The quantum hydrodynamic formulation of Dirac equation and its generalized stochastic and non-linear analogs

TL;DR

This paper derives a quantum hydrodynamic formulation of the Dirac equation using real variables, explores its low-velocity limit, and discusses the Lorentz invariance of the quantum potential affecting non-locality.

Contribution

It introduces a novel hydrodynamic representation of the Dirac equation based on action matrices and wave function moduli, extending previous models.

Findings

Derivation of hydrodynamic equations from Dirac formalism

Connection to Pauli equation in low velocity limit

Discussion of Lorentz invariance of quantum potential

Abstract

The quantum hydrodynamic like equations as a function of two real sets of variables, the 4x4 action matrix and the 4 dimensional wave function modulus vector of the Dirac equation, are derived in the present work. The paper shows that in the low velocity limit the equations lead to the hydrodynamic representation of the Pauli equation for charged particle with spin given by Janossy and by Bialynicki.The Lorentz invariance of the relativistic quantum potential that generates the non-local behavior of the quantum mechanics is discussed.