In the Matsubara formalism, the basic idea (due to Felix Bloch[1]) is that the expectation values of operators in a canonical ensemble may be written as expectation values in ordinary quantum field theory[2] where the configuration is evolved by an imaginary time
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
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.
and, through the de Broglie relation, to a discretized thermal energy spectrum
This has been shown to be a useful tool in studying the behavior of quantum field theories at finite temperature.
[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.
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.
[15] In fact all that is needed is one section running along the real time axis, as the route to the end point,
[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.
[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.