What is the Feynman path integral formulation?

The path-integral formulation of quantum mechanics generalizes the classical action principle by replacing a single unique trajectory with a sum, or functional integral, over all quantum-mechanically possible trajectories. This method computes a quantum amplitude and offers advantages such as easier coordinate changes and natural use of the Lagrangian. Developed fully by Richard Feynman in 1948, it builds on foundational ideas from Norbert Wiener and Paul Dirac.

What we know

• The path-integral formulation replaces the classical notion of a single trajectory with a sum or functional integral over an infinity of possible trajectories. [1, 2]
• The basic idea can be traced back to Norbert Wiener's Wiener integral, which was extended to the Lagrangian in quantum mechanics by Paul Dirac in a 1933 paper. [2]
• The complete method of the path integral was developed in 1948 by Richard Feynman, building on preliminaries from his doctoral work under John Archibald Wheeler. [2]
• Advantages of the formulation include easier achievement of manifest Lorentz covariance and the relative ease of guessing the correct form of the Lagrangian. [2]

Still open

• The unitarity of the S-matrix remains obscure within the path-integral formulation.

Where to go next

• whoHow did Paul Dirac influence the path integral?
• the bigger pictureWhat is the action principle in classical mechanics?
• how it workedHow does the Schrödinger equation relate to diffusion?
• compared withWhat are the other mathematical formulations of quantum mechanics?
• what followedHow is the path integral used in lattice gauge theory?
• an unexpected connectionHow does quantum mechanics connect to stochastic processes?The path integral relates quantum and stochastic processes because the Schrödinger equation acts as a diffusion equation with an imaginary diffusion constant, connecting it to random walks.

Sources

• [1]path integral (Wikidata Q898323) · CC0 1.0
• [2]Path-integral formulation · CC BY-SA 4.0
• [3]Mathematical formulation of quantum mechanics · CC BY-SA 4.0
• [4]Phase-space formulation · CC BY-SA 4.0
• [5]Matrix mechanics · CC BY-SA 4.0