Relativistic Classical and Quantum Mechanics: Clock Synchronization and Bound States, Center of Mass and Particle Worldlines, Localization Problems and Entanglement

Abstract

Parametrized Minkowski theories allow to describe isolated systems in global non-inertial frames in Minkowski space-time (defined as M/oller-admissible $3+1$ splittings with clock synchronization and radar 4-coordinates) , with the transitions among frames described as gauge transformations. The restriction to the intrinsic inertial rest frame allows to formulate a new relativistic classical mechanics for N-particle systems compatible with relativistic bound states. There is a complete control on the relativistic collective variables (Newton-Wigner center of mass, Fokker-Pryce center of inertia, M/oller center of energy) and on the realization of the Poincare’ algebra (with the explicit form of the interaction-dependent Lorentz boosts). The particle world-lines are found to correspond to the ones of predictive mechanics and localization problems are clarified. The model can be consistently quantized avoiding the instantaneous spreading of the center-of-mass wave packets (Hegerfeldt theorem), because the non-local non-covariant center of mass is a non-measurable quantity. The basic difference with non-relativistic quantum mechanics is that the composite N-particle system cannot be represented as the tensor product of single particle Hilbert spaces, but only as the tensor product of the center-of-mass Hilbert space with the one of relative motions. This spatial non-separability (due to the Lorentz signature of space-time) makes relativistic entanglement much more involved than the non-relativistic one. Some final remarks on the emergence of classicality from quantum theory are done.