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In every textbook, the path integral of the Green's function is written in imaginary-time. I wonder whether we could write real-time green function in the path integral form.

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You mean the (equilibrium) statistical mechanics textbooks, which deal mainly with the statistical sum, and therefore need only the imaginary path integral.

The real-time path integral is usually derived in quantum mechanics textbooks, where it is sometimes also shown to be the direct solution of the Schrödinger equation with a source term, i.e. a true Green's function in the mathematical sense. Feymann's own QM book is probably a good place to look.

Finally, non-equilibrium statistical mechanics routinely uses Green's functions in real time. For the path integral formulation of the Keldysh formalism you may look here.

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PS: I notice that if we can distort the time integral line (to include imaginary part), then it is possible to establish the path integral formulation of real-time green function (at least for fermions). So I edited my question so that the time integral has to be defined on real axis.

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All right after discussing with professor, I’ll answer the question myself. The crucial point is that the real-time green function is defined on zero temperature, so the contribution to the two point function will only come from the ground state, and the factor $e^{-\beta H}$ will be thrown away. As a result, the remaining time ordered operator will appear as some evolution operators not in a time ordered place. (Without considering time contour as in @Vadim answer or other more complicated situations) So it can not be written as a path integral.

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