Functional integration: Difference between revisions

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Revision as of 07:29, 23 October 2022

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{{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] [Df] \equiv \int\limits_int_{-\infty}^\infty \cdots \int\limits_int_{-\infty}^\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] [Df] \equiv \int\limits_int_{-\infty}^\infty \cdots \int\limits_int_{-\infty}^\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>