View source for Functional integration

{{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.

==Functional integration==
{{Confusing|section|date=January 2014}}
{{unreferenced section|date=March 2017}}
Whereas standard [[Riemann integral|Riemann integration]] sums a function ''f''(''x'') over a continuous range of values of ''x'', functional integration sums a [[functional (mathematics)|functional]] ''G''[''f''], which can be thought of as a "function of a function" over a continuous range (or space) of functions ''f''. Most functional integrals cannot be evaluated exactly but must be evaluated using [[perturbation methods]]. The formal definition of a functional integral is
<math display="block">
\int G[f]\; \mathcal{D}[f] \equiv \int_{\mathbb{R}}\cdots \int_{\mathbb{R}} G[f] \prod_x df(x)\;.
</math>

However, in most cases the functions ''f''(''x'') can be written in terms of an infinite series of [[orthogonal functions]] such as <math>f(x) = f_n H_n(x)</math>, and then the definition becomes
<math display="block">
\int G[f] \; \mathcal{D}[f] \equiv \int_{\mathbb{R}} \cdots \int_{\mathbb{R}}  G(f_1; f_2; \ldots) \prod_n df_n\;,

</math>

which is slightly more understandable. The integral is shown to be a functional integral with a capital <math>\mathcal{D}</math>. Sometimes the argument is written in square brackets <math>\mathcal{D}[f]</math>, to indicate the functional dependence of the function in the functional integration measure.

==Examples==
Most functional integrals are actually infinite, but often the limit of the [[quotient]] of two related functional integrals can still be finite. The functional integrals that can be evaluated exactly usually start with the following [[Gaussian integral]]:

:<math>
\frac{\displaystyle\int \exp\left\lbrace-\frac{1}{2} \int_{\mathbb{R}}\left[\int_{\mathbb{R}} f(x)  K(x;y) f(y)\,dy +  J(x)  f(x)\right]dx\right\rbrace \mathcal{D}[f]}
     {\displaystyle\int \exp\left\lbrace-\frac{1}{2} \int_{\mathbb{R}^2} f(x)  K(x;y)  f(y) \,dx\,dy\right\rbrace \mathcal{D}[f]} =
 \exp\left\lbrace\frac{1}{2}\int_{\mathbb{R}^2} J(x) \cdot K^{-1}(x;y) \cdot J(y) \,dx\,dy\right\rbrace\,,
</math>

in which <math>
K(x;y)=K(y;x)
</math>. By functionally differentiating this with respect to ''J''(''x'') and then setting  to 0 this becomes an exponential multiplied by a monomial in ''f''. To see this, let's use the following notation:

<math>
 G[f,J]=-\frac{1}{2} \int_{\mathbb{R}}\left[\int_{\mathbb{R}} f(x)  K(x;y) f(y)\,dy +  J(x)  f(x)\right]dx\, \quad,\quad W[J]=\int \exp\lbrace G[f,J]\rbrace\mathcal{D}[f]\;. 
</math>

With this notation the first equation can be written as:

<math>
\dfrac{W[J]}{W[0]}=\exp\left\lbrace\frac{1}{2}\int_{\mathbb{R}^2} J(x)  K^{-1}(x;y)  J(y) \,dx\,dy\right\rbrace.
</math>

Now, taking functional derivatives to the definition of <math>
W[J]
</math> and then evaluating in <math>
J=0
</math>, one obtains:

<math>
\dfrac{\delta }{\delta J(a)}W[J]\Bigg|_{J=0}=\int f(a)\exp\lbrace G[f,0]\rbrace\mathcal{D}[f]\;,
</math>

<math>
\dfrac{\delta^2 W[J]}{\delta J(a)\delta J(b)}\Bigg|_{J=0}=\int f(a)f(b)\exp\lbrace G[f,0]\rbrace\mathcal{D}[f]\;,
</math>

<math>
\qquad\qquad\qquad\qquad\vdots
</math>

which is the result anticipated. More over, by using the first equation one arrives to the useful result:

:<math>
    \dfrac{\delta^2}{\delta J(a)\delta J(b)}\left(\dfrac{W[J]}{W[0]}\right)\Bigg|_{J=0}=
 K^{-1}(a; b)\;;
</math>

Putting these results together and backing to the original notation we have:

<math>
\frac{\displaystyle\int f(a)f(b)\exp\left\lbrace-\frac{1}{2} \int_{\mathbb{R}^2} f(x)  K(x;y) f(y)\, dx\,dy\right\rbrace \mathcal{D}[f]}
     {\displaystyle\int \exp\left\lbrace-\frac{1}{2} \int_{\mathbb{R}^2} f(x)  K(x;y)  f(y) \,dx\,dy\right\rbrace \mathcal{D}[f]} =
 K^{-1}(a;b)\,.
</math>

Another useful integral is the functional [[delta function]]:

:<math>
\int \exp\left\lbrace \int_{\mathbb{R}} f(x) g(x)dx\right\rbrace \mathcal{D}[f] = \delta[g] = \prod_x\delta\big(g(x)\big),
</math>

which is useful to specify constraints. Functional integrals can also be done over [[Grassmann number|Grassmann-valued]] functions <math>\psi(x)</math>, where <math>\psi(x) \psi(y) = -\psi(y) \psi(x)</math>, which is useful in quantum electrodynamics for calculations involving [[fermions]].

==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>\Pr\big(w(s + t), t \mid 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.

* Itō and Stratonovich calculus

===The Feynman integral===

* Trotter formula, or [[Lie product formula]].
* The Kac idea of Wick rotations.
* Using x-dot-dot-squared or i S[x] + x-dot-squared.
* The Cartier DeWitt–Morette relies on integrators rather than measures

===The Lévy integral===

* [[Fractional quantum mechanics]]
* [[Fractional Schrödinger equation]]
* [[Lévy process]]
* [[Fractional statistical mechanics]]

==See also==
*[[Path integral formulation|Feynman path integral]]
*[[Partition function (quantum field theory)]]
*[[Saddle point approximation]]

==References==
{{Reflist}}

==Further reading==
*[http://www.scholarpedia.org/Path_integral    Jean Zinn-Justin (2009), ''Scholarpedia'' '''4'''(2):8674].
* [[Hagen Kleinert|Kleinert, Hagen]], ''Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets'', 4th edition, World Scientific (Singapore, 2004); Paperback {{ISBN|981-238-107-4}}  '' (also available online: [http://www.physik.fu-berlin.de/~kleinert/b5 PDF-files])''
*{{ cite journal|author-link=Nick Laskin|arxiv=0811.1769|doi=10.1103/PhysRevE.62.3135|title=Fractional quantum mechanics|year=2000|last1=Laskin|first1=Nick|journal=Physical Review E|volume=62|issue=3|pages=3135–3145|pmid=11088808 |bibcode = 2000PhRvE..62.3135L |s2cid=15480739 }}
*{{ cite journal|author-link=Nick Laskin|arxiv=quant-ph/0206098 |doi=10.1103/PhysRevE.66.056108|title=Fractional Schrödinger equation|year=2002|last1=Laskin|first1=Nick|journal=Physical Review E|volume=66|issue=5|bibcode = 2002PhRvE..66e6108L|pmid=12513557|page=056108 |s2cid=7520956 }}
*{{SpringerEOM |id=Integral_over_trajectories |title=Integral over trajectories |author-first=R. A. |author-last=Minlos}}
* O. G. Smolyanov, E. T. Shavgulidze. ''Continual integrals''. Moscow, Moscow State University Press, 1990. (in Russian). http://lib.mexmat.ru/books/5132
*[[Victor Popov]], Functional Integrals in Quantum Field Theory and Statistical Physics, Springer 1983
*[[Sergio Albeverio]], Sonia Mazzucchi, A unified approach to infinite-dimensional integration, Reviews in Mathematical Physics, 28, 1650005 (2016)






[[Category:Integral calculus]]
[[Category:Functional analysis]]
[[Category:Mathematical physics]]
[[Category:Quantum mechanics]]
[[Category:Quantum field theory]]