Functional integration

Whereas normal integration sums a function, f(x), over a continuous range of values of x, functional integration sums a functional, G[f], over a continuous range of functions, f. Most functional integrals cannot be solved exactly but must be solved using perturbation methods. The formal definition is:

$\int{ G[f] [Df] } \equiv \int\limits_{-\infty}^\infty{ ... \int\limits_{-\infty}^\infty{ G[f] } }\prod_x df(x)$

However in most cases the functions f(x) can be written in terms of an infinite series of orthogonal functions such as $f(x) = f_n H_n(x)$ and then the definition becomes:

$\int{ G[f] [Df] } \equiv \int\limits_{-\infty}^\infty{ ... \int\limits_{-\infty}^\infty{ G(f_1,f_2,..) } }\prod_n df_n$

which is slightly more understandable. The integral is shown to be a functional integral with a capital D. Sometimes it is written in square brackets [Df] or D[f] to indicate f is a function.

Examples

Most functional integrals are actually infinite but the quotient of two functional integrals can be finite. The functional integrals that can be solved exactly usually start with the following Gaussian integral:

$\frac{ \int{ e^{i \int{ -\frac{1}{2}f(x).K(x,y).f(y) dxdy} %2B \int{ J(x).f(x) dx} }[Df] } } { \int{ e^{i \int{ -\frac{1}{2}f(x).K(x,y).f(y) dxdy} } [Df] } } = e^{i \frac{1}{2}\int{ J(x).K^{-1}(x,y).J(y) dxdy } }$

By functionally differentiating this with respect to J(x) and then setting J to 0 this becomes an exponential multiplied by a polynomial in f. For example setting $K(x,y)=\Box\delta(x-y)$ we find:

$\frac{ \int{ f(a) f(b) e^{i \int{ f(x) \Box f(x) dx^4}} }[Df] }{ \int{ e^{i \int{ f(x) \Box f(x) dx^4}} }[Df] } = K^{-1}(a,b) = \frac{1}{|a-b|^2}$

where a,b and x are 4-dimensional vectors. This comes from the formula for the propagation of a photon in quantum electrodynamics. Another useful integral is the functional delta function:

$\int{ e^{i \int{ f(x) g(x) dx}} }[Df] = \delta[g] = \prod_x\delta( g(x) )$

which is useful to specify constraints. Functional integrals can also be done over Grassmann-valued functions $\psi(x)$ where $\psi(x)\psi(y)=-\psi(y)\psi(x)$ which is useful in quantum electrodynamics for calculations involving fermions.

Inverse of functional derivative

Functional integration can be thought of as the inverse operation to functional differentiation. The functional derivative of the functional G is written as:

$\frac{\delta}{\delta f(x)} G[f]$

In symbolic algebra software

Most symbolic algebra packages such as Maple or Mathematica do not support functional (path) integration as standard although additional packages can be constructed for them.

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.

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:

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

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

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

The Lévy integral

References

^ Daniell, P. J. (1919-07). "Integrals in An Infinite Number of Dimensions". The Annals of Mathematics. Second Series 20 (4): 281–288. ISSN 0003-486X. JSTOR 1967122.

Functional integration

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