Vertex theory of many-fermion systems

Abstract

A general theory of the weak-response of many-fermion systems is formulated in terms of the longitudinal vertex function, Γ. By including effects of exchange the vertex theory embeds the conventional dielectric theory, as expressed by the momentum-energy space factorization Γ = ϵ −1 Γ . Here Γ is the proper vertex function and ϵ −1 is the generalized dielectric constant. The integral equation for Γ is rederived via weak-response theory, then factored into separate equations for Γ and ϵ −1. Basic properties of Γ, particularly gauge-invariance, are discussed. A nonperturbative approximation, called the υ′ approximation, is introduced in which the interaction kernel of the integral equation for Γ is replaced by a suitably chosen energy-independent effective interaction υ′. The υ′ approximation is shown to be a generalization of the time dependent Hartree-Fock approximation. By using the υ′ approximation as an input to a more exact theory expressions are obtained for the quasi-particle lifetime and the coupling of the quasi particle to the collective mode. Some new physical consequences of the theory are (a) the momentum dependence of the quasi-particle scattering amplitude in a weak external potential, (b) the existence of an intrinsic longitudinal coupling constant renormalization associated with the static long wavelength limit of Γ , and (c) the formal occurrence of a “spurious” pole of Γ coupled with a zero of ϵ −1 for a sufficiently strong repulsive interaction υ′. The last bears a resemblance to a phenomenon recently noted in elementary-particle theory. A preliminary note on some of this work employing a simplified nongauge-invariant version of the υ′ approximation was published earlier. Detailed results on the solution for Γ in the υ′ approximation and in lowest order perturbation theory will be published in a second paper.