Bohmian Mechanics at Space-Time Singularities. I. Timelike Singularities

TL;DR

This paper extends Bohmian mechanics to curved space-times with timelike singularities, introducing a stochastic model with particle creation and annihilation at the singularity, exemplified by the Reissner-Nordstrom geometry.

Contribution

It develops a Bohmian framework incorporating particle creation and annihilation at timelike singularities using boundary conditions on the wave function.

Findings

Explicit equations for particle dynamics near singularities.

A stochastic law for particle creation and annihilation.

Model applicable to quantum field theories with singularities.

Abstract

We develop an extension of Bohmian mechanics to a curved background space-time containing a singularity. The present paper focuses on timelike singularities. We use the naked timelike singularity of the super-critical Reissner-Nordstrom geometry as an example. While one could impose boundary conditions at the singularity that would prevent the particles from falling into the singularity, we are interested here in the case in which particles have positive probability to hit the singularity and get annihilated. The wish for reversibility, equivariance, and the Markov property then dictates that particles must also be created by the singularity, and indeed dictates the rate at which this must occur. That is, a stochastic law prescribes what comes out of the singularity. We specify explicit equations of a model involving an interior-boundary condition on the wave function at the singularity, which can be used also in other versions of quantum theory besides Bohmian mechanics. As the resulting theory involves particle creation and annihilation, it can be regarded as a quantum field theory, and the stochastic process for the Bohmian particles is analogous to Bell-type quantum field theories.