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As in standard many-body textbook (at least in my class), we use real-time green function when temperatures is zero, and we use imaginary-time green function when the temperature is finite.

My question is: Is this just a convenient definition or is there some other reasons that we have to use different green functions?

To be more specific, what I mean by “convenient” is that, when the temperature is finite, we have to compute the partition function to get the expectation value, but the evolution operator $exp [-i H t]$ can not be combined with the partition function factor $exp[- \beta H]$ so we can not conveniently compute the result. Is this $the \, reason$ ?

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It is indeed a matter of convenience. However, as I noted in the answer to your other question, non-equilibrium statistical mechanics makes use of the Green's functions with a time-contour that has both real and imaginary time parts. It is a somewhat advanced topic, however, there is rich literature on the subject: Kadanoff & Baym's book, the book by Jauho, the review by Rammer and Smith, and the path integral formulation here.

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