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.

In the Matsubara (imaginary-time) formalism,the basic idea (due to Felix Bloch) is that the expectation values of operators in a thermal ensemble:may be written as expectation values in ordinary quantum field theory where the configuration is evolved by an imaginary time . One can therefore switch toa spacetime with Euclidean signature, where theabove trace (Tr) leads to the requirement that all bosonic and fermionicfields be periodic and antiperiodic, respectively, with respect tothe Euclidean time direction with periodicity . This allowsone 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 bydiscrete imaginary (Matsubara) frequencies. Real-time observables can be retrievedby analytic continuation, or by using instead a so-called real-time formalism.The latter involves replacing a straight time contour from (large negative) realinitial time $t\_i$ to by one that first runs to (large positive) real time $t\_f$ and then suitably back to . The piecewise compositionof the resulting complex time contour leads to a doubling of fields and more complicatedFeynman rules, but obviates the need of analytic continuations of the imaginary-time formalism.

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