Quasiparticles and Projection Operators

Abstract

A formal procedure for expressing the $T$ matrix in terms of a reduced $T$ matrix is developed. The reduced $T$ matrix results when the original interaction or propagator which appears in the $T$-matrix integral equation is replaced by a reduced interaction or propagator. This reduction procedure provides a very neat derivation of the projection-operator formalism of Feshbach and the quasiparticle formalism of Weinberg. The two formalisms are compared. The projection-operator formalism appears to offer some advantages over the quasiparticle formalism. The expressions that appear have a more direct physical interpretation. For the bound-state problem, the projection-operator formalism leads to a perturbation expansion for the energy which is a generalization of the Wigner-Brillouin perturbation expansion. For the problem of using an elementary-particle state to represented a bound state of the system, the projection-operator formalism leads to an exact correspondence instead of the approximate one provided by the quasiparticle formalism. From this result we conclude that the bound state and the elementary-particle state are completely equivalent ways to describe a discrete state of a system.