Time-independent theory of one-particle Green’s functions

Abstract

A new time-independent theory of Green’s functions is presented, which is based on a Fock space Liouvillean and its resolvent. Unlike current time-independent approaches in this formalism there is no need to introduce a complete operator manifold like that of Manne and Dalgaard for inner projection, nor for invoking a ‘‘killer condition.’’ A perturbative construction of the Green’s functions involves a perturbation expansion of both the resolvent of the Liouvillean and of the wave function. The approach is very general and is by no means limited to a closed-shell reference function. For functions of the latter type a very compact approach is possible in a particle–hole formalism, which automatically leads to the Feynman–Dyson series. An essential point in its derivation is that the perturbation expansion of the resolvent and of the wave function lead to diagrams with the same topology but with different energy denominators but that such diagrams can be added up to a single one, which may contain both ‘‘Rayleigh–Schrödinger’’ and ‘‘Brillouin–Wigner’’ denominators. It is shown that a consistent theory can be based on ‘‘principal-value’’ Green’s functions. The one-particle Green’s function is treated in some detail, the polarization propagator is only briefly discussed.