleans. Blog entry.

Week 7 (Nov. 21) 
The untyped lambdacalculus, continued.
"Building a computer" inside the free cartesian closed
category on an object X with X = hom(X,X).
Operations on booleans. The "ifthenelse" construction.
Addition and multiplication of Church numerals. Defining functions
recursively: the astounding Fixed Point Theorem.
Blog entry.

Week 8 (Nov. 30) 
The Fixed Point Theorem and the diagonal argument (after Tom
Payne). Cantor's "negative" use of diagonalization to
prove that infinite bit strings cannot be enumerated, versus
Curry's "positive" use of diagonalization to get a fixed point for
any lambdaterm in the untyped lambda calculus.
Preview of coming attractions: quantum computation, and
categorifying everything so far to see computation as a process.
Blog entry.

F. William Lawvere, Diagonal arguments and cartesian closed categories, in Category Theory, Homology Theory, and their Applications II, eds. Dold and Eckmann, Springer Lecture Notes in Mathematics 92, 1969, pp. 134145.

Noson Yanofsky, A
universal approach to selfreferential paradoxes, incompleteness
and fixed points.
Here are some other things to read. It's good to start with these:
These go into a bit more depth:
For some basic category theory definitions, try this:
When you're done with these, try the
continuation of this seminar in
Winter 2007.
© 2006 John Baez and Derek Wise
baez@math.removethis.ucr.andthis.edu