Quantum Gravity Seminar - Fall 2006

John Baez and Derek Wise

These courses are also available via my blog at the n-Category 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.

the Lagrangian approach to classical mechanics,

the path-integral 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 hand-written 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 5 (Oct. 31) - The canonical 1-form α on T*X. Symplectic structures. Why a symplectic structure should be a nondegenerate 2-form (so we get time evolution from a Hamiltonian) and closed (so time evolution preserves this 2-form). The action expressed in terms of the canonical 1-form. Blog entry. Homework: show that if α is the canonical 1-form on the cotangent bundle of a manifold, then ω = -dα is a nondegenerate 2-form.

is a nondegenerate 2-form. Answers by Alex Hoffnung.

Week 6 (Nov. 7) - The canonical 1-form. 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 1-form 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. Supplementary reading: Carlo Rovelli, Covariant hamiltonian formalism for field theory: Hamilton-Jacobi equation on the space G.

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 p-dimensional surfaces in spacetime (p ≥ 0). The concept of "p-velocity". The canonical p-form on the extended phase space Λp T*M, where M is spacetime. Blog entry.

Week 8 (Nov. 28) - From particles to membranes, continued. A coordinate-free definition of p-velocity. The action for a charged point particle in general relativity, versus the action for a charged membrane. The electromagnetic field versus its p-form 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 2-form 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:

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.

John Baez and Mike Stay, Physics, Topology, Logic and Computation: a Rosetta Stone. Available in PDF and Postscript