Abstract
An $S$-matrix theory is developed for a system of strongly interacting particles in which unstable particles are included. The particular system considered is $\ensuremath{\pi}+N\ensuremath{\leftrightarrow}\ensuremath{\pi}+N$, $\ensuremath{\pi}+N\ensuremath{\leftrightarrow}\ensuremath{\rho}+N$, and $\ensuremath{\rho}+N\ensuremath{\leftrightarrow}\ensuremath{\rho}+N$. Only the one-pion-exchange interaction is included. The relationship to the strip approximation and the application to the higher resonances in pion-nucleon scattering are discussed. Complex singularities are evaluated, and their relationship to the generalized unitarity condition for the pion-pion system is stressed. An extended $N{D}^{\ensuremath{-}1}$ method is used to develop a system of nonsingular, uncoupled, Fredholm integral equations, from which the transition amplitudes can be evaluated.
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