Feynman integral

(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.

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 ℏ 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 also 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 also Least-action principle.