A Dyson Equation Approach to the Calculation of Correlation Functions

We derive an integral equation for the linear response of a Fermi liquid which is the analog of Dyson's equation for the one-particle Green's function. The corresponding particle-hole self-energy depends only on the frequency and the wave number of the applied field, and we are able to give an explicit expression for it. The solution of the response function resembles ordinary RPA, only the bare interaction is replaced by this self-energy. Current approximations are easily rederived within our technique. Criteria to get approximations which conserve particle number, momentum, and energy are equivalent to those of Kadanoff and Baym.