Simple Proof for Global Existence of Bohmian Trajectories

TL;DR

This paper proves that under certain conditions, Bohmian trajectories exist for all times, extending previous results to particles with spin, magnetic fields, and Dirac wavefunctions, using a simpler proof approach.

Contribution

It provides a simpler proof of global existence of Bohmian trajectories and extends the result to more complex quantum systems.

Findings

Almost all Bohmian trajectories exist globally under specified conditions.

The proof is simplified compared to previous methods.

Conditions on the current vector field ensure almost-sure global existence.

Abstract

We address the question whether Bohmian trajectories exist for all times. Bohmian trajectories are solutions of an ordinary differential equation involving a wavefunction obeying either the Schroedinger or the Dirac equation. Some trajectories may end in finite time, for example by running into a node of the wavefunction, where the law of motion is ill-defined. The aim is to show, under suitable assumptions on the initial wavefunction and the potential, global existence of almost all solutions. We provide a simpler and more transparent proof of the known global existence result for spinless Schroedinger particles and extend the result to particles with spin, to the presence of magnetic fields, and to Dirac wavefunctions. Our main new result are conditions on the current vector field on configuration-space-time which are sufficient for almost-sure global existence.