Metric and Dispersive Dynamics of Phase Perturbations in Bohmian Quantum Mechanics

By linearizing the hydrodynamic equations associated with the Madelung transformation in the context of Bohmian dynamics, we derive an effective metric for the phase perturbation around a background state. We show that the metric previously obtained previously by the same authors arises naturally as a particular case of a more general structure, in which the complete functional variation of the quantum potential is retained and external classical potentials can be incorporated. This generalization preserves the dependence of the quantum potential on the density and its spatial derivatives, generating additional terms that introduce dispersive corrections and act as sources in the effective Klein-Gordon equation. To organize these contributions, we introduce differential operators that separate the local sector responsible for the effective geometry from the corrections associated with the complete functional structure of the quantum potential. We identify the conditions under which these additional terms become negligible, recovering homogeneous propagation and the effective geometry for the Gaussian wave packet of a free particle.