Madelung Structure of the Dirac Equation

TL;DR

This work derives a covariant Madelung reformulation of the Dirac equation by expressing Dirac spinors in polar form and recasting the dynamics as a hydrodynamic-like system with a continuity equation for the velocity density and a Hamilton-Jacobi/guidance equation for momentum, augmented by two relativistic quantum potentials built from the spinor degrees of freedom. In 1+3 dimensions the momentum is encoded as P^μ = m cos β u^μ plus first-order derivative terms of φ^2 and β, while a curl of the velocity density appears in lower dimensions and adds a supplementary classical-like constraint; the framework is shown to be equivalent to the Dirac equations, with conservation laws and second-order relations explored. The analysis clarifies how relativistic structure resolves the Wallstrom objection by admitting inherently multi-valued spinor fields and identifies Goldstone-like internal variables as hidden variables, embedding the Madelung picture within a covariant, spinor-based Bohmian viewpoint. The results extend the Madelung program beyond nonrelativistic spinless quantum mechanics, offering a consistent hydrodynamic interpretation of spinor fields and a foundation for trajectory-based formulations in curved spacetimes. Overall, the paper provides a detailed, dimension-spanning covariant bridge between Dirac dynamics and Madelung-type hydrodynamics with potential implications for relativistic Bohmian mechanics and quantum matter in curved geometries.

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.