FeynmanIntegral

FeynmanIntegral is a term used to describe integrals that arise in the Feynman path integral formulation of quantum mechanics and quantum field theory. Named after Richard P. Feynman, these integrals express quantum amplitudes as a sum over histories or field configurations.

In non-relativistic quantum mechanics, the transition amplitude between configurations is given by a path integral over all possible trajectories, weighting each path by exp(i S[path]/ħ). In quantum field theory, this generalizes to functional integrals over field configurations.

In perturbation theory, Feynman diagrams encode terms in the expansion; each diagram corresponds to a Feynman integral over internal momenta, with propagators and interaction vertices.

These integrals are often divergent and require regularization and renormalization. Regularization introduces a regulator to tame divergences; renormalization absorbs infinities into redefined parameters.

The Euclidean version, obtained by Wick rotation to imaginary time, yields better mathematical control and connects to statistical mechanics via Feynman-Kac-type relations.

Although highly successful in physics, mathematical rigor of Feynman integrals remains an active area; several rigorous formulations exist in constructive quantum field theory and axiomatic approaches.

See also: path integral, Feynman diagram, renormalization, dimensional regularization.