Madelung Structure of the Dirac Equation

TL;DR

This paper reformulates the Dirac equation into a Madelung-like system, linking quantum potentials and relativistic dynamics, and discusses conservation laws and multi-valued solutions.

Contribution

It introduces a Madelung structure for the Dirac equation, connecting quantum potentials with relativistic velocity fields in a novel way.

Findings

Reformulation of Dirac equations into a Madelung-like system

Identification of quantum potentials in relativistic context

Discussion of conservation laws and multi-valued solutions

Abstract

We consider the Dirac equations in polar form proving that they can equivalently be re-configured into a system of equations consisting of derivatives of the velocity density plus the Hamilton-Jacobi equation, giving the momentum in terms of relativistic quantum potentials (i.e. displaying first-order derivatives of the two degrees of freedom of the spinor field): this system is said to have Madelung structure. Conservation laws, second-order equations and multi-valuedness are also discussed.