Functional integration: Difference between revisions

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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_{-\inftymathbb{R}}^\infty \cdots \int_{-\inftymathbb{R}}^\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}[f] \equiv \int_{-\inftymathbb{R}}^\infty \cdots \int_{-\inftymathbb{R}^\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.<math>

W[J]

</math>

==Examples==

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<math>

\dfrac{W[J]}{W[0]}=\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>

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<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>

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*[[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.