Caloron

At zero temperature, instantons are the name given to solutions of the classical equations of motion of the Euclidean version of the theory under consideration, and which are furthermore localized in Euclidean spacetime. They describe tunneling between different topological vacuum states of the Minkowski theory. One important example of an instanton is the BPST instanton, discovered in 1975 by Alexander Belavin, Alexander Markovich Polyakov, Albert Schwartz and Yu S. Tyupkin.{{cite journal

| last = Belavin | first = A

| author-link = Alexander Belavin

|author2=Polyakov

|author2-link=Alexander Markovich Polyakov

|author3=Albert Schwartz

|author3-link=Albert Schwarz

|author4=Tyupkin |author4-link=Yu. S. Tyupkin | title = Pseudoparticle solutions of the Yang–Mills equations

| journal = Physics Letters B

| volume = 59 | issue = 1 | pages = 85

| year = 1975

| doi = 10.1016/0370-2693(75)90163-X

|bibcode = 1975PhLB...59...85B }} This is a topologically stable solution to the four-dimensional SU(2) Yang–Mills field equations in Euclidean spacetime (i.e. after Wick rotation).

Finite temperatures in quantum field theories are modeled by compactifying the imaginary (Euclidean) time (see thermal quantum field theory).See {{Harvcoltxt|Das|1997}} for a derivation of this formalism. This changes the overall structure of spacetime, and thus also changes the form of the instanton solutions. According to the Matsubara formalism, at finite temperature, the Euclidean time dimension is periodic, which means that instanton solutions have to be periodic as well.

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:{{cite journal| last1 = Harrington |author2=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 }}

:$A\_\backslash mu^a(x)\; =\; \backslash bar\backslash eta\_\{\backslash mu\backslash nu\}^a\; \backslash Pi(x)\; \backslash partial\_\backslash nu\; \backslash Pi^\{-1\}(x)\; \backslash quad\backslash text\{with\}\; \backslash quad\; \backslash Pi(x)\; =\; 1+\backslash frac\{\backslash pi\backslash rho^2T\}r\; \backslash frac\{\backslash sinh(2\backslash pi\; rT)\}\{\backslash cosh(2\backslash pi\; rT)-\backslash cos(2\backslash pi\; \backslash tau\; T)\}\; \backslash \; ,$

where $\backslash bar\backslash eta\_\{\backslash mu\backslash nu\}^a$ 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, $\backslash tau$ is the Euclidean time and T is the temperature. This solution was found based on a periodic multi-instanton solution first suggested by Gerard 't Hooft{{Harvcoltxt|Shifman|1994|p=122}} and published by Edward Witten.{{cite journal | last1 = Witten | author-link = 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}}

{{Reflist}}

• {{cite book

|last = Das

|first = Ashok

|title = Finite Temperature Field Theory

|publisher = World Scientific

|year = 1997

|isbn = 981-02-2856-2 }}

• {{cite book

|last = Shifman

|title = Instantons in Gauge Theory

|publisher = World Scientific

|year = 1994

|isbn = 981-02-1681-5 }}

• {{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 |s2cid=119496217}}
• {{cite arXiv|eprint=hep-th/0511125|author1=Daniel Nogradi|title=Multi-calorons and their moduli|year=2005}}
• {{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|year=2006}}
• {{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|issue=1–3|arxiv=hep-ph/0610426|bibcode = 2007NuPhB.774..268G |s2cid=119471511}}

Category:Gauge theories

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