Classical and Non-Relativistic Limits of a Lorentz-Invariant Bohmian Model for a System of Spinless Particles

TL;DR

This paper analyzes and extends a Lorentz-invariant Bohmian model for spinless particles, exploring its classical and non-relativistic limits, and clarifying the physical meaning of the evolution parameter.

Contribution

It extends a Lorentz-invariant Bohmian model to include electromagnetic interactions and clarifies the physical interpretation of its temporal parameter.

Findings

The model recovers Einsteinian mechanics in the classical limit.

In the non-relativistic limit, it resembles standard Bohmian mechanics with a modified time parameter.

The parameter { extsigma} tends to the particle's proper time in the classical limit.

Abstract

A completely Lorentz-invariant Bohmian model has been proposed recently for the case of a system of non-interacting spinless particles, obeying Klein-Gordon equations. It is based on a multi-temporal formalism and on the idea of treating the squared norm of the wave function as a space-time probability density. The particle's configurations evolve in space-time in terms of a parameter {\sigma}, with dimensions of time. In this work this model is further analyzed and extended to the case of an interaction with an external electromagnetic field. The physical meaning of {\sigma} is explored. Two special situations are studied in depth: (1) the classical limit, where the Einsteinian Mechanics of Special Relativity is recovered and the parameter {\sigma} is shown to tend to the particle's proper time; and (2) the non-relativistic limit, where it is obtained a model very similar to the usual non-relativistic Bohmian Mechanics but with the time of the frame of reference replaced by {\sigma} as the dynamical temporal parameter.