Hydrodynamic representation and Energy Balance for Dirac and Weyl fermions in curved space-times

TL;DR

This paper develops a hydrodynamic framework for Dirac and Weyl fermions in curved space-times, deriving equations that describe their energy and particle balance in a unified geometric setting.

Contribution

It introduces a generalized Madelung transformation to derive hydrodynamic equations for fermions in curved space-times, including Dirac and Weyl equations, with new energy balance insights.

Findings

Derived Dirac-Euler equations with continuity and Bernoulli equations

Established particle balance equations for fermions

Extended hydrodynamic representation to Weyl (chiral) fermions

Abstract

Using a generalized Madelung transformation, we derive the hydrodynamic representation of the Dirac equation in arbitrary curved space-times coupled to an electromagnetic field. We obtain Dirac-Euler equations for fermions involving a continuity equation and a first integral of the Bernoulli equation. Comparing between the Dirac and Klein-Gordon equations we obtain the balance equation for fermion particles. We also use the correspondence between fermions and bosons to derive the hydrodynamic representation of the Weyl equation which is a chiral form of the Dirac equation.