Path integral formulation

The path integral formulation of quantum mechanics, developed by Richard Feynman in 1948, is an alternative to the operator-based formalism of Heisenberg and Schrödinger. Rather than evolving a quantum state through unitary operators, the path integral computes the probability amplitude for a transition by summing — or integrating — over all possible trajectories between initial and final configurations, weighted by a phase factor determined by the classical action. The formulation is not merely a curiosity. It is the natural language for quantum field theory, where the paths become field configurations and the integral extends over all possible field histories.

The path integral's deepest conceptual contribution is the democratization of classical paths. In the classical limit, the phase oscillates rapidly, and destructive interference cancels all paths except those where the action is stationary — the classical trajectories. Quantum mechanics is therefore not a separate theory from classical mechanics but a generalization in which all paths contribute, with the classical paths emerging as the interference-selected survivors. This same logic appears in the renormalization group: the low-energy effective description is selected from the space of all possible theories by the flow equations, just as classical paths are selected from the space of all trajectories by the stationary phase condition.

The path integral remains mathematically ill-defined in four-dimensional interacting field theories. The integral is not Lebesgue-integrable; it is defined only perturbatively or on a lattice. This is not a failure of the concept but a measure of the frontier: the path integral is the right idea, and the mathematics is still catching up.