Projection Operator Approach to the Self-Energy

Abstract

Feshbach's projection operator formalism is extended to the description of the self-energy. This necessitates the introduction of "extended'' projection operators. They act within an "extended'' Hilbert space in which the number of nucleons is not fixed. The compact formula derived for the self-energy is formally similar to Feshbach's original expression of the "generalized'' optical-model potential. The theory is formulated in the nuclear case, but it also applies to atomic systems. It covers both the "retarded'' and the "time-ordered'' Green's ?functions, and the "proper'' and "improper'' self-energies. It is first worked out in a stationary formalism, in order to better exhibit its analogy with Feshbach's original theory of the generalized optical-model potential. The main results are then also derived in a time- dependent framework; since the number of nucleons is not fixed, the definition of the Møller operators requires due caution. It is shown that, in finite systems, Dyson's equation does not uniquely determine the self-energy, in contrast to common assumption. However, the difference between the various possibilities has little practical consequence. We exhibit the relationship between the present approach and a recent "configuration interaction formulation of the Dyson equation.'' Contact is also established with the "linked-cluster'' perturbation expansion of the self-energy in powers of the strength of the nucleon-nucleon interaction.