Everything about Functional Integral totally explained
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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's 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 can't 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's 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 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 measuresFurther Information
Get more info on 'Functional Integral'.
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