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Functional integration is a collection of results in mathematics and physics where the domain of an integral is no longer a region of space, but a space of functions. Functional integrals arise in probability, in the study of partial differential equations and in Feynman's approach to the quantum mechanics of particles and fields.
In an ordinary integral 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 process of integration consists of adding the values of the integrand at each point of the domain of integration. Making this procedure rigorous requires a limiting procedure, where the domain of integration is divided into smaller and smaller regions. 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 are the topic of research in the beginning of the 21st century.
Functional integration was introduced by Wiener in 1921 in his studies of Brownian motion. He developed a rigorous method —now known as the Wiener measure— for assigning a probability to a particle's random path. 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 according to its classical properties.
Functional integration is central to quantization techniques in theoretical physics. The algebraic properties of functional integrals are used to develop series used to calculate properties in quantum electrodynamics and the standard model.
The problem of functional integration
Integration of functions is a summation. If the domain of integration is the square [0, 1] × [0, 1], the integral is computed by breaking the region into small rectangles. Each rectangle serves as the base of a prism whose height is any value of the function within the rectangle. The integral is the sum of the volumes (base × height) of all the prisms. If the rectangles are small enough and the function smooth, the process converges.
A functional is a function that associates a function to a number. This is in distinction to common functions that associate numbers to numbers. Examples of functionals include the functional that is one for any function, or the functional that returns the integral of the function over a domain.
By analogy with integration of functions, functional integration is a summation procedure where the domain of integration is a space of functions and the functional integral an addition of cylinders (just like the prisms in ordinary integration) with the functional as the height and some amount (or measure) of function space as the base. The “area” of the functions can be represented by Dω , the functional by F[ω] (square brackets are often used to distinguish functionals from common functions), the space of functions by I and the functional integral by
The result of integrating a functional is to be a number. Developing a definition to equation (1)
that has properties similar to ordinary integration
is the problem of the definition of functional integration. There is no general theory to make sense of this formal expression as there is for the conventional integration
In functional integration there are different spaces to consider:
- The functions ω are defined on a ν-dimensional space called space-time. This is how many dimensions are used to specify a point in the domain of ω. Space-time is often assumed to be a subset of Rν, the Euclidean space with ν dimensions. In applications of functional integration, the functions ω represent particle paths (in which case ν = 1) or a physical field such as the vector potential (in which case ν = 4).
- The space for the range of the functions ω also varies depending on applications. This space is locally a subset of Rκ. In applications it can be R³, as in the quantum mechanics; or a more complicated space, such as a tangent bundle as in the case of quantum chromodynamics.
- The domain of integration is a function space, and likely to be infinite dimensional.
A definition of functional integration, by analogy with common integration, is expected to satisfy certain properties. Functional integration should be itself a linear functional, such that ∫Dω (F+α G) = ∫ Dω F + α ∫ Dω G. Volumes in functional space should be invariant under translation. A ball in functional space centered around a function f or that same function plus a constant should result in the same value for the functional integral. Also, if the functional space happens to be finite dimensional, then the functional integral should be related to the ordinary integration. These conditions are impossible to satisfy for functional integrals.
Attempting to directly generalize the notion of volume in functional space has not led to a useful theory of functional integration. Discretization of the functional integral in equation (1) could be an approach towards its definition. For the case of one-dimensional paths (ν=1 and κ=1), the functional integration is replaced by an n-dimensional integral and the functional is computed from the value of the path ω and n points. The functional integral would then be the value of the n-dimensional integral in the limit of n going to infinity:
The “size” of a function space can be computed from this expression by using the simple functional F
]=1. Choosing a function space where each ωi
varies over limited range of length W
, the n
-dimensional integral is equal to Wn
. This will diverge to infinity or converge to zero in the limit. Building a theory of integration when the value of the integral can only be zero or infinity is not very interesting.
Most of the cylinders that contribute to the functional integral (2) correspond to discontinuous functions. In Brownian motion or in the path integral formulation of quantum mechanics, the paths are continuous. Both applications did not generalize the notion of volume to functional spaces, as in equation (1), but rather generalized the notion of a Gaussian integral. In applications, the functional being integrated is related to an action functional S arising from classical mechanics. Action functionals can be written as the sum of two terms, S0 + Si , with S0 involving the derivative of the function squared. For example, for the case of one-dimensional paths
Smooth paths lead to small values of the functional, and large variations of the path (as if almost discontinuous) lead to large values of the functional. Introducing a term exp(−S0
) into the functional integral should dampen the effects of discontinuous paths. This leads to Gaussian functional integrals.
Instead of generalizing the notion of volume to infinite dimensions, the Gaussian integral can be generalized. If M is a positive n×n symmetric matrix, and x and J are n-dimensional vectors, the basic Gaussian integral can be used to show that