Functional integration

Functional integration is a collection of results in mathematics and physics where the domain of an integral is no longer an ordinary region of space, but a space of functions. Functional integrals appear in probability, in the study of partial differential equations, and in the path integral formulation to the quantum mechanics of particles and fields. While the term suggests an extension of ordinary integration, the lack of a translation-invariant measure on infinite-dimensional spaces means that functional integrals must be defined by more subtle methods or interpreted formally. In an ordinary integral (in the sense of Lebesgue integration), there is a function to be integrated (the integrand) and a region of space over which to integrate the function (the domain of integration). The integral represents the limit of a sum obtained by dividing the region into smaller parts, evaluating the function on each, and adding up the results. For each small region, the value of the integrand cannot vary much, so it may be replaced by a single value. In a functional integral the domain of integration is a space of functions. For each function, the integrand returns a value to add up. Making this procedure rigorous poses challenges that continue to be topics of current research. Functional integration was developed by Percy John Daniell in an article of 1919 and Norbert Wiener in a series of studies culminating in his articles of 1921 on Brownian motion. They developed a rigorous method (now known as the Wiener measure) for assigning a probability to a particle's random path. Richard Feynman developed another functional integral, the path integral, useful for computing the quantum properties of systems. In Feynman's path integral, the classical notion of a unique trajectory for a particle is replaced by an infinite sum of classical paths, each weighted differently...