Functional integration: Difference between revisions

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==Functional Integration==

{{Confusing|section|date=January 2014}}

Whereas standard 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>

\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>

\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 ''D''. Sometimes it is written in square brackets: [''Df''] or ''D''[''f''], to indicate that ''f'' is a function.

==Examples==