Projection operators, quantum field theory, and the pion-nucleon system

An analysis is carried out of a π-N-Δ field theory using an extension of the Feshbach projection operator technique in conjunction with a set of identities which relate Green's functions acting in subspaces whose projection operators are related by the action of the meson creation and annihilation operators. Exact relations are derived for the propagators, self-energies, and vertex functions that arise in the field theory. A separation of the pion-nucleon elastic scattering amplitudes into their one-particle irreducible and one-particle reducible parts is obtained. An exact set of coupled integral equations for the pion-nucleon amplitudes is derived. When a certain approximation for the effective potentials in these equations is made, it is found that the one-particle irreducible amplitudes satisfy a set of three-particle equations that are exactly analogous to the quasiparticle equations of Alt, Grassberger, and Sandhas. Moreover, the one-particle reducible parts can be obtained from the solutions of the three-particle equations. These equations are nonlinear in that the kernels are determined by the solutions. An approximate set of linear equations is obtained whose solutions satisfy two-particle and three-particle unitarity and have reasonable analytic structure. An iteration scheme for improving on the linear approximation is presented