Macroscopic Causality and Permanence of Smoothness for Two-Particle Scattering∗

Recent developments in the formulation of causality restrictions on the S matrix are reviewed, with attention focused on the behavior of matrix elements of the translation operator between suitably localized in and out states. Rapid decrease for large translations outside the timelike velocity cone of the center of momentum follows from Poincaré invariance and boundedness of S, as a result of a generalization of a theorem of Jost and Hepp. At present, rapid decrease can be proved in the Haag-Ruelle scattering theory, when the in state is translated to large positive times, but not for the remaining timelike directions, where thresholds of intermediate particles play a role. In the case of two-particle reactions, we show that rapid decrease for timelike directions is equivalent to permanence of smoothness of the p-space wave function, as an application of rapid convergence properties of the angular momentum expansion. Work supported by the National Science Foundation. Present address: Randall Laboratory of Physics, The University of Michigan, Ann Arbor, Michigan 48109 –1040. July 3, 2011: This is a LTEX version of extended notes for lectures presented by the author at the Fourth Boulder Conference on Particles and High Energy Physics, 1967. It has cosmetic changes from the published version in Lectures in Theoretical Physics, XB, (Gordon and Breach, New York, 1968), pp. 357–376.