In quantum field theory, the Feynman/time ordered Green's function takes the form $$D_F(p) \sim \frac{1}{p^2 - m^2 + i \epsilon}$$ and the $i \epsilon$ reflects the fact that the Green's function is not unique, but must be fixed by boundary conditions. Different choices of boundary conditions yield different placements of the $i \epsilon$, such as $$\frac{1}{(p_0-i\epsilon + \mathbf{p})^2 - m^2}, \quad \frac{1}{(p_0+i\epsilon + \mathbf{p})^2 - m^2}, \quad \frac{1}{p^2 - m^2 - i \epsilon}$$ which correspond to retarded, advanced, and "anti-time ordered". The placement of the $i \epsilon$ determines things like the allowed direction of Wick rotation.
Quantum mechanics is simply a quantum field theory in zero spatial dimensions, so everything here should also apply to quantum mechanics. But the correspondence is a bit obscure. At the undergraduate level, Green's functions just don't come up in quantum mechanics, because we usually talk about the propagator $$K(x_f, t_f, x_i, t_i) = \langle x_f | U(t_f, t_i) | x_i \rangle$$ instead (see here, here for propagators vs. Green's functions, which are often confused). At this level it's not as useful to talk about a Green's function, because we don't add source terms to the Schrodinger equation like we would for fields.
In a second course on quantum mechanics, we learn about scattering theory, where Green's functions are useful. But some of them are missing. The four choices in quantum field theory come from the fact that there are both positive and negative frequency solutions to relativistic wave equations like the Klein-Gordan equation, but in quantum mechanics this doesn't hold: for a positive Hamiltonian the Schrodinger equation has only positive frequency/energy solutions. So it looks like there are only two Green's functions, the retarded and advanced one, which are indeed the two commonly encountered (see here). These correspond to adding an infinitesimal damping forward and backward in time.
Are there four independent Green's functions in quantum mechanics or just two? What happened to the Feynman Green's function? Can one define it, and if so what is it good for?
Edit: see here for a very closely related question.
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