Path integral formulation: Difference between revisions

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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 walk]]s.<ref>{{cite web |last=Vinokur |first=V. M. |date=2015-02-27 |url=https://www.gc.cuny.edu/CUNY_GC/media/CUNY-Graduate-Center/PDF/ITS/Vinokur_Spring2015.pdf |title=Dynamic Vortex Mott Transition}}</ref>

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]].<ref>{{harvnb|Chaichian|Demichev|2001}}</ref> This idea was extended to the use of the [[Lagrangian (field theory)|Lagrangian]] in quantum mechanics by [[Paul Dirac]], who gave the ideas that would lead to path integral formulation in his 1933 article.<ref>{{harvnb|Dirac|1933}}</ref><ref>{{harvnb|Van Vleck|1928}}</ref><ref name=":0">{{cite arXiv |eprint=1004.3578 |class=physics.hist-ph |first=Jeremy |last=Bernstein |title=Another Dirac |date=2010-04-20}}</ref> The complete method was developed in 1948 by [[Richard Feynman]].<ref>{{Cite journal |last=Feynman |first=R. P. |date=1948-04-01 |title=Space-Time Approach to Non-Relativistic Quantum Mechanics |url=https://link.aps.org/doi/10.1103/RevModPhys.20.367 |journal=Reviews of Modern Physics |volume=20 |issue=2 |pages=367–387 |doi=10.1103/RevModPhys.20.367}}</ref>. Some preliminaries were worked out earlier in his doctoral work under the supervision of [[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 (field theory)|Lagrangian]] (rather than a [[Hamiltonian (quantum mechanics)|Hamiltonian]]) as a starting point.

[[File:Feynman paths.png|alt=|thumb|These are five of the infinitely many paths available for a particle to move from point A at time t to point B at time t’(>t). Paths which self-intersect or go backwards in time are not allowed.]]