Path Integral Formulation

The path integral formulation of quantum mechanics is a description of quantum theory which generalizes the action principle of classical mechanics. It replaces the classical notion of a single, unique trajectory for a system with a sum, or functional integral, over an infinity of possible trajectories to compute a quantum amplitude.

The basic idea of the path integral formulation can be traced back to Norbert Wiener, who introduced the Wiener integral for solving problems in diffusion and Brownian motion. This idea was extended to the use of the Lagrangian in quantum mechanics by P. A. M. Dirac in his 1933 paper. The complete method was developed in 1948 by Richard Feynman. Some preliminaries were worked out earlier, in the course of his doctoral thesis work with John Archibald Wheeler. The original motivation stemmed from the desire to obtain a quantum-mechanical formulation for the Wheeler-Feynman absorber theory using a Lagrangian (rather than a Hamiltonian) as a starting point.

This formulation has proven crucial to the subsequent development of theoretical physics, because it is manifestly symmetric between time and space. Unlike previous methods, the path-integral allows a physicist to easily change coordinates between very different canonical descriptions of the same quantum system.

The path integral also relates quantum and stochastic processes, and this provided the basis for the grand synthesis of the 1970s which unified quantum field theory with the statistical field theory of a fluctuating field near a second-order phase transition. The Schrödinger equation is a diffusion equation with an imaginary diffusion constant, and the path integral is an analytic continuation of a method for summing up all possible random walks. For this reason path integrals were used in the study of Brownian motion and diffusion a while before they were introduced in quantum mechanics.

The path integral formulation of quantum mechanics has been advanced in 1999 to the Lévy flights path integral. The original idea was to substitute the well-known Feynman path integral over Brownian-like paths with a path integral over Lévy-like quantum paths. The outcome of implementation of this idea is an alternative path integral approach to quantum mechanics, which results in a new fundamental quantum equation - the fractional Schrödinger equation.

Upon transition to imaginary time the fractional Schrödinger equation becomes fractional diffusion equation. For this reason the Lévy path integral is a powerful tool to study anomalous diffusion phenomena, fractional kinetics and stochastic processes with involvement of the Lévy flights. The Lévy path integral formulation in imaginary time domain brings a deep insight into fundamental relationship between statistical physics and stochastic processes that exhibit long-range dependency.