Functional Integral

A functional integral is an integral over a space of functions or fields. Here a field is a function defined on spacetime, rather than a field in the algebraic sense. A functional integral is commonly written

where denotes integration over field configurations and the square brackets emphasize that depends on the entire function . A finite-dimensional approximation replaces the field by finitely many variables and the functional integral by an ordinary multiple integral.

In Euclidean quantum field theory, a typical functional integral weights each field configuration by , where is the action. Functional integrals in quantum field theory are frequently formal; particular constructions, such as Gaussian measures and the Wiener measure, can make suitable cases rigorous (Glimm and Jaffe 1987). Examples of equations formulated using functional integrals include the Dyson-Schwinger equations and the Wetterich equation.