Thermal quantum field theory

{{Short description|Quantum field theory at non-zero temperatures}}

In theoretical physics, thermal quantum field theory (thermal field theory for short) or finite temperature field theory is a set of methods to calculate expectation values of physical observables of a quantum field theory at finite temperature.

In the Matsubara formalism, the basic idea (due to Felix Bloch{{Cite journal

|author=Bloch, F.

|title=Zur Theorie des Austauschproblems und der Remanenzerscheinung der Ferromagnetika

|journal=Z. Phys.

|volume=74

|issue=5–6

|pages=295–335

|year=1932

|doi=10.1007/BF01337791

|bibcode = 1932ZPhy...74..295B |s2cid=120549836

}}) is that the expectation values of operators in a canonical ensemble

:$\backslash langle\; A\backslash rangle=\backslash frac\{\backslash mbox\{Tr\}\backslash ,\; [\backslash exp(-\backslash beta\; H)\; A]\}\{\backslash mbox\{Tr\}\backslash ,\; [\backslash exp(-\backslash beta\; H)]\}$

may be written as expectation values in ordinary quantum field theory{{Cite book|author=Jean Zinn-Justin|title=Quantum Field Theory and Critical Phenomena|publisher=Oxford University Press|year=2002|isbn=978-0-19-850923-3}} where the configuration is evolved by an imaginary time $\backslash tau\; =\; i\; t(0\backslash leq\backslash tau\backslash leq\backslash beta)$. One can therefore switch to a spacetime with Euclidean signature, where the above trace (Tr) leads to the requirement that all bosonic and fermionic fields be periodic and antiperiodic, respectively, with respect to the Euclidean time direction with periodicity $\backslash beta\; =\; 1/(kT)$ (we are assuming natural units $\backslash hbar\; =\; 1$). This allows one to perform calculations with the same tools as in ordinary quantum field theory, such as functional integrals and Feynman diagrams, but with compact Euclidean time. Note that the definition of normal ordering has to be altered.{{Cite journal

|author=T.S. Evans and D.A. Steer

|title= Wick's theorem at finite temperature

|journal=Nucl. Phys. B

|volume=474

|issue=2

|pages=481–496

|year=1996

|doi=10.1016/0550-3213(96)00286-6

|arxiv = hep-ph/9601268 |bibcode = 1996NuPhB.474..481E |s2cid= 119436816

}}

In momentum space, this leads to the replacement of continuous frequencies by discrete imaginary (Matsubara) frequencies $v\_n\; =\; n\; /\; \backslash beta$ and, through the de Broglie relation, to a discretized thermal energy spectrum $E\_n\; =\; 2\; n\; \backslash pi\; k\; T$. This has been shown to be a useful tool in studying the behavior of quantum field theories at finite temperature.

D.A. Kirznits JETP Lett. 15 (1972) 529.

D.A. Kirznits and A.D. Linde, Phys. Lett. B42 (1972) 471; it Ann. Phys.

101 (1976) 195.{{Cite journal

|author=Weinberg, S.

|title= Gauge and Global Symmetries at High Temperature

|journal=Phys. Rev. D

|volume=9

|issue=12

|pages=3357–3378

|year=1974

|doi=10.1103/PhysRevD.9.3357

|bibcode = 1974PhRvD...9.3357W }}{{Cite journal

|author=L. Dolan, and R. Jackiw

|title=Symmetry behavior at finite temperature

|journal=Phys. Rev. D

|volume=9

|issue=12

|pages=3320–3341

|year=1974

|doi=10.1103/PhysRevD.9.3320

|bibcode = 1974PhRvD...9.3320D }}

It has been generalized to theories with gauge invariance and was a central tool in the study of a conjectured deconfining phase transition of Yang–Mills theory.

C. W. Bernard, Phys. Rev. D9 (1974) 3312.

D.J. Gross, R.D. Pisarski and L.G. Yaffe, Rev. Mod. Phys. 53 (1981) 43. In this Euclidean field theory, real-time observables can be retrieved by analytic continuation.{{Cite journal

|author=T.S. Evans

|title= N-Point Finite Temperature Expectation Values at Real Times

|journal=Nucl. Phys. B

|volume=374

|issue=2

|pages=340–370

|year=1992

|doi=10.1016/0550-3213(92)90357-H

|arxiv=hep-ph/9601268

|bibcode = 1992NuPhB.374..340E |s2cid= 120072328

}} The Feynman rules for gauge theories in the Euclidean time formalism, were derived by C. W. Bernard.    

The Matsubara formalism, also referred to as imaginary time formalism, can be extended to systems with thermal variations.{{Cite journal |last=S. Ganesh |date=2022 |title=Quantum theory, thermal gradients and the curved Euclidean space |url=https://doi.org/10.1142/S0217751X22501251 |journal=International Journal of Modern Physics A |volume=37 |issue=17 |doi= 10.1142/S0217751X22501251|arxiv=2206.13324|bibcode=2022IJMPA..3750125G |s2cid= 250073218}} In this approach, the variation in the temperature is recast as a variation in the Euclidean metric. Analysis of the partition function leads to an equivalence between thermal variations and the curvature of the Euclidean space.{{Cite journal |last=Ganesh |first=S |date=2023-02-16 |title=5D thermal field theory, Einstein field equations and spontaneous symmetry breaking |url=https://iopscience.iop.org/article/10.1088/1361-6382/acb24c |journal=Classical and Quantum Gravity |volume=40 |issue=4 |pages=045008 |arxiv=2301.04827v1 |doi=10.1088/1361-6382/acb24c |issn=0264-9381}}

The alternative to the use of fictitious imaginary times is to use a real-time formalism which come in two forms.{{Cite journal

|author=N.P. Landsman and Ch.G. van Weert

|title= Real- and imaginary-time field theory at finite temperature and density

|journal=Physics Reports

|volume=145

|issue=3–4

|pages=141–249

|year=1987

|doi=10.1016/0370-1573(87)90121-9

|bibcode = 1987PhR...145..141L }} A path-ordered approach to real-time formalisms includes the Schwinger–Keldysh formalism and more modern variants.{{Cite journal|author=A.J. Niemi, G.W. Semenoff|title=Finite Temperature Quantum Field Theory in Minkowski Space|journal=Annals of Physics|volume=152|issue=1|pages=105–129|year=1984|doi=10.1016/0003-4916(84)90082-4|bibcode = 1984AnPhy.152..105N }} The latter involves replacing a straight time contour from (large negative) real initial time $t\_i$ to $t\_i\; -\; i\backslash beta$ by one that first runs to (large positive) real time $t\_f$ and then suitably back to $t\_i\; -\; i\backslash beta$.{{Cite arXiv

|author=Zinn-Justin, Jean

|title=Quantum field theory at finite temperature: An introduction

|year=2000

|eprint=hep-ph/0005272

}} In fact all that is needed is one section running along the real time axis, as the route to the end point, $t\_i\; -\; i\backslash beta$, is less important.{{Cite journal

|author=T.S. Evans

|title= New Time Contour for Equilibrium Real-Time Thermal Field-Theories

|journal=Phys. Rev. D

|volume=47

|issue=10

|pages=R4196–R4198

|year=1993

|doi=10.1103/PhysRevD.47.R4196

|pmid= 10015491

|arxiv = hep-ph/9310339 |bibcode = 1993PhRvD..47.4196E |s2cid= 119486408

}} The piecewise composition of the resulting complex time contour leads to a doubling of fields and more complicated Feynman rules, but obviates the need of analytic continuations of the imaginary-time formalism. The alternative approach to real-time formalisms is an operator based approach using Bogoliubov transformations, known as thermo field dynamics.{{Cite journal

|author1=H. Chiu

|author2=H. Umezawa

|authorlink2=Hiroomi Umezawa

|title= A unified formalism of thermal quantum field theory

|journal=International Journal of Modern Physics A

|volume=9

|issue=14

|pages=2363 ff

|year=1993

|doi=10.1142/S0217751X94000960

|bibcode = 1994IJMPA...9.2363C}}

As well as Feynman diagrams and perturbation theory, other techniques such as dispersion relations and the finite temperature analog of Cutkosky rules can also be used in the real time formulation.{{Cite journal|author=R.L. Kobes, G.W. Semenoff|title=Discontinuities of Green Functions in Field Theory at Finite Temperature and Density|journal=Nucl. Phys. B|volume=260|issue=3–4|pages=714–746|year=1985|doi=10.1016/0550-3213(85)90056-2|bibcode = 1985NuPhB.260..714K }}{{Cite journal|author=R.L. Kobes, G.W. Semenoff|title=Discontinuities of Green Functions in Field Theory at Finite Temperature and Density |journal=Nucl. Phys. B|volume=272|issue=2|pages=329–364|year=1986|doi=10.1016/0550-3213(86)90006-4|bibcode = 1986NuPhB.272..329K }}

An alternative approach which is of interest to mathematical physics is to work with KMS states.