Editing Functional integration (section)

{{Short description|Integration over the space of functions}}
{{distinguish|functional integration (neurobiology)}}

'''Functional integration''' is a collection of results in [[mathematics]] and [[physics]] where the domain of an [[integral]] is no longer a region of space, but a [[Function space|space of functions]]. Functional integrals arise in [[probability]], in the study of [[partial differential equations]], and in the [[path integral formulation|path integral approach]] to the [[quantum mechanics]] of particles and fields.

In an ordinary integral (in the sense of [[Lebesgue integration]]) there is a function to be integrated (the integrand) and a region of space over which to integrate the function (the domain of integration). The process of integration consists of adding up the values of the integrand for each point of the domain of integration.  Making this procedure rigorous requires a limiting procedure, where the domain of integration is divided into smaller and smaller regions.  For each small region, the value of the integrand cannot vary much, so it may be replaced by a single value. In a functional integral the domain of integration is a space of functions.  For each function, the integrand returns a value to add up.  Making this procedure rigorous poses challenges that continue to be topics of current research.

Functional integration was developed by [[Percy John Daniell]] in an article of 1919<ref>{{Cite journal
| volume = 20
| issue = 4
| pages = 281–288
| last = Daniell
| first = P. J.
| title = Integrals in An Infinite Number of Dimensions
| journal = The Annals of Mathematics
| series = Second Series| date = July 1919
| jstor = 1967122
| doi = 10.2307/1967122
}}</ref> and [[Norbert Wiener]] in a series of studies culminating in his articles of 1921 on [[Brownian motion]].  They developed a rigorous method (now known as the [[Wiener measure]]) for assigning a probability to a particle's random path.  [[Richard Feynman]] developed another functional integral, the [[path integral formulation|path integral]], useful for computing the quantum properties of systems.  In Feynman's path integral, the classical notion of a unique trajectory for a particle is replaced by an infinite sum of classical paths, each weighted differently according to its classical properties.

Functional integration is central to quantization techniques in theoretical physics.  The algebraic properties of functional integrals are used to develop series used to calculate properties in [[quantum electrodynamics]] and the [[standard model]] of particle physics.