# Can the real-time Green's function be written in the form of path integral on the real axis? [closed]

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.

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.