Thermal quantum field theory

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

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In the Matsubara (imaginary-time) formalism, the basic idea (due to Felix Bloch) is that the expectation values of operators in a thermal ensemble

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

may be written as expectation values in ordinary quantum field theory where the configuration is evolved by an imaginary time $t = i β$ . 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 $β = 1 / (k T)$ . 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. In momentum space, this leads to the replacement of continuous frequencies by discrete imaginary (Matsubara) frequencies. Real-time observables can be retrieved by analytic continuation, or by using instead a so-called real-time formalism. The latter involves replacing a straight time contour from (large negative) real initial time $t i$ to $t i - i β$ by one that first runs to (large positive) real time $t f$ and then suitably back to $t i - i β$ . 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.