Scattering theory in relativistic quantum mechanics

We construct a relativistic quantum scattering theory in a framework originally suggested by Stueckelberg, where the dynamical evolution of a system in space-time is described by means of an invariant parameter $\tau$. The wave operator for the reduced motion of a two-body system is related to measureable cross sections. The optical theorem is proved, and it is shown that in the nonrelativistic limit the cross section has the same interpretation as in the usual nonrelativistic scattering theory. A perturbation expansion for the S matrix is obtained, and its form is compared with that of the perturbative structure of quantum constraint Hamiltonian dynamics and quantum field theory. The problem of electromagnetic scattering of two charged particles is formulated and it is shown, for a heavy target, that the Rutherford cross section is obtained to lowest order.