The Feynman-Kac Formula

Abstract

The main questions raised so far concerned the Brownian motion as a real phenomenon. The problem of relating it to quantum mechanics was only touched upon. Now that we have the material to penetrate deeper into the subject, we shall, in the present chapter, give an account of the functional integral description of interacting quantum mechanical systems using Brownian motion as a tool. In doing so we no longer mention the Brownian particle: it is treated merely as a formal object and is not considered any part of the observable reality. Also, quantum mechanical probability should not be confused with the statistics of the underlying Brownian motion.

The physicist cannot understand the mathematician’s care in solving an idealized physical problem. The physicist knows the real problem is much more complicated. It has already been simplified by intuition which discards the unimportant and often approximates the remainder. Richard Feynman