Mandelstam Representation and Regge Poles with Absorptive Energy-Dependent Potentials

Abstract

The Mandelstam representation in potential theory is proved for superpositions of absorptive, energy-dependent Yukawa potentials. The dependence of the potential $V(r, s)$ on the energy $s$ is such that $V(r, s)$ is analytic in the $s$ plane cut from $s_0$ to infinity, where $s_0$ is a threshold for inelastic processes. Such a potential is implied by the causality condition that the scattered wave cannot precede the incident wave. Further, it is shown that certain interactions nonlocal in both space and time can also be reduced to the above type of local, energy-dependent potential. The postulate of absorptivity [$\mathrm{Im}V(r, s)\leqq 0$] is crucial for the existence of the usual Mandelstam representation. The relationship of the absorptive potentials in the Schrödinger equation and in dispersion theory to unitarity is discussed. An analysis of partial waves with complex angular momenta is given from the point of view of the Green's function, and it is shown that absorptive energy-dependent causal potentials behave qualitatively like real energy-independent potentials, so far as Regge pole trajectories are concerned. Regge poles for emissive potentials can behave anomalously. Absorptive potentials may, however, give rise to new singularities in unphysical regions.