Thermal quantum field theory

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

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

may be written as

${\displaystyle \tau =it(0\leq \tau \leq \beta )}$. One can therefore switch to a fields be periodic and antiperiodic, respectively, with respect to the Euclidean time direction with periodicity ${\displaystyle \beta =1/(kT)}$ (we are assuming natural units ${\displaystyle \hbar =1}$). This allows one to perform calculations with the same tools as in ordinary quantum field theory, such as In

momentum space

, this leads to the replacement of continuous frequencies by discrete imaginary (Matsubara) frequencies ${\displaystyle v_{n}=n/\beta }$ and, through the

de 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.[4]^{[5][6] [7]} 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.^[8][9] In this Euclidean field theory, real-time observables can be retrieved by analytic continuation.^[10] The Feynman rules for gauge theories in the Euclidean time formalism, were derived by C. W. Bernard.^[8]    

The Matsubara formalism, also referred to as imaginary time formalism, can be extended to systems with thermal variations.^[11][12] 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.^[11][12]

The alternative to the use of fictitious imaginary times is to use a real-time formalism which come in two forms.^[13] A path-ordered approach to real-time formalisms includes the Schwinger–Keldysh formalism and more modern variants.^[14] The latter 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 ${\displaystyle t_{i}-i\beta }$.^[15] 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.^[16] 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.^[13][17] 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.^[18][19]

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

See also

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

References

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