Functional integration

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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.

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[edit] 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

\int_{I} D\omega \, F[\omega]\,. (1)

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:

\int D\omega \, F(\omega) = \lim_{n \rightarrow \infty} \int d\omega_1 \int d\omega_2 \ldots \int d\omega_n F(\omega_1, \ldots , \omega_n) (2)

The “size” of a function space can be computed from this expression by using the simple functional F[ω]=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

S_0[\omega] =\int d\tau \, \frac{\dot{\omega}^2}{2} .

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.

[edit] Gaussian integration

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

\int Dx \, e^{-\frac{1}{2} x \cdot M \cdot x + x \cdot J } = \frac{1}{\sqrt{|\det{M}|}} \, e^{ \frac{1}{2} J \cdot M^{-1} \cdot J }

The integration variables have been abbreviated with Dx=(2π )-n/2dx1...dxn.

The determinant and the matrix operations in the result of the many-dimensional Gaussian can be interpreted in terms of infinite-dimensional objects. This result then can be used as the basis for the definition of a functional integral. The action functional S of a path x(t) can be approximated by a discretization of the time domain of the path into n+1 pieces of length a. At these time values the path assumes the values x0, ..., xn+1. The two end points x0 and xn+1 remain fixed and are not part of the n-fold integration. The discretized action S can be put in the form xΔx + xJ. The matrix Δ is similar to a discretized Laplace operator, the vector J will have only two non-zero entries and the vector x runs from x1 to xn. The determinant of Δ and the inverse matrix Δ-1 can be evaluated as a function of the number of discretization steps n. The determinant is n+1. The inverse matrix is a matrix with entries of order one divided by the determinant.

Using a limit of iterated Gaussians it is possible to define a functional integral for paths (ν=1). With the notation O(1) to indicate a matrix with entries of order one, the iterated Gaussian integral

\int Dx \, e^{-\frac{1}{2} x \cdot M \cdot x + x \cdot J } = \left(\frac{a^{n+1}}{n+1}\right)^{1/2} e^{\frac{1}{2 a (n+1)} O(1)}

will converge if the entire expression is divided by a(n+1)/2.

In the case of a spacetime of more than one dimension, generalizations of the Gaussian integral will not converge. This is at the root of many of the difficulties in quantum field theory, as the Gaussian integral corresponds to the quantum field theory with no interactions. The difficulty in generalizing the Gaussian integral is that the growth of the determinant in the result cannot be removed by a rescaling. (The determinant of the operator in ν spacetime dimensions grows as 2nν.) In physics it is common to refer to the divergence due to the determinant as an infrared divergence and the divergence due to the limit of the discretization a going to zero as an ultraviolet divergence.

Due to the behavior of Gaussian integrals, in one spacetime dimension (ν=1) it becomes possible to define a functional integral of the form

\int Dx e^{-S[x]} .

In the applications of functional integration to quantum mechanics the action S[x] is pure imaginary and the exponential is an oscillating function. This leads to further difficulties relating to conditional convergence. In spacetime dimensions greater than one (ν ≥ 2), it is not known how to define a functional integral without resorting to regularization procedures that do away with many of the properties of integration.

[edit] Approaches to path integrals

Functional integrals where the space of integration are paths (ν = 1) can be defined in many different ways. The definitions fall in two different classes: the constructions derived from Wiener's theory yield an integral based on a measure; whereas the constructions following Feynman's path integral do not. Even within these two broad divisions, the integrals are not identical, that is, they are defined for different classes of functions.

[edit] The Wiener integral

In the Wiener integral a probability is assigned to a class of Brownian motion paths. The class consists of the paths w that are known to go through a small region of space at a given time. The passage through different regions of space is assumed independent of each other and the distance between any two points of the Brownian path is assumed to be Gaussian distributed with a variance that depends on the time t and on a diffusion constant D:

\mathrm{Prob}(w(s+t),t|w(s),s) = \frac{1}{\sqrt{2\pi D t}} \exp\left({-\frac{\|w(s+t) - w(s)\|^2}{2Dt}} \right)

The probability for the class of paths can be found by multiplying the probabilities of starting in one region and then being at the next. The Wiener measure can be developed by considering the limit of many small regions.

  • Ito and Stratonovich calculus

[edit] The Feynman integral

  • Trotter formula
  • The Kac idea of Wick rotations.
  • Using x-dot-dot-squared or i S[x] + x-dot-squared.
  • The Cartier DeWitt-Morette relies on integrators rather than measures

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