08 January 2008

A composition law for discrete-time QM

One of the best ways to understand quantum mechanics — bear with me — is as a one-dimensional quantum field theory. No, it's not backwards, and we really should think of QM as inherently one-dimensional: there's one dimension of time. The configuration space of a particle is finite-dimensional; the size of the space of paths the particle could take — and Feynman says that a particle takes every possible path — is largely determined by number of "time" dimensions in the problem, since a path has a point in configuration space for every moment in time.

In any case, I'm going to think of it that way. And then I'm going to think about path integration: the transition amplitude between two configurations is some poorly-defined integral over the infinite-dimensional space of paths connecting those configurations. Rather than trying to compute in infinite dimensions, physicists since Feynman have formally expanded the integral asymptotically, and interpreted the coefficients combinatorially as diagrams. (And interpreted the diagrams as describing actual events, which is a matter for them physicists rather than for us mathematicians to discuss.)

These formal power series should — indeed, must, if the formalism is to make sense — satisfy a particular gluing axiom. But it seems that no one has gone to the trouble to verify that in fact they do; there is no proof available in the literature. In his fall-semester QFT class, Nicolai Reshetikhin suggested that someone try to fix this, and I volunteered. Silly me.

Long story short, after making progress on a related issue, and then avoiding the project for more than a month, I've worked the 0-dimensional analogue, where we approximate the time interval by a sequence of (equally-spaced) discrete points. If you would like to read about it, the paper is available here.

Update: The second, and likely final, installment is now available. It concludes with the gluing rules for arbitrary perturbative quantum field theories.

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