Quantum Gravity Seminar - Fall 2006

John Baez and Derek Wise

• Quantization and Cohomology (Tuesdays)
• Classical versus Quantum Computation (Thursdays)

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.

Quantization and Cohomology

• 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 1 (Oct. 3) - How the dynamics of p-branes 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 1-form (prelude). Blog entry.
  • Homework: A Spring in Imaginary Time.
  • Answers to homework by Garett Leskowitz.
  • Answers by Jeffrey Morton.
  • Answers by Erin Pearse.
  • Answers by Alex Hoffnung.
  • Answers by Mike Stay.
• 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 1-form on a cotangent bundle, and how this gives a symplectic structure. Blog entry.
  Homework: show the symplectic structure ω = dq_i ∧ dpⁱ on the cotangent bundle gives ω(v_H, -) = dH, where the Hamiltonian vector field v_H is given by
  v_H = ∂H/∂p_i ∂/∂qⁱ - ∂H/∂qⁱ ∂/∂p_i .
  • Answers by Alex Hoffnung.
• 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.
  • 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 Λ^p 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:

• John Baez, Classical mechanics.
• Graeme Segal, Notes on symplectic manifolds and quantization, also available in PDF form.

Classical versus Quantum Computation

• 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

• Week 0 (Sept. 28) - Computation, the lambda calculus and cartesian closed categories: an overview. Blog entry.
  • Homework: exercises 1.1 and 1.2 in Selinger's Lectures notes on the lambda calculus.
  • Answers to homework by Mike Stay.
  • Answers by Toby Bartels.
  • Answers by Jeff Morton.
  • Answers by Garett Leskowitz.
  • Answers by Alex Hoffnung.
• Week 1 (Oct. 5) - Types and operations. Categories as theories. Monoidal categories versus categories with finite products - also known as "cartesian categories". Blog entry.
  • Homework: show that every category with finite products can be made into a monoidal category - see above notes for details.
  • Answers by J. Prasad Senesi.
  • Answers by Mike Stay.
  • Answers by Alex Hoffnung.

• 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.
  • James Dolan, Holodeck strategies and the lambda-calculus (a continuation of the above lecture).
  • Todd Trimble, Holodeck games and CCCs.
  • John Baez, "week240" of This Week's Finds in Mathematical Physics (a gentle introduction to cartesian closed categories and holodeck games).

• Week 4 (Oct. 26) - Currying and uncurrying, evaluation and coevaluation. Basic aspects of the "quantum lambda-calculus": so far, the fragment of the lambda-calculus 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 lambda-calculus. From lambda-terms to string diagrams. Internalizing composition. The "untyped" lambda-calculus. Church numerals and booleans. Blog entry.
• Week 7 (Nov. 21) - The untyped lambda-calculus, continued. "Building a computer" inside the free cartesian closed category on an object X with X = hom(X,X). Operations on booleans. The "if-then-else" 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 lambda-term 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. 134-145.
  • Noson Yanofsky, A universal approach to self-referential paradoxes, incompleteness and fixed points.

• Mark Chu-Carroll, Lambda calculus and category theory.
• Wikipedia, Lambda calculus.

• John Baez, Cartesian closed categories and the λ-calculus.
• Mark Chu-Carroll, Lambda calculus and category theory.
• Peter Selinger, Lecture notes on the lambda calculus.
• Phil Scott, Some aspects of categories in computer science.

• John Baez, Some definitions everyone should know.

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