Quantum Gravity Seminar - Winter 2007
John Baez and Derek Wise
This quarter's seminar will be a continuation
of the Fall 2006 seminar. So,
we will again discuss two subjects:
As usual, John Baez lectured
and Derek Wise
took beautiful notes which you can see 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
n-Category
Café.
This is a new experiment, and I hope you try it.
Each week's notes comes with a blog
entry where you can ask questions, make comments, and chat with
other people following the course.
The week numbers for these notes continue where
last
quarter's notes leave off. The course continues in the
Spring.
There are also 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 last
Fall's lectures, we discussed the Lagrangian and Hamiltonian
approaches to the classical mechanics of point particles, and sketched
how these could be generalized to strings and higher-dimensional
membranes by a process that we'll ultimately see as
categorification. You can read what we did here:
-
John Baez and Apoorva Khare, with figures by Christine Dantas,
Course Notes on Quantization and Cohomology, Fall 2006,
in
PDF and
Postscript.
This quarter we're digging deeper into the process of quantization!
Here are this quarter's notes:
-
John Baez and Apoorva Khare, with figures by Christine Dantas,
Course Notes on Quantization and Cohomology, Winter 2007,
in
PDF and
Postscript.
You can also see Derek's hand-written notes, week by week.
Each week's notes come with a blog entry where you can read
discussions and ask your own questions. There's also some
supplementary reading material:
-
Week 10 (Jan. 16) -
A quick review of
classical versus quantum mechanics, in both the Lagrangian and Hamiltonian
approaches. What are path integrals, really?
How do we quantize a classical Hamiltonian to obtain a quantum one?
Blog entry.
-
Week 11 (Jan. 23) - Action as a functor
from a category of "configurations" and "paths" to the real numbers
(viewed as a one-object category). Three things physicists do with
this functor: find its critical points, find its minima, and integrate
its exponential. The analogy between the (classical) principal of
least action and the (quantum) principal of path integration. The
underlying analogy between the real numbers equipped the operations
min and +, and the complex numbers with operations + and ×. Blog
entry.
-
Week 12 (Jan. 30) - Classical, quantum
and statistical mechanics as "matrix mechanics". In
quantum mechanics we use linear algebra over the ring of complex
numbers; in classical mechanics everything is formally the same, but
we instead use the rig Rmin = R ∪ {+∞} where
the addition is min and the multiplication is +.
As a warmup for bringing statistical mechanics into the picture -
and linear algebra over yet another rig - we recall how the dynamics
of particles becomes the statics of strings after Wick rotation.
Blog
entry.
-
Week 13 (Feb. 6) - Statistical mechanics
and deformation of rigs. Statistical mechanics (or better, "thermal
statics") as matrix mechanics
over a rig RT that depends on the temperature T.
As T → 0, the rig RT reduces to Rmin and
thermal statics reduces to classical statics, just as
quantum dynamics reduces to classical dynamics as Planck's constant
approaches zero.
Tropical mathematics, idempotent analysis and Maslov dequantization.
Blog
entry.
-
Week 14 (Feb. 13) - An example of
path-integral quantization: the free particle on a line (part 1).
Blog
entry.
-
Week 15 (Feb. 20) -
The free particle on a line (part 2).
Showing the path-integral approach agrees with the Hamiltonian
approach. Fourier transforms and Gaussian integrals.
Blog
entry.
-
Week 16 (Feb. 27) -
More examples of path-integral quantization. The particle
in a potential on the real line. The Lie-Trotter Theorem.
The particle in a potential on a complete Riemannian manifold.
Back to general questions: how do we get a Hilbert space from
a category equipped with an action functor? The problem of
Cauchy surfaces.
Blog
entry.
-
Week 17 (Mar. 6) -
Hilbert spaces and operator algebras from categories.
Under what conditions can we
obtain a Hilbert space from a category C equipped with an
"action" functor S: C → R? The importance
of time reversal: groupoids versus ∗-categories (also known
as †-categories). The 'category algebra' of C.
Blog
entry.
-
Week 18 (Mar. 13) -
The big picture. Building a Hilbert space from
a finite category C equipped with an "amplitude" functor
A: C → U(1). Example: discretized version of the free particle
on the line. Generalizing from particles to strings by categorifying everything in sight.
Building a 2-Hilbert space from a finite 2-category C
equipped with a 2-functor A: C → U(1)Tor, where
U(1)Tor is the 2-group of U(1)-torsors.
Blog
entry.
Supplementary reading:
When you're done with all this stuff, go on to
the
Spring 2007
continutation of this seminar!
In this continuation of the
Fall 2006
seminar on classical versus quantum computation, we explained more
precisely the relation between cartesian closed categories and the
lambda calculus, and tried to categorify it.
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 lectures notes.
Each week's notes comes with a blog
entry where you can ask questions and make comments:
This seminar continues, but with a drastic shift towards
homotopy theory, in Spring 2007.
© 2007 John Baez and Derek Wise
baez@math.removethis.ucr.andthis.edu