Why is it necessary to use dressed single particle Green's function in QFT bound state problem (Bethe–Salpeter equation)?

Question

On p. 333 in book Quantum Electrodynamics by Walter Greiner, Joachim Reinhardt or other references, they claim that in Bethe–Salpeter equation, we have to use dressed single particle Green's function, including all self interaction with photon.

We want to remark that in principle one should not use the free Feynman propagators $S_F$ when calculating (6.11). Instead one should use “dressed” propagators that contain the interaction with their own photon field to all orders (cf. Chap. 5). Then one has taken into account all self-interaction graphs in (6.10). In the same way one should use the exact photon propagators and vertex functions when calculating $K$. The renormalization problems related to that will not be discussed here.

Why is this necessary and is this calculation without duplication or repeated diagrams in one-body Green function (GF)?

Answer 1

It is necessary to sum over all possible histories/Feynman diagrams. In the Bethe–Salpeter equation, we consider all $2\to 2$ diagrams.

In particular, as a subset, we must include all pairs of $1\to 1$ diagrams.

[Specifically, Greiner & Reinhardt apparently assume the 2 fermion-lines cannot be broken (read: no creation/annihilation of external particles). The kernel $K$ in eq. (6.7) apparently only includes diagrams that connect the 2 fermion-lines.]

To systematically streamline the combinatorics it is natural to use the 2-particle irreducible (2PI) formalism.