Fourier Decomposition

Abstract

In this chapter we wish to show that the Brownian bridge may be reconstructed from a denumerable set of independent Gaussian random variables other than the ones used in the bisection algorithm considered in Sect. 3.5. For it is possible to give meaning to the concept of Fourier decomposition of Brownian motion and to prove statistical independence of the relevant (random) Fourier coefficients. The motive for introducing this new structural element is that it allows us to approximate or even evaluate path integrals in a more systematic fashion.

This sum-over-histories way of looking at things is not really so mysterious, once you get used to it. Like other profoundly original ideas, is has become slowly absorbed into the fabric of physics, so that now after thirty years it is difficult to remember why we found it at the beginning so hard to grasp. Freeman Dyson