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February 15, 2007

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?

Posted at February 15, 2007 8:40 PM UTC

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2 Comments & 2 Trackbacks

Read the post Quantization and Cohomology (Week 15)
Weblog: The n-Category Café
Excerpt: An example of path integral quantization (continued).
Tracked: February 21, 2007 1:38 AM

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)hom(x,y)?

Posted by: jim stasheff on February 22, 2007 1:55 AM | Permalink | Reply to this

Re: Quantization and Cohomology (Week 14)

Jim wrote:

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

I answered that question — without any profound insight — back on page 4 of week 11’s lecture (see item 2).

What I said was that to integrate over hom(x,y)hom(x,y), this set had better be equipped with the structure of a measure space. Or, at least it should be a ‘generalized measure space’: a gadget I defined a while back, which more conveniently handles some the path integrals that people know how to make rigorous.

Of course, the big problem is that in most really interesting quantum theories, people don’t know how to make the path integrals rigorous.

Indeed, even the pathetically simple example I’m tackling here requires that we go beyond the usual Lebesque theory of integration — as next week’s lecture shows. (See especially the comments on the blog.)

That’s where generalized measure spaces swing into action…

Posted by: John Baez on February 22, 2007 2:11 AM | Permalink | Reply to this
Read the post QFT of Charged n-Particle: The Canonical 1-Particle
Weblog: The n-Category Café
Excerpt: On the category of paths whose canonical Leinster measure reproduces the path integral measure appearing in the quantization of the charged particle.
Tracked: March 19, 2007 9:02 PM

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