The Geometric Origin of the Madelung Potential

TL;DR

This paper explores the geometric origins of the quantum potential, linking it to scalar curvature in the metric of extended particles, and discusses its implications for wave equations and quantization.

Contribution

It reveals that the quantum potential arises from scalar curvature and connects quantization to SO(2)-reduction of the Lorentz frame bundle.

Findings

Quantum potential linked to scalar curvature of the metric.

Quantization of circulation explained via SO(2)-reduction.

Reformulation of Madelung equations using exterior differential forms.

Abstract

Madelung's hydrodynamical forms of the Schrodinger equation and the Klein-Gordon equation are presented. The physical nature of the quantum potential is explored. It is demonstrated that the geometrical origin of the quantum potential is in the scalar curvature of the of the metric that defines the kinetic energy density for an extended particle and that the quantization of circulation (Bohr-Sommerfeld) is a consequence of associating an SO(2)-reduction of the Lorentz frame bundle with wave motion. The Madelung equations are then cast in a basis-free form in terms of exterior differential forms in such a way that they represent the equations for a timelike solution to the conventional wave equations whose rest mass density satisfies a differential equation of the "Klein-Gordon minus nonlinear term" type. The role of non-zero vorticity is briefly examined.