Scattering in Relativistic Particle Mechanics.

The problem of direct interaction in relativistic particle mechanics has been extensively studied and a variety of models has been proposed avoiding the conclusions of the so-called no-interaction theorems. In this thesis we study scattering in the relativistic two-body problem. We use our results to analyse gauge invariance in Hamiltonian constraint models and the uniqueness of the symplectic structure in manifestly covariant relativistic particle mechanics. We first present a general geometric framework that underlies approaches to relativistic particle mechanics. This permits a model-independent and geometric definition of the notions of asymptotic completeness and of Moller and scattering operators. Subsequent analysis of these concepts divides into two parts. First, we study the kinematic properties of the scattering transformation, i.e. those properties that arise solely from the invariance of the theory under the Poincare group. We classify all canonical (symplectic) scattering transformations on the relativistic phase space for two free particles in terms of a single function of the two invariants of the theory. We show how this function is determined by the center of mass time delay and scattering angle and vice versa. The second part of our analysis of the relativistic two-body scattering problem is devoted to the dynamical properties of the scattering process. Hence, we turn to two approaches to relativistic particle mechanics: the Hamiltonian constraint models and the manifestly covariant formalism. Using general geometric arguments, we prove "gauge invariance" of the scattering transformation in the Todorov -Komar Hamiltonian constraint model. We conclude that the scattering cross sections of the Todorov-Komar models have the same angular dependence as their non-relativistic counterpart, irrespective of a choice of gauge. This limits the physical relevance of those models. We present a physically non -trivial Hamiltonian constraint model, starting from Fokker's action principle. To study the Moller operators in the manifestly covariant approach, we extend techniques developed for dealing with non-relativistic two-body scattering and determine precise conditions on the dynamical vectorfields under which the Moller operators can be proven to exist. We then show how Moller operators can be used to construct the Hamiltonian structure in the manifestly covariant approach. Finally, we turn our attention to the quantization of the models discussed. We determine a notion of position in a model for the quantum mechanical treatment of the free relativistic particle that does not violate causality. This result must be compared to recent proofs of the fact that the notions of strict localization and of causality are not mutually compatible in relativistic quantum mechanics. (Abstract shortened with permission of author.).