Editing Path integral formulation (section)

{{Short description|Formulation of quantum mechanics}}
{{about|a formulation of quantum mechanics|integrals along a path, also known as line or contour integrals|line integral}}
{{Quantum mechanics|cTopic=Formulations}}
The '''path integral formulation''' is a description in [[quantum mechanics]] that generalizes the [[stationary action principle]] of [[classical mechanics]].  It replaces the classical notion of a single, unique classical trajectory for a system with a sum, or [[functional integral]], over an infinity of quantum-mechanically possible trajectories to compute a [[probability amplitude|quantum amplitude]].

This formulation has proven crucial to the subsequent development of [[theoretical physics]], because manifest [[Lorentz covariance]] (time and space components of quantities enter equations in the same way) is easier to achieve than in the operator formalism of [[canonical quantization]]. Unlike previous methods, the path integral allows one to easily change [[coordinates]] between very different [[canonical coordinates|canonical]] descriptions of the same quantum system. Another advantage is that it is in practice easier to guess the correct form of the [[Lagrangian (field theory)|Lagrangian]] of a theory, which naturally enters the path integrals (for interactions of a certain type, these are ''coordinate space'' or ''Feynman path integrals''), than the [[Hamiltonian (quantum mechanics)|Hamiltonian]]. Possible downsides of the approach include that [[unitarity]] (this is related to conservation of probability; the probabilities of all physically possible outcomes must add up to one) of the [[S-matrix]] is obscure in the formulation. The path-integral approach has proven to be equivalent to the other formalisms of quantum mechanics and quantum field theory. Thus, by ''deriving'' either approach from the other, problems associated with one or the other approach (as exemplified by Lorentz covariance or unitarity) go away.<ref>{{harvnb|Weinberg|2002|loc=Chapter 9.}}</ref>

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]]. In physics, it is 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.

[[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).]]