Functional integration: Difference between revisions

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==Approaches to path integrals==

{{expand section|date=October 2009}}

Functional integrals where the space of integration consists of paths (''ν'' = 1) can be defined in many different ways. The definitions fall in two different classes: the constructions derived from [[Wiener process|Wiener's theory]] yield an integral based on a [[Measure (mathematics)|measure]];, whereas the constructions following Feynman's path integral do not. Even within these two broad divisions, the integrals are not identical, that is, they are defined differently for different classes of functions.

===The Wiener integral===

In the [[Wiener process|Wiener integral]], a probability is assigned to a class of [[Brownian motion]] paths. The class consists of the paths ''w'' that are known to go through a small region of space at a given time. The passage through different regions of space is assumed independent of each other, and the distance between any two points of the Brownian path is assumed to be [[Normal distribution|Gaussian -distributed]] with a [[variance]] that depends on the time ''t'' and on a diffusion constant ''D'':

:<math> \mathrm{Pr}\big(w(s + t), t | w(s), s\big) = \frac{1}{\sqrt{2\pi D t}} \exp\left({-\frac{\|w(s+t) - w(s)\|^2}{2Dt}} \right).</math>

The probability for the class of paths can be found by multiplying the probabilities of starting in one region and then being at the next. The Wiener measure can be developed by considering the limit of many small regions.