Cluster Decomposition ofS-Matrix Elements

This paper discusses the physical basis for the cluster-decomposition properties of momentum-space $S$-matrix elements. The starting point is the proposition that in the limit of wide separation the probability for two separate processes should factor as the product of the individual probabilities. It is shown that factorization of the probabilities implies factorization of the corresponding $S$-matrix elements; i.e., any phase factors which might appear are at most constants which can be adjusted to unity. This provides the basis for the proof given recently by Wichmann and Crichton of the momentum-space cluster properties. It is shown that the resulting decomposition equations may contain unwanted factors which depend on the phase of the one-particle $S$-matrix elements. In order to minimize the effect of these factors the decomposition equations can be rewritten so that the unitarity equations for the "connected parts" (which are supposed to be the basis of a dynamical $S$-matrix theory) take their usual form without spurious phase factors.