Probability and complex quantum trajectories: Finding the missing links

TL;DR

This paper develops a comprehensive complex probability density in quantum mechanics that unifies complex trajectories with real-line Born probabilities, explaining classical confinement of particles.

Contribution

It introduces a globally defined, normalisable complex probability density compatible with quantum trajectories, extending previous work to include regions previously not conserved.

Findings

Complex probability density is defined over the entire complex plane.

The total probability remains conserved despite local non-conservation.

The scheme explains classical particle confinement near the real axis.

Abstract

It is shown that a normalisable probability density can be defined for the entire complex plane in the modified de Broglie-Bohm quantum mechanics, which gives complex quantum trajectories. This work is in continuation of a previous one that defined a conserved probability for most of the regions in the complex space in terms of a trajectory integral, indicating a dynamical origin of quantum probability. There it was also shown that the quantum trajectories obtained are the same characteristic curves that propagate information about the conserved probability density. Though the probability density we now adopt for those regions left out in the previous work is not conserved locally, the net source of probability for such regions is seen to be zero in the example considered, allowing to make the total probability conserved. The new combined probability density agrees with the Born's probability everywhere on the real line, as required. A major fall out of the present scheme is that it explains why in the classical limit the imaginary parts of trajectories are not observed even indirectly and particles are confined close to the real line.