Finslerian geometrization of quantum mechanics in the hydrodynamical representation

TL;DR

This paper develops a Finslerian geometric framework, specifically using Kropina metrics, to describe quantum hydrodynamics in the Madelung form, including electromagnetic effects, and applies it to navigation problems.

Contribution

It introduces a novel Finslerian geometric approach to quantum hydrodynamics, incorporating electromagnetic fields, and connects it to the Zermelo navigation problem.

Findings

Finslerian geometry describes quantum motion via Kropina metrics.

Electromagnetic effects are incorporated into the geometric formalism.

Solutions to the Zermelo navigation problem are obtained within this framework.

Abstract

We consider a Finslerian type geometrization of the non-relativistic quantum mechanics in its hydrodynamical (Madelung) formulation, by also taking into account the effects of the presence of the electromagnetic fields on the particle motion. In the Madelung representation the Schr\"{o}dinger equation can be reformulated as the classical continuity and Euler equations of classical fluid mechanics in the presence of a quantum potential, representing the quantum hydrodynamical evolution equations. The equation of particle motion can then be obtained from a Lagrangian similar to its classical counterpart. After the reparametrization of the Lagrangian it turns out that the Finsler metric describing the geometric properties of quantum hydrodynamics is a Kropina metric. We present and discuss in detail the metric and the geodesic equations describing the geometric properties of the quantum motion in the presence of electromagnetic fields. As an application of the obtained formalism we consider the Zermelo navigation problem in a quantum hydrodynamical system, whose solution is given by a Kropina metric. The case of the Finsler geometrization of the quantum hydrodynamical motion of spinless particles in the absence of electromagnetic interactions is considered in detail, and the Zermelo navigation problem for this case is also discussed.