# 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 *q*_{0} to a point *q*_{1} in time *t* as the expression below, where *S*(path)

*q*

_{0}to

*q*

_{1}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, ℏ → 0. Formal application of the method of stationary phase to the above expression says that the significant paths for small ℏ will be the paths of stationary action. One thereby recovers classical mechanics in the hamiltonian stationary action formulation. *See* Least-action principle