## Quantum Gravity Seminar - Spring 2004

### Quantization and Categorification

#### John Baez

Here are some lecture notes on quantization and categorification. They are a continuation of the Fall 2003 and Winter 2004 notes. As before, these notes were written by Derek Wise based on lectures by John Baez. Besides the lecture notes you'll also find homework problems, which contain mini-essays on related topics. Working these problems is a good way to learn math and physics. If you get stuck, you can also read answers to the problems. Different answers shed different light on the problems... so look at more than one!

The goal of this course was to understand the category-theoretic foundations of the combinatorics of Feynman diagrams. To keep things simple, we considered only Feynman diagrams for the perturbed quantum harmonic oscillator, instead of a full-fledged quantum field theory. But to handle even this simple case, we needed some interesting category theory.

So, in the 1st week, we introduced three incredibly important but underappreciated concepts in mathematics: stuff, structure and properties. Most mathematical gadgets are defined by specifying some stuff equipped with structure satisfying some properties. For example, a group is a set equipped with some operations satisfying some equations. The interesting thing is that this threefold division can be made completely precise using category theory! My student Toby Bartels wrote a nice pedagogical paper about this. There is a lot more to say about how these notions are related to homotopy theory; for a taste of that, read the discussion involving Toby, John, James Dolan, and David Corfield.

In the 2nd week, we used these concepts to define a "stuff type" as any type of stuff that you can equip a finite set with. We also showed how to compute a formal power series from a stuff type, called its "generating function". The generating function of a stuff type counts the number of ways you can put this stuff on an n-element set, so generating functions play an important role in combinatorics. But they're also states of the quantum harmonic oscillator! We thus get an interesting connection between combinatorics and quantum theory - a further development of the main themes of this course.

When we speak of "counting the number of ways" you can equip an n-element set with some type of stuff, don't be fooled! In general, there's not a set, but rather a groupoid of ways to put stuff on a finite set. So, to count it we need the notion of "groupoid cardinality", introduced in the 6th week of the Winter quarter. The cardinality of a groupoid can be a fraction or even irrational!

In the 3rd week of this quarter, we saw how to compose stuff types. This takes us further towards the grand dream of a theory of "sets with an irrational number of elements". For example, we saw how to construct a groupoid whose cardinality is:

ecosh(1) = 4.678982327...

In the homework for the 3rd week, we got groupoids whose cardinality is related to the Riemann zeta function!

In the 4th week, we saw how to take the inner product of stuff types. This is a categorified version of the inner product on the Hilbert space of the harmonic oscillator - also known as "Fock space". In the 5th week we introduced "stuff operators", which are a categorified version of operators on Fock space. In particular, we described a stuff operator analogous to the annihilation operator.

In the 7th week, we reviewed perturbation theory for the quantum harmonic oscillator. In the 8th and 9th weeks, we saw how to compute the time evolution of this system using Feynman diagrams - both in basic quantum mechanics and the new categorified version we're exploring here!

Finally, in the 10th and last week of the course, we studied Feynman diagrams arising when the perturbation of the harmonic oscillator involves "Wick" or "normal-ordered" powers of the position operator.

The perturbed quantum harmonic oscillator like a quantum field theory in miniature - it's actually a quantum field theory where space is 0-dimensional. So, in this course we only took the first step towards categorifying more interesting field theories, where space has more dimensions. There's a lot more to be done in this subject! Learn about it and join the fun.

All of these are PDF files:

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:

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.
After this course was finished, John's student Jeffrey Morton wrote a paper explaining the ideas in this course and developing them further by figuring out how to include the all-important concept of phases into categorified quantum mechanics:
The ideas in this course were also generalized to a program of replacing Hilbert spaces by groupoids and operators by spans of groupoids. This program is called "groupoidification":

The following papers are also very relevant. The first takes a while to download, because it's over 600 pages long! The second was written by a student of Ross Street, who'd gotten interested in stuff types when John lectured about them in Sydney:

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.

© 2004 John Baez and Derek Wise
baez@math.removethis.ucr.andthis.edu