Functional Integrals

Abstract

A functional integral is an integral over an infinite-dimensional space, usually a space of functions in one or several variables. One generally defines a functional integral as a limit of ordinary multiple integrals. For example, consider a functional F(φ) on the space of functions φ(t) of a single variable t, with a ≤ t ≤ b. We can restrict F to the space of continuous functions that are linear in each of the segments [t ₀, t ₁],..., [t _{N −1} t _N ], with t _i = a + i(b − a)/N. This space is finite-dimensional, since every such function is determined by its values at a = t ₀, t ₁,..., t _N = b. If we call these values φ ₀,...,φ _N , the space is parametrized by (φ ₀,...,φ _N ), and we can consider the integral J _N of F with respect to φ _l,...,φ _N . The integral of F(φ) is naturally defined as the limit of the approximations J _N , as N → ∞:

$$J = \int {F[\varphi ]} IId\varphi (\tau ) = \mathop {\lim }\limits_{N \to \infty } {J_N}.$$

(23.1)