Renormalized ladder-type expansions for many-particle propagators

For a system of indistinguishable fermions with pairwise interactions, we consider renormalized, perturbative expansions for the three- and four-particle propagators (Green’s functions) in terms of exact one-particle propagators, along with a pair interaction that may be bare (as it appears in the Hamiltonian) or dressed (polarized). Care is taken to provide a rigorous foundation for the diagrammatic representations of these perturbation series; in particular, it is demonstrated how each topologically distinct diagram represents an embedding (in three-dimensional space) of numerous separate terms in the perturbation series. Within a renormalized ladder approximation for the three- and four-particle propagators, we derive diagram weights (which differ from the two-particle diagram weights) in order to limit the perturbation series to topologically distinct, permutationally independent diagrams. These results lay the foundation for perturbative approximations that decouple the propagator and density matrix equations of motion.