QG Seminar: Winter 2004

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Quantum Gravity Seminar - Winter 2004

Quantization and Categorification

John Baez and Derek Wise

Here are some lecture notes on quantization and categorification. They are a continuation of the Fall 2003 notes. As before, these notes have been written by Derek Wise, based on lectures by John Baez.

In the fall session we talked about how to quantize and then categorify the simplest physical system of all: the harmonic oscillator. Lots of people know how to quantize the harmonic oscillator, but fewer know that if you then categorify it you recover Joyal's theory of "structure types", which is important in combinatorics. This gives a purely combinatorial explanation of the discreteness arising in the quantum harmonic oscillator!

In this quarter we'll generalize this discussion to include quantum field theory, delve deeper into the category theory behind structure types, and introduce "stuff types" - which appear when you categorify the theory of Feynman diagrams.

Besides the lecture notes you can also get homework problems, which contain mini-essays on related topics. You can also get answers to some of these problems. Different answers shed different light on the problems, and some include cool extra stuff!

All of these are PDF files:

Week 1 - Review of structure types. This quarter's plan. Fibonacci numbers.
- Tribonacci Numbers - Homework for week 1.
- Answers to Tribonacci number questions by Miguel Carrión Álvarez.
- Answers to Tribonacci number questions by Erin Pearse.
- Answers to Tribonacci number questions by Derek Wise.
- Partition Functions, Partition Numbers and Making Change - an extra homework for the highly energetic student.
Week 2 - The quantum harmonic oscillator with many degrees of freedom: counting states of a given energy.
- Counting Derangements - Homework for week 2.
- Answers to derangement questions by Miguel Carrión Álvarez.
- Answers to derangement questions by Jeffrey Morton.
- Answers to derangement questions by Derek Wise.
- Answers to derangement questions by Erin Pearse, featuring a purely combinatorial proof of the recurrence satisfied by subfactorials!
Week 3 - The quantum harmonic oscillator with many degrees of freedom: a square of functors commuting up to natural isomorphism. (Only one lecture this week, since Danny Stevenson spoke about gerbes on Thursday.)
- Counting Partitions - Homework for week 3.
- Answers to partition questions by Derek Wise, featuring a bijective proof of Euler's theorem on partitions into odd parts versus partitions into distinct parts - and some cool extra identities!
- Answers to partition questions by Jeffrey Morton, featuring yet another proof of Euler's theorem.
- Answers to partition questions by Miguel Carrión Álvarez.
- Answers to partition questions by Erin Pearse.
Week 4 - Violin string theory: quantizing the open string with Dirichlet boundary conditions. Partition numbers and the partition function of the quantum string.
- k-Colorings as Categorified Coherent States - Homework for week 4.
- Answers to k-coloring questions by Miguel Carrión Álvarez.
- Answers to k-coloring questions by Erin Pearse.
- Answers to k-coloring questions by Jeffrey Morton.
- Answers to k-coloring questions by Derek Wise.
Week 5 - Two roles of generating functions in quantum theory: as states of the harmonic oscillator, and as partition functions. Partition functions in classical and quantum statistical mechanics.
- Euler's Proof that 1 + 2 + 3 + ... = -1/12 - Homework for week 5.
- Answers to Euler's proof questions by Erin Pearse.
- Answers to Euler's proof questions by Jeffrey Morton.
- Answers to Euler's proof questions by Miguel Carrión Álvarez.
- Answers to Euler's proof questions by Derek Wise.
Week 6 - Generating functions from partition functions: the analogy between thermodynamic systems and structure types. Groupoids as "sets with fractional cardinality". How to evaluate a structure type at a set and get a groupoid - a categorified version of evaluating a power series at a natural number and getting a real number.
- Bernoulli Numbers - Homework for week 6.
- Answers to Bernoulli number questions by Jeffrey Morton, featuring a careful study of the difficulties of categorifying the Bernoulli numbers!
- Answers to Bernoulli number questions by Miguel Carrión Álvarez.
- Answers to Bernoulli number questions by Erin Pearse.
- Answers to Bernoulli number questions by Derek Wise.
Week 7 - Applying the structure type F to the set Z and getting the groupoid F(Z) of "F-structured, Z-colored sets": examples. Categorified hyperbolic trigonometry. Half-colored sets, double factorials, hypercubes and the square root of e.
- The Riemann Zeta Function - Homework for week 7.
- Answers to Riemann zeta function questions by Miguel Carrión Álvarez, featuring two approaches to estimating the growth rate of the Bernoulli numbers!
- Answers to Riemann zeta function questions by Jeffrey Morton, featuring an analysis of why it's hard to categorify the function z/(1 - e^z).
- Answers to Riemann zeta function questions by Erin Pearse.
- Answers to Riemann zeta function questions by Derek Wise.
Week 8 - Applying a structure type F to a groupoid Z and getting a groupoid F(Z) - a categorified version of evaluating a power series at a real number and getting a real number. The necessary technology: taking the weak quotient of a groupoid by a group acting on it. How to define the weak quotient by a universal property.
Week 9 - How to construct the weak quotient. F(Z) as the groupoid of "F-structured finite sets with elements labelled by objects of the groupoid Z". Examples: the groupoid of "finite sets labelled by k-element sets", which has cardinality e^1/k!. The groupoid of "finite sets with elements labelled by finite sets", which has cardinality e^e.
- pp. 87 & 88 in color - much better pedagogically, but adding these color pages makes the files huge, so we have included them as a separate file.
Week 10 - Composing structuring types - a categorified version of composing functions given by power series. Examples: the structure type "being a finite set with elements labelled by 2-element sets", which has generating function e^z²/2!. Categorified hyperbolic trigonometry revisited. Exponentiation and connected structures. The need for stuff types.

Mike Stay has kindly created an index to these notes:

Quantization and Categorification: Index

For the continuation of this course, check out the Spring 2004 notes! You can also find links to more references there.

If you discover any errors in the course notes please email me, and we'll try to correct them. We're keeping a list of errors that haven't been fixed yet.

Here's a terse outline of the whole course:

Categorifying Quantum Mechanics

Yet another view of the same material can be found in this paper:

John Baez and James Dolan, From finite sets to Feynman diagrams, in Mathematics Unlimited - 2001 and Beyond, vol. 1, edited by Björn Engquist and Wilfried Schmid, Springer-Verlag, Berlin, 2001, pp. 29-50.

If you're interested in structure types and generating functions, you'll also enjoy these books:

Herbert Wilf, Generatingfunctionology, Academic Press, Boston, 1994. Available for free at http://www.cis.upenn.edu/~wilf/.
F. Bergeron, G. Labelle, and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, Cambridge U. Press, 1998.

You can also download TeX or LaTeX files of the homework problems and some solutions, if for some bizarre reason you want them. However, the authors keep all rights to this work, except when stated otherwise.