Mandelstam Representation and Regge Poles with Absorptive EnergyDependent Potentials
Abstract
The Mandelstam representation in potential theory is proved for superpositions of absorptive, energydependent 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, energydependent potential. The postulate of absorptivity [ImV(r, s)<=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 energydependent causal potentials behave qualitatively like real energyindependent 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.
 Publication:

Physical Review
 Pub Date:
 November 1962
 DOI:
 10.1103/PhysRev.128.1474
 Bibcode:
 1962PhRv..128.1474C