Three-particle equations for the pion-nucleon system

An analysis is carried out of a -N- field theory using projection operator techniques developed recently by the author. Relationships are derived between matrix elements of resolvents formed from bare N and states and various three-particle amplitudes. These relations are exactly analogous to those obtained in potential theory between the three-particle Greens function and the transition operators of Alt, Grassberger, and Sandhas. When the three-particle amplitudes are integrated over with appropriate weighting functions, the elastic scattering and production amplitudes are obtained. The weighting functions are the analogs of bound state wave functions for the N and subsystems. A separation of the production amplitude into its one-fermion irre- ducible and reducible parts is derived. It is shown that the three-particle amplitudes are solutions of AlT, Grassberger, and Sandhas type of equations as extended by Kowalski to allow for a three-body interaction. When the AlT, Grassberger, and Sandhas quasiparticle method is applied to these equations, the two-particle equations developed previously by the author are recovered. If the one-fermion irreducible part of the three-body interaction is neglected a closed set of coupled nonlinear integral equations for all of the quantities of interest is obtained.