Exploration ofS-Matrix Theory

The possibility of constructing an $S$-matrix theory from postulates concerning unitarity, analyticity, connectedness, the $i\epsilon$ prescription and the spin-statistics connection is explored. The existence and residues of the physical region poles are shown to follow from the connected unitarity equations. The validity of certain fundamental theorems known from field theory. Hermitian analyticity, extended unitarity, the existence of antiparticles, the substitution law for crossed processes and the $\mathrm{TCP}$ theorem is reduced, in simple cases, to the question of whether the $S$-matrix singularity structure permits specific distortions of certain paths. These distortions are shown to be possible in a "model" singularity structure consisting of the normal thresholds, and depend only upon simple properties of these singularities. It is explained that it is logically impossible to deduce the complete singularity structure without the results we are trying to prove. A suggested resolution of this difficulty is to set up a scheme of successive iterations in singularity structure to be justified by selfconsistency. Then our work is the first step in such a scheme.