Imaginary-time Green's Function and Nonequilibrium Green's Function I've been learning Green's Function approach recently, following Radi Jishi's book. It says that the imaginary-time Green's function (Matsubara function) cannot be used in non-equilibrium situations, which makes me wonder why because I don't see where we impose the equilibrium condition when we derive the perturbative expansion for Matsubara function. I do understand that the contour-ordered Green's function can deal with non-equilibrium, but can someone tell me why we usually don't apply Matsubara function formalism to studying non-equilibrium phenomena? Thanks a lot!
 A: Right after I post this question, I found a book by Rammer that talks about this question. Here is the excerpt:
"The imaginary-time Green’s functions can also be used to study non-equilibrium
states by letting the external potential depend on the imaginary time. The Matsubara technique is then a bit cumbersome, but can be used to derive exact equations,
say, the Dyson equation for real-time Green’s functions. In fact this was the method
used originally to study non-equilibrium superconductivity in the quasi-classical approximation. However, for general non-equilibrium situations, the necessary
analytical continuation in arbitrarily many Matsubara frequencies becomes nontrivial
(and are usually left out of textbooks), and are more involved than using the realtime technique. Furthermore, when approximations are made, the real-time results
obtained upon analytical continuation can be spurious. However, the main disadvantage of the imaginary-time formalism is that it lacks physical transparency."
Hope this can be helpful to those who share the same confusion as I did.
