Path integral formulation

Summary

The path integral formulation describes quantum evolution as a sum over all paths a system could take. Each path $q(\cdot)$ contributes the phase $e^{iS[q]/\hbar}$, where $S$ is the classical action. The transition amplitude is the "integral" of these phases over all paths with fixed end points. No Lebesgue-type measure on paths makes this literal. So the integral is defined as a limit of finite-dimensional integrals over broken-line paths. After Wick rotation it becomes a genuine integral, given by the Feynman–Kac formula. It is the standard tool for perturbative quantum field theory, and it shows how classical mechanics appears when $\hbar \to 0$, by stationary phase.