Complex Trajectories and Dynamical Origin of Quantum Probability

TL;DR

This paper explores complex quantum trajectories derived from a modified de Broglie-Bohm framework, revealing that quantum probability emerges from the underlying dynamics and analyzing probability conservation in the complex plane.

Contribution

It introduces a complex trajectory formulation that explains the origin of quantum probability and examines probability conservation in the complex domain.

Findings

Normalisable probability density can be defined over the entire complex plane.

Probability is not always locally conserved in the complex plane.

No significant complex motion occurs in the classical regime.

Abstract

Complex quantum trajectories, which were first obtained from a modified de Broglie-Bohm quantum mechanics, demonstrate that Born's probability axiom in quantum mechanics originates from dynamics itself. We show that a normalisable probability density can be defined for the entire complex plane, though there may be regions where the probability is not locally conserved. Examining this for some simple examples such as the harmonic oscillator, we also find why there is no appreciable complex extended motion in the classical regime.