Relativistic potential models as systems with constraints and their interpretation

Recent proposals of classical relativistic two-particle systems with an interaction potential instead of a force field are studied. These are local in an evolution parameter and are formulated as generalized gauge theories within Dirac's formalism for constrained Hamiltonian systems simulating reparametrization invariance. It is shown that the solutions of the equations of motion in general are world sheets instead of world lines. Requiring world lines we derive a set of conditions generalizing the world-line conditions usually given in the literature. Only a limited set of gauge constraints is shown to satisfy these conditions, notably when the evolution parameter is chosen to be the time of the rest frame of the system. A property of this gauge is that the total momentum at laboratory time is in general not just the sum of the momenta of the particles; the missing part is normally viewed as a momentum carried by a force field