Flux Continuity and Probability Conservation in Complexified Bohmian Mechanics

TL;DR

This paper investigates probability conservation in complexified Bohmian mechanics, revealing that probability is generally not conserved along trajectories, despite their usefulness in quantum system analysis.

Contribution

It provides a generalized framework for understanding flux continuity and demonstrates that probability conservation does not hold in complexified Bohmian trajectories.

Findings

Probability conservation generally does not hold in complexified Bohmian mechanics.

A generalized expression for wavefunction conjugation in complex variables is derived.

Complexified flux continuity is not satisfied in most cases.

Abstract

Recent years have seen increased interest in complexified Bohmian mechanical trajectory calculations for quantum systems, both as a pedagogical and computational tool. In the latter context, it is essential that trajectories satisfy probability conservation, to ensure they are always guided to where they are most needed. In this paper, probability conservation for complexified Bohmian trajectories is considered. The analysis relies on time-reversal symmetry considerations, leading to a generalized expression for the conjugation of wavefunctions of complexified variables. This in turn enables meaningful discussion of complexified flux continuity, which turns out not to be satisfied in general, though a related property is found to be true. The main conclusion, though, is that even under a weak interpretation, probability is not conserved along complex Bohmian trajectories.