### Quantization and Cohomology (Week 14)

#### Posted by John Baez

This week in our course on Quantization and Cohomology, I decided it was time to do an example for the students who’d never seen path integrals:

- Week 14 (Feb. 13) - An example of path-integral quantization: the free particle on a line (part 1).

Last week’s notes are here; next week’s notes are here.

I didn’t give myself enough time to prepare this class, so I wasn’t very happy with how it turned out. In general, one can do path integrals for a particle on the line by doing an integral over piecewise-linear paths with a certain bunch of pieces, then taking the limit where the pieces get very small. The miraculous feature of the *free* particle is that you get the exact answer *even before* taking the limit. One says ‘the stationary phase approximation is exact’.

In this class I showed how this followed from a simple equation involving integral kernels — but I spent so much time getting the notation straightened out that we didn’t get around to checking the equation! So, I’ll do it next time. I’m tempted to avoid doing the calculation directly (a certain Gaussian integral, nothing hard but a bit annoying), and use a sneaky trick.

Part of why I was dissatisfied is that there must be some infinitely beautiful way to tackle this classic problem, and I don’t feel I’ve found it. Maybe someone out there knows the best way?

## Re: Quantization and Cohomology (Week 14)

Your opening line refers to any category but in that context what does it mean to integrate over $hom(x,y)$?