Meson Pair Theory

Abstract

A comprehensive treatment of meson pair theory is presented from the point of view of the M\o{}ller scattering matrix and the $S$-matrix. First, the solutions of the classical field equations are used to exhibit the Heisenberg operator for the meson field terms of the in operator of Yang and Feldman. The orthogonality and completeness of this set of one-particle scattering functions is demonstrated. Both the M\o{}ller matrix and the $S$-matrix are constructed explicitly as ordered operators by taking advantage of the fact that they generate known linear transformations. Schwinger's dynamical principle is used both for these purposes and to find the nucleon self-energy. Explicit proofs of the unitarity of the scattering matrices are given. The M\o{}ller matrix is used as generating function for all Fock-space amplitudes of the system. In this way are exhibited (a) the Green's function renormalization constant (${Z}_{2}$ of Dyson), (b) the mesic proper field of the source, and (c) that matrix which describes scattering of a single meson. The latter, distinct from the classical one-particle scattering matrix, is obtained by solving a singular integral equation and shown to provide the same meson scattering amplitude. It is seen that only the one-meson amplitude of the one-meson state is singular on the energy shell, agreement with the simple one-channel form of the $S$-matrix. An alternative treatment of the theory from the point of view of the equations of motion of the source is presented an appendix.