# Feynman integral

## Feynman integral

A technique, also called the sum over histories, which is basic to understanding and analyzing the dynamics of quantum systems. It is named after fundamental work of Richard Feynman. The crucial formula gives the quantum probability density for transition from a point q0 to a point q1 in time t as the expression below, where S(path)

is the classical mechanical action of a trial path, and &planck; is the rationalized Planck's constant. The integral is a formal one over the infinite-dimensional space of all paths which go from q0 to q1 in time t. Feynman defines it by a limiting procedure using approximation by piecewise linear paths.

Feynman integral ideas are especially important in quantum field theory, where they not only are a useful device in analyzing perturbation series but are also one of the few nonperturbative tools available. See Quantum field theory

An especially attractive element of the Feynman integral formulation of quantum dynamics is the classical limit, &planck; → 0. Formal application of the method of stationary phase to the above expression says that the significant paths for small &planck; will be the paths of stationary action. One thereby recovers classical mechanics in the hamiltonian stationary action formulation. See Least-action principle

## Feynman integral

[′fīn·mən ′int·ə·grəl]
(quantum mechanics)
A term in a perturbation expansion of a scattering matrix element; it is an integral over the Minkowski space of various particles (or over the corresponding momentum space) of the product of propagators of these particles and quantities representing interactions between the particles.
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In perturbative QFT, one has to consider Feynman graphs, and to associate to each such a graph a Feynman integral (further related to quantities actually measured in physical experiments).
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The Feynman integral corresponding to the photon loop is given by
In this computation of the Feynman integrals, the dimensional regularization method in the conventional QED theory is also used.
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It is a well-known fact that the Feynman integral is a generalized function [6] [9].
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H]-function associated with a certain class of Feynman integrals, J.
Mishra, On product of hypergeometric functions, general class of multivariable polynomials and a generalized hypergeometric series associated with Feynman integrals, Bull.

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