### Quantization and Cohomology (Week 15)

#### Posted by John Baez

This week in our course on Quantization and Cohomology, we finished off the path-integral quantization of the free particle:

- Week 15 (Feb. 20) - The free particle on a line (part 2). Showing the path-integral approach agrees with the Hamiltonian approach. Fourier transforms and Gaussian integrals.

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

Avoiding an unpleasant direct proof that the stationary phase approximation is exact for the free particle on a line, we instead took an indirect route which covered all the basic facts about Fourier transforms and Gaussians. The Wick rotation was cleverly disguised as a little maneuver to compute the integral

$\int_{-\infty}^\infty e^{-i u^2/2} d u$

which, you will note, fails to converge absolutely!

## Re: Quantization and Cohomology (Week 15)

I am currently teaching a similar subject and thus are naturally very interested in how you introduce these subjects.

However I was a bit disappointed when I realised you were cheating: Your analytic continuation looks very clever but of course a priori there is no reason why you can interchange taking the integral and doing the limit k->i. You could have argued similarly and taken the limit k->-i and obtained a mess. There is another difference: As it is unitary, you can compute the quantum mechnical propagator for both positive and negative t but the heat kernel (the euclidean version) only exists for t>0. You could see this as a manifestation of QM being time reversal invariant but thermodynamics being not (because of the second law).