Intheoretical physics,thermal quantum field theory (thermal field theory for short) orfinite temperature field theory is a set of methods to calculate expectation values of physical observables of aquantum field theory at finitetemperature.

There are three main formalisms used to describe finite-temperature states:^[1]

• Matsubara formalism, based on evolving the system in imaginary time.
• Schwinger–Keldysh formalism, based on the real-time evolution, allowing the treatment of non-equilibrium processes.
• Umezawa formalism (thermo field dynamics), which is based on real-time evolution, and introduces a doubledHilbert space to represent thermal states.^[2]

In the Matsubara formalism, the basic idea (due toFelix Bloch^[3]) is that the expectation values of operators in acanonical ensemble

${\displaystyle ⟨A⟩=\frac{\text{Tr}[\exp (-\beta H)A]}{\text{Tr}[\exp (-\beta H)]}}$

may be written asexpectation values in ordinaryquantum field theory^[4] where the configuration is evolved by animaginary time${\displaystyle \tau =it(0\le \tau \le \beta )}$. One can therefore switch to aspacetime withEuclidean signature, where the above trace (Tr) leads to the requirement that allbosonic andfermionic fields be periodic and antiperiodic, respectively, with respect to the Euclidean time direction with periodicity${\displaystyle \beta =1/(kT)}$ (we are assumingnatural units${\displaystyle \hslash =1}$). This allows one to perform calculations with the same tools as in ordinary quantum field theory, such asfunctional integrals andFeynman diagrams, but with compact Euclidean time. Note that the definition of normal ordering has to be altered.^[5]

Inmomentum space, this leads to the replacement of continuous frequencies by discrete imaginary (Matsubara) frequencies${\displaystyle v_n=n/\beta }$ and, through thede Broglie relation, to a discretized thermal energy spectrum${\displaystyle E_n=2n\pi kT}$. This has been shown to be a useful tool in studying the behavior of quantum field theories at finite temperature.^[6][7][8][9]

It has been generalized to theories with gauge invariance and was a central tool in the study of a conjectured deconfiningphase transition ofYang–Mills theory.^[10][11] In this Euclidean field theory, real-time observables can be retrieved byanalytic continuation.^[12] The Feynman rules for gauge theories in the Euclidean time formalism, were derived by C. W. Bernard.^[10]

An alternative approach which is of interest tomathematical physics is to work withKMS states.

A path-ordered approach to real-time formalisms includes theSchwinger–Keldysh formalism and more modern variants.^[13] It involves replacing a straight time contour from (large negative) real initial time${\displaystyle t_i}$ to${\displaystyle t_i-i\beta }$ by one that first runs to (large positive) real time${\displaystyle t_f}$ and then suitably back to${\displaystyle t_i-i\beta }$.^[14] In fact all that is needed is one section running along the real time axis, as the route to the end point,${\displaystyle t_i-i\beta }$, is less important.^[15] Thepiecewise composition of the resulting complex time contour leads to a doubling of fields and more complicated Feynman rules, but obviates the need ofanalytic continuations of the imaginary-time formalism.

As well as Feynman diagrams and perturbation theory, other techniques such as dispersion relations and the finite temperature analog ofCutkosky rules can also be used in the real time formulation.^[16][17]

The alternative approach to real-time formalisms is an operator based approach usingBogoliubov transformations, known asUmezawa formalism orthermo field dynamics.^[2][18]

• Matsubara summation
• Polyakov loop
• Quantum thermodynamics
• Quantum statistical mechanics

• Quantum field theory
• Statistical mechanics

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