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| In [[mathematical physics]], a '''caloron''' is the finite temperature generalization of an [[instanton]].
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| ==Finite temperature and instantons==
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| At zero temperature, instantons are the name given to solutions of the classical [[field equation|equations of motion]] of the Euclidean version of the theory under consideration, and which are furthermore localized in Euclidean [[spacetime]]. They describe [[quantum tunneling|tunneling]] between different topological [[vacuum state]]s of the Minkowski theory. One important example of an instanton is the [[BPST instanton]], discovered in 1975 by [[Alexander Belavin|Belavin]], [[Alexander Markovich Polyakov|Polyakov]], [[Albert Schwarz|Schwartz]] and [[Yu. S. Tyupkin|Tyupkin]].<ref>{{cite journal
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| | last = Belavin | first = A
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| | authorlink = Alexander Belavin
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| | coauthors = [[Alexander Markovich Polyakov|Polyakov]], [[Albert Schwarz|Albert Schwartz]] and [[Yu. S. Tyupkin|Tyupkin]]
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| | title = Pseudoparticle solutions of the Yang–Mills equations
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| | journal = [[Physics Letters B]]
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| | volume = 59 | issue = 1 | pages = 85
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| | year = 1975
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| | doi = 10.1016/0370-2693(75)90163-X
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| |bibcode = 1975PhLB...59...85B }}</ref> This is a [[topology|topologically]] stable solution to the four-dimensional SU(2) [[Yang–Mills theory|Yang–Mills]] field equations in Euclidean spacetime (i.e. after [[Wick rotation]]).
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| Finite temperatures in quantum field theories are modeled by compactifying the imaginary (Euclidean) time (see [[thermal quantum field theory]]).<ref>See {{Harvcoltxt|Das|1997}} for a derivation of this formalism.</ref> This changes the overall structure of spacetime, and thus also changes the form of the instanton solutions. At finite temperature, the Euclidean time dimension is periodic{{why|date=September 2012}}, which means that instanton solutions have to be periodic as well.
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| ==In SU(2) Yang–Mills theory==
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| In SU(2) [[Yang–Mills theory]] at zero temperature, the instantons have the form of the [[BPST instanton]]. The generalization thereof to finite temperature has been found by Harrington and Shepard:<ref>{{cite journal| last = Harrington | coauthors = Shepard | year = 1978 | title = Periodic Euclidean Solutions and the Finite Temperature Yang–Mills Gas | journal = [[Physical Review D]] |volume=17 | issue = 8 | page = 2122 | doi = 10.1103/PhysRevD.17.2122 | first1 = Barry|bibcode = 1978PhRvD..17.2122H }}</ref>
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| :<math> A_\mu^a(x) = \bar\eta_{\mu\nu}^a \Pi(x) \partial_\nu \Pi^{-1}(x) \quad\text{with} \quad \Pi(x) = 1+\frac{\pi\rho^2T}r \frac{\sinh(2\pi rT)}{\cosh(2\pi rT)-\cos(2\pi rT)} \ ,</math>
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| where <math>\bar\eta_{\mu\nu}^a</math> is the anti-[['t Hooft symbol]], ''r'' is the distance from the point ''x'' to the center of the caloron, ''ρ'' is the size of the caloron, and ''T'' is the temperature. This solution was found based on a periodic multi-instanton solution first suggested by [[Gerardus 't Hooft|'t Hooft]]<ref>{{Harvcoltxt|Shifman|1994|p=122}}</ref> and published by [[Edward Witten|Witten]].<ref>{{cite journal | last = Witten | authorlink = Edward Witten | journal = [[Physical Review Letters]] | volume = 38 | issue = 3 | year = 1977 | pages = 121 | title = Some Exact Multi-Instanton Solutions of Classical Yang–Mills Theory | doi = 10.1103/PhysRevLett.38.121 | first1 = Edward | bibcode=1977PhRvL..38..121W}}</ref>
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| ==References and notes==
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| {{reflist}}
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| ==Bibliography==
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| *{{cite book
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| |last = Das
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| |first = Ashok
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| |title = Finite Temperature Field Theory
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| |publisher = [[World Scientific]]
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| |year = 1997
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| |isbn = 981-02-2856-2 |ref = harv}}
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| *{{cite book
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| |last = Shifman
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| |title = Instantons in Gauge Theory
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| |publisher = [[World Scientific]]
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| |year = 1994
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| |isbn = 981-02-1681-5 |ref = harv}}
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| *{{cite journal|author1=Dmitri Diakonov|author2=Nikolay Gromov|doi=10.1103/PhysRevD.72.025003|title=SU(N) caloron measure and its relation to instantons|year=2005|volume=72|issue=2|pages=025003|journal=[[Physical Review D]]|arxiv=hep-th/0502132|bibcode = 2005PhRvD..72b5003D }}
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| *{{cite arxiv|eprint=hep-th/0511125|author1=Daniel Nogradi|title=Multi-calorons and their moduli|class=hep-th|year=2005}}
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| *{{cite arxiv|eprint=hep-th/0609019|author1=Shnir|title=Self-dual and non-self dual axially symmetric caloron solutions in SU(2) Yang-Mills theory|class=hep-th|year=2006}}
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| *{{cite journal|author1=Philipp Gerhold|author2=Ernst-Michael Ilgenfritz|author3=Michael Müller-Preussker|doi=10.1016/j.nuclphysb.2007.04.003|year=2007|title=Improved superposition schemes for approximate multi-caloron configurations|pages=268–297|volume=774|journal=[[Nuclear Physics B]]|arxiv=hep-ph/0610426|bibcode = 2007NuPhB.774..268G }}
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| [[Category:Quantum field theory]]
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