Self-consistent Green function equations and the hierarchy of approximations for the four-point propagator

The equation of motion for the Green function is combined with the Bethe-Salpeter equation for the scattering amplitude yielding a concise and formally closed system of three equations that encapsulates the essence of Green function theory. Two of the three equations formally resemble a Dyson-like relation. We prove that this formally simple set is exactly equivalent to Hedin's equations. Our derivation therefore constitutes an alternative to Hedin's derivation which is based on functional derivatives. Furthermore, we briefly discuss how approximations can be introduced as a hierarchy of approximations to the four-point Green function.