Two-body propagators with dynamic effective interactions in a many-fermion medium: I. General structure

Abstract

Excitations of the many-body medium cause dynamic effective interactions between fermions. In actual calculations these interactions are almost always replaced by static approximations. In this paper a Green function approach is outlined which preserves the dynamic structure of the interactions. The central quantity is the four-vertex or two-fermion scattering amplitude Γ in the medium, which depends on three energy variables s, t, and u. Integral equations connect Γ to irreducible effective interactions in the three channels. With static interactions the s- and t-channels correspond to particle-particle and particle-hole configurations, respectively. Dynamic interactions mix these configurations. They also introduce poles in the crossed channels: the exchange of a phonon is a t-pole in the s-channel while pairons induce s-poles in the t-channel. Poles of the one-body propagator G( ω) and the two-body propagators K( s) and K( t) correspond to states of the A ± 1, A ± 2 and A-particle systems. The dynamic coupling between channels results in an intricate connection between various aspects of the many-fermion problem.