Functional integration: Difference between revisions

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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^{\exp\left\lbrace-\frac{i1}{2} \intint_{\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]}

{\displaystyle\int e^{\exp\left\lbrace-\frac{i1}{2} \intint_{\mathbb{R}^2} f(x) \cdot K(x,;y) \cdot f(y) \,dx\,dy}\right\rbrace \mathcal{D}[Dff]} =

e^{-\exp\left\lbrace\frac{i1}{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]} =

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

</math>

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

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>

\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 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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*[[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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*{{ 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)

*[[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.