Quantum Gravity Seminar - Fall 2003

Quantization and Categorification

John Baez and Derek Wise

Andre Joyal invented his theory of "espèces des structures" - translated as "species" or "structure types" - in order to understand more deeply how people use generating functions to count structures on finite sets. It turns out that just as a natural number is a watered-down or "decategorified" version of a finite set, a generating function is a decategorified version of a structure type.

Recently, James Dolan and John Baez realized that structure types and more general "stuff types" can also be used to more deeply understand the role of annihilation and creation operators, Feynman diagrams and the like in quantum theory. It turns out that some of the mysteries of quantum mechanics are really just decategorified versions of simple facts about structures on finite sets. For example, the fact that position and momentum don't commute has a purely combinatorial interpretation! Ultimately, it boils down to the fact that there's one more way to put a ball in a box and then take one out than to take one out and then put one in.

John has been dying to talk about this stuff for a while, so in the fall of 2003 he began lecturing on "Quantization and Categorification" in his Quantum Gravity Seminar here at U. C. Riverside.

From this page you can download copies of handwritten notes of this seminar, taken by Derek Wise. The notes are available week by week, but note there is some overlap in the files, since Derek hadn't yet adopted the practice yet of starting a new "week" on a fresh page:

The notes are also available as a single large pdf file. Here it is:

Mike Stay has kindly created an index to these notes:

The notes are a mixture of what John wrote on the board during the seminar, what he said but didn't write down, and sometimes whatever Derek interpreted him to be saying! If you discover any errors 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:

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


© 2003 John Baez and Derek Wise