Expansion of the scattering matrix in terms of multiple impulsive peripheral interactions

Abstract

A new perturbation expansion is developed for the S-matrix. Each term of the expansion corresponds to a Feynman graph which describes a multiple peripheral collision. The complete effects of the short range forces are included in the first approximation. The peripheral interaction expansion is based on the analytic properties of the ordinary perturbation expansion and may be looked upon as a rearrangement of the terms of that expansion. It provides a simple perturbation-theory picture for all the concrete results that have been derived by formulating dynamical theories in terms of dispersion relations. The Feynman graphs encountered in the peripheral interaction expansion are those of the ordinary expansion with all self-energy parts omitted. This omission is possible because each line of a graph represents a set of quanta which correspond not only to the discrete states, but also to all states lying in the continuum which have a limited angular momentum. Each vertex corresponds to an elementary interaction which has a strength equal to the appropriate exact S-matrix element. Renormalization of the successive terms of the peripheral interaction is the most complicated problem which is encountered, although it is similar to renormalization of the ordinary expansion. In the treatment of this problem one encounters all possible successive graphical contractions. With the aid of these topological constructions it is possible to state the general renormalization rule quite concisely.