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 thenoften 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 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 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 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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* 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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*[[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=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
*[[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.