Bohmian Mechanics for a Degenerate Time Foliation

TL;DR

This paper extends the hypersurface Bohm--Dirac model to degenerate time foliations, demonstrating that a Bohm-type law of motion and equivariant distribution remain valid even when multiple hypersurfaces share common regions.

Contribution

It introduces a generalized framework for Bohmian mechanics in relativistic space-time with degenerate foliations, maintaining the core properties of the model.

Findings

The Bohmian law of motion is well-defined in degenerate foliations.

The $||$ distribution remains equivariant in this setting.

The model's consistency is preserved despite degeneracies.

Abstract

The version of Bohmian mechanics in relativistic space-time that works best, the hypersurface Bohm--Dirac model, assumes a preferred foliation of space-time into spacelike hypersurfaces (called the time foliation) as given. We consider here a degenerate case in which, contrary to the usual definition of a foliation, several leaves of the time foliation have a region in common. That is, if we think of the time foliation as a 1-parameter family of hypersurfaces, with the hypersurfaces moving towards the future as we increase the parameter, a degenerate time foliation is one for which a part of the hypersurface does not move as we increase the parameter. We show that the hypersurface Bohm--Dirac model still works in this situation; that is, we show that a Bohm-type law of motion can still be defined, and that the appropriate $|\psi|^2$ distribution is still equivariant with respect to this law.