Cluster Decomposition Properties of the $S$ Matrix

Conditions on the $S$ matrix which arise from the assumption that it describes interactions which are at least approximately local are discussed. Particular conditions of this kind, which may be called cluster decomposition properties, are formulated and the implications of these conditions for the structure of the $S$ matrix are studied. The discussion is restricted to the case of a world in which there is only one kind of particle, namely a spinless boson of finite mass. The considerations presented apply equally well to relativistic, as well as to nonrelativistic scattering theories. It is not assumed that the $S$ matrix can be derived within the framework of a strictly local field theory, nor is it assumed that the $S$-matrix elements possess any particular properties of analyticity. As an illustration it is pointed out that the cluster decomposition properties assumed hold good in the conventional perturbation theory approach to field theory.