Relation between Real and Imaginary Time Response Functions

In the last chapter we used the Hartree approximation to describe nonequilibrium phenomena. Unfortunately, we cannot directly write more complicated approximations in the real time domain because we have no simple boundary conditions that can act as a guide in determining g₂(U). Therefore, we have, at this stage, no complete theory for determining the physical response function g(U). [As we saw in Chapter 4, simple physical arguments do not suffice to determine approximations for the two-particle Green’s function; it is necessary to use the boundary conditions to determine the range of the time integrations in, e.g., (4-6) and (4-7).]