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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
In an ordinary integral (in the sense of [[Lebesgue integration
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
{{Confusing|section|date=January 2014}}
{{unreferenced section|date=March 2017}}
Whereas standard
<math display="block">
\int G[f]\; \mathcal{D}[
:<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}[
:<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
==Examples==
Most functional integrals are actually infinite, but
:<math>
\frac{\displaystyle\int
{\
</math>
in which <math>
By functionally differentiating this with respect to ''J''(''x'') and then setting 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 to 0 this becomes an exponential multiplied by a
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:
\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>
:<math>
\int
</math>
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* The Kac idea of Wick rotations.
* Using x-dot-dot-squared or i S[x] + x-dot-squared.
* The Cartier
===The Lévy integral===
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*[[Partition function (quantum field theory)]]
*[[Saddle point approximation]]
*{{SpringerEOM |id=Integral_over_trajectories |title=Integral over trajectories |author-first=R. A. |author-last=Minlos}}▼
==References==
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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=
*{{ 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)
*[[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.
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