Volume 9, Issue 3, pages 131-157
In this paper, by using a hyperfinite dimensional space and time lattice (ST -lattice) of nonstandard analysis, we present a variant of Bohmian interpretation of quantum mechanics which we call probabilistic Bohmian mechanics, PBM. We describe the model for non-relativistic quantum mechanics in detail and elaborate on its relativistic extension. The assumption of quantum equilibrium does not exist in the PBM model and its absence is compensated for by assuming that the particles are moving with infinite speed on a space time lattice according to probability density, which in PBM is regarded as density of space and time position states. In relativistic extension the PBM model assumes that Lorentz symmetry of space and time is just a wave phenomenon related to the space time symmetry of wave equations on real standard axes, whereas the motion of particles with infinite speed can only be described within the hyperfinite dimensional time and space lattice of preferred space time foliation (PST -lattice). In the PBM model the particle trajectories do not exist on the standard real time and space axes of any Lorentz frame and thus the assumption of wave function collapse is necessary. The wave function collapse is considered as a two-step process of decoherence (on standard real axes) and the subsequent particle jump (on the ST -lattice). For the PBM model, as an objective collapse theory, the causality problem, related to wave function collapse, is addressed by the preferred space-time foliation.
