Path integral formulation

The path integral formulation was developed by physicist Richard Feynman in 1948, offering a third approach to quantum mechanics alongside Schrödinger's wave mechanics and Heisenberg's matrix mechanics. Feynman's PhD advisor John Wheeler helped him formalize the idea, which later became foundational to quantum field theory.

While the Schrödinger equation evolves a wave function forward through time, the path integral calculates the probability amplitude of a particle's transition between two points by summing over every possible path connecting them. In the classical limit, the path of least action dominates, recovering Newtonian mechanics as a special case.

Each possible path contributes a phase proportional to the classical action—the integral of the Lagrangian along that path—divided by Planck's constant. Paths far from the classical trajectory interfere destructively and cancel out, explaining why only the classical path is observed at macroscopic scales.

Path integrals are central to quantum field theory and the Standard Model of particle physics, and they are essential for calculating scattering amplitudes in QED and QCD. They are also widely used in statistical mechanics, condensed matter physics, and string theory.