Functional integration: Difference between revisions

Revision as of 23:09, 24 April 2023

Revision as of 23:17, 24 April 2023

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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>

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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>