Functional integration: Difference between revisions

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{{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 (mathematics)|___domain]] of an [[integral]] is no longer a [[manifold|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|ordinary integral]]) 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

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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 Integrationintegration==

{{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}[Dff] \equiv \int\limits_int_{-\inftymathbb{R}}^\infty \cdots \intint_{\limits_mathbb{-\inftyR}}^\infty G[f] \prod_x df(x)\;.

:<math>

\int G[f] [Df] \equiv \int\limits_{-\infty}^\infty \cdots \int\limits_{-\infty}^\infty 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}[Dff] \equiv \int\limits_int_{-\inftymathbb{R}}^\infty \cdots \int\limits_int_{-\inftymathbb{R}^\infty} G(f_1,; f_2,; \ldots) \prod_n df_n\;,

:<math>

\int G[f] [Df] \equiv \int\limits_{-\infty}^\infty \cdots \int\limits_{-\infty}^\infty 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 itthe argument is written in square brackets: [''Df''] or ''<math>\mathcal{D''}[''f'']</math>, to indicate thatthe ''f''functional isdependence aof 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.{{clarify|date=March 2017}} The functional integrals that can be solvedevaluated exactly usually start with the following [[Gaussian integral]]:

:<math>

\frac{\displaystyle\int e^{i \int exp\left\lbrace-\frac{1}{2} \int_{\mathbb{R}}\left[\int_{\mathbb{R}} f(x) \cdot K(x,;y) \cdot f(y) \,dx\,dy + \int J(x) \cdot f(x) \,right]dx}\right\rbrace \mathcal{D}[Dff]}

{\int e^{i displaystyle\int \exp\left\lbrace-\frac{1}{2} \int_{\mathbb{R}^2} f(x) \cdot K(x,;y) \cdot f(y) \,dx\,dy}\right\rbrace \mathcal{D}[Dff]} =

e^{i \exp\left\lbrace\frac{1}{2}\intint_{\mathbb{R}^2} J(x) \cdot K^{-1}(x,;y) \cdot J(y) \,dx\,dy}.\right\rbrace\,,

</math>

in which <math>

By functionally differentiating this with respect to ''J''(''x'') and then setting ''J'' to 0 this becomes an exponential multiplied by a polynomial in ''f''. For example, setting <math>K(x, y) = \Box\delta(x - y)</math>, we find:

K(x;y)=K(y;x)

</math>. By functionally differentiating this with respect to ''J''(''x'') and then setting ''J'' to 0 this becomes an exponential multiplied by a polynomialmonomial in ''f''. ForTo example,see setting <math>K(xthis, y)let's =use \Box\delta(x - y)</math>,the wefollowing findnotation:

:<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}=

\frac{\int f(a) f(b) e^{i \int f(x) \Box f(x) \,dx^4} [Df]}

K^{-1}(a; b)\;;

{\int e^{i \int f(x) \Box f(x) \,dx^4} [Df]} =

</math>

K^{-1}(a, b) = \frac{1}{|a - b|^2},

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>

where ''a'', ''b'' and ''x'' are 4-dimensional vectors. This comes from the formula for the propagation of a photon in quantum electrodynamics. Another useful integral is the functional [[delta function]]:

:<math>

\int e^{i\exp\left\lbrace \intint_{\mathbb{R}} f(x) g(x)dx\right\rbrace \,dxmathcal{D} [Dff] = \delta[g] = \prod_x\delta\big(g(x)\big),

</math>

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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 \,\big|\,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.

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* The Kac idea of Wick rotations.

* Using x-dot-dot-squared or i S[x] + x-dot-squared.

* The Cartier DeWitt-MoretteDeWitt–Morette relies on integrators rather than measures

===The Lévy integral===

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==See also==

*[[Path integral formulation|Feynman path integral]]

*[[Partition function (quantum field theory)]]

*[[Saddle point approximation]]

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*[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=31353135–3145|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

*Popov[[Victor V.N.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)

*[[John R. Klauder|Klauder, John]]. "[https://www.phys.ufl.edu/functional-integration/ Lectures on Functional Integration]." ''University of Florida.'' [https://web.archive.org/web/20240708182058/http://www.phys.ufl.edu/functional-integration/ Archived] on July 8th, 2024.