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 |access-date=2018-12-15 |archive-date=2017-08-12 |archive-url=https://web.archive.org/web/20170812032227/http://www.gc.cuny.edu/CUNY_GC/media/CUNY-Graduate-Center/PDF/ITS/Vinokur_Spring2015.pdf |url-status=dead }}</ref>

The path integral has impacted a wide array of sciences, including [[polymer physics]], quantum field theory, [[string theory]] and [[cosmology]]. BornIn out ofphysics, it areis a foundation for [[lattice gauge theory]] and [[quantum chromodynamics]].<ref name=":02" /> It has been called the "most powerful formula in physics",<ref>{{Cite web |last=Wood |first=Charlie |date=2023-02-06 |title=How Our Reality May Be a Sum of All Possible Realities |url=https://www.quantamagazine.org/how-our-reality-may-be-a-sum-of-all-possible-realities-20230206/ |access-date=2024-06-21 |website=[[Quanta Magazine]]}}</ref> with [[Stephen Wolfram]] also declaring it to be the "fundamental mathematical construct of modern quantum mechanics and quantum field theory".<ref>{{Cite web |last=Wolfram |first=Stephen |author-link=Stephen Wolfram |date=2020-04-14 |title=Finally We May Have a Path to the Fundamental Theory of Physics… and It’s Beautiful |url=https://writings.stephenwolfram.com/2020/04/finally-we-may-have-a-path-to-the-fundamental-theory-of-physics-and-its-beautiful/ |access-date=2024-06-21 |website=writings.stephenwolfram.com |language=en}}</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]], whose 1933 paper gave birth to path integral formulation.<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><ref name=":02">{{cite arXiv |eprint=2003.12683 |class=physics.hist-ph |first=N. D. |last=Hari Dass |title=Dirac and the Path Integral |date=2020-03-28}}</ref> The complete method was developed in 1948 by [[Richard Feynman]].{{sfn|Feynman|1948}} 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.