Antisymmetrized, translationally invariant theory of the nucleon optical potential

Earlier work showed how a nucleon optical model wave function could be defined as a projection of a many-nucleon scattering state within a translationally invariant second quantized many-body theory. In this paper, an optical potential operator that generates this optical model wave function is defined through a particular off-shell extension of the elastic transition operator. The theory is expressed explicitly in terms of the many-nucleon Hamiltonian in a mixed representation in which localized target nucleus states feature. No reference to a mean-field concept is involved in the definition. It is shown that the resulting optical model operator satisfies the requirements of rotational invariance and translational invariance and has standard behavior under the time-reversal transformation. The contributions to the optical potential from two different exchange mechanisms are expressed in terms of an effective Hamiltonian involving a nucleon-number-conserving one-body interaction. In the weak-binding limit, the method reduces to a version of Feshbach's projection operator formulation of the optical potential with a truncated nucleon-nucleon potential including exchange terms and recoil corrections. Definitions of the nucleon single-particle Green's function and the corresponding Dyson self-energy modified by corrections for translational invariance are presented, and different definitions of the optical potential operator are compared.