Stochastic quantization: Difference between revisions

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In [[theoretical physics]], '''stochastic quantization''' is a method for modelling [[quantum mechanics]], introduced by [[Edward Nelson]] in 1966,<ref>{{Cite journal | last1 = Nelson | first1 = E. | title = Derivation of the Schrödinger Equation from Newtonian Mechanics | doi = 10.1103/PhysRev.150.1079 | journal = Physical Review | volume = 150 | issue = 4 | pages = 1079–1085 | year = 1966 |bibcode = 1966PhRv..150.1079N }}</ref><ref>{{Cite journal | last1 = Fényes | first1 = I. |author-link1= Imre Fényes| doi = 10.1007/BF01338578 |bibcode=1952ZPhy..132...81F| title = Eine wahrscheinlichkeitstheoretische Begründung und Interpretation der Quantenmechanik| journal = Zeitschrift für Physik| volume = 132 | pages = 81–106| year = 1952 | issue = 1 | s2cid = 119581427 }}</ref><ref>{{Cite journal | last1 = De La Peña-Auerbach | first1 = L. |author-link1=Luis de la Peña| title = A simple derivation of the Schroedinger equation from the theory of Markoff processes | doi = 10.1016/0375-9601(67)90639-1 | journal = Physics Letters A | volume = 24 | issue = 11 | pages = 603–604 | year = 1967 |bibcode = 1967PhLA...24..603D }}</ref> and streamlined by [[Giorgio Parisi]] and Yong-Shi Wu.<ref name="Parisi">{{cite journal|last = Parisi|first = G|year = 1981|title = Perturbation theory without gauge fixing|journal = Sci. Sinica|volume = 24|pages = 483|author2 = Y.-S. Wu}}</ref>

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==Description==

In physics, '''stochastic quantization''' is a method for modelling quantum mechanics, interoduced by Edward Nelson in 1966,<ref>{{cite doi|10.1103/PhysRev.150.1079|noedit}}; {{cite doi|10.1007/BF01338578|noedit}}; {{cite doi|10.1016/0375-9601(67)90639-1|noedit}}</ref> and streamlined by [[Giorgio Parisi|Parisi]] and Wu.<ref name=Parisi>{{cite journal|last=Parisi|first=G|coauthors=Y.-S. Wu|journal=Sci. Sinica|year=1981|volume=24|pages=483}}</ref>

ItStochastic quantization serves to quantize [[Euclidean field theory|Euclidean field theories]],<ref name=damgaard>{{cite journal|last=DAMGAARDDamgaard |first=Poul|coauthorsauthor2=Helmuth HUFFELHuffel |title=STOCHASTICStochastic Quantization QUANTIZATION|journal=Physics Reports|year=1987|volume=152|issue=5&6|pages=227-398227–398|url=http://homepage.univie.ac.at/helmuth.hueffel/PhysRep.pdf|accessdateaccess-date=8 March 2013|bibcode = 1987PhR...152..227D |doi = 10.1016/0370-1573(87)90144-X |hdl=1721.1/3101|hdl-access=free}}</ref> and is used for numerical applications, such as [[numerical simulation]]s of [[Gauge theory|gauge theories]] with [[fermion]]s. This serves to address the problem of [[fermion doubling]] that usually occurs in these numerical calculations.

It serves to quantize [[Euclidean field theory|Euclidean field theories]],<ref name=damgaard>{{cite journal|last=DAMGAARD|first=Poul|coauthors=Helmuth HUFFEL|title=STOCHASTIC QUANTIZATION|journal=Physics Reports|year=1987|volume=152|issue=5&6|pages=227-398|url=http://homepage.univie.ac.at/helmuth.hueffel/PhysRep.pdf|accessdate=8 March 2013}}</ref> and is used for numerical applications, such as [[numerical simulation]]s of [[Gauge theory|gauge theories]] with [[fermion]]s.

Stochastic quantization takes advantage of the fact that a Euclidean quantum field theory can be modeled as the [[Thermodynamic equilibrium|equilibrium limit]] of a [[statistical mechanics|statistical mechanical system]] coupled to a [[heat bath]]. In particular, in the [[Path integral formulation|path integral]] representation of a Euclidean quantum field theory, the path integral measure is closely related to the [[Boltzmann distribution]] of a statistical mechanical system in equilibrium. In this relation, Euclidean [[Green's functions]] become [[correlation function]]s in the statistical mechanical system. A statistical mechanical system in equilibrium can be modeled, via the [[ergodic hypothesis]], as the [[stationary distribution]] of a [[stochastic process]]. Then the Euclidean path integral measure can also be thought of as the stationary distribution of a stochastic process; hence the name stochastic quantization.

== See also ==

* [[Supersymmetric theory of stochastic dynamics]]

* [[Stochastic quantum mechanics]]

==References==

{{reflist}}

[[Category:Stochastic processes]]

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[[Category:Stochastic processes]]