Metric Formulation of Free Particle Dynamics in Bohmian Mechanics

Abstract

In this work, we propose a geometric description of the quantum dynamics of a free particle based on phase perturbations. Using the hydrodynamic formalism and Bohmian quantum mechanics, and assuming a certain class of conditions/restrictions on the quantum potential, we derive an effective pseudo-Riemannian metric whose conformal factor is determined by the probability density, allowing the quantum dynamics to be reinterpreted in terms of the geodesic propagation of perturbations. This framework establishes a geometric structure that connects the causal propagation of such perturbations to the conditions under which they can influence Bohmian trajectories, providing a new perspective on the dynamical independence of different branches of the wave function and their possible interaction through the guiding equation. We apply this approach to a Gaussian wave packet, analyzing the resulting effective geometry and the associated causal horizon, as well as the conditions under which perturbations compatible with the adopted probability density cease to exert significant influence on the dynamics in question, so that the particle’s trajectory becomes effectively insensitive to additional fluctuations of the wave function.