Trajectory construction of Dirac evolution

TL;DR

This paper develops an exact trajectory-based model for the Dirac equation, extending quantum trajectory methods to relativistic spinor fields and demonstrating Lorentz covariance and potential inclusion.

Contribution

It introduces a novel trajectory formulation for the Dirac equation, including conserved densities and Lorentz covariance, expanding quantum trajectory models to relativistic spinor dynamics.

Findings

Derived an exact formula for Dirac spinor propagation.

Established Lorentz covariance of trajectory equations.

Extended results to include external potentials.

Abstract

We extend our programme of representing the quantum state through exact stand-alone trajectory models to the Dirac equation. We show that the free Dirac equation in the angular coordinate representation is a continuity equation for which the real and imaginary parts of the wavefunction, angular versions of Majorana spinors, define conserved densities. We hence deduce an exact formula for the propagation of the Dirac spinor derived from the self-contained first-order dynamics of two sets of trajectories in 3-space together with a mass-dependent evolution operator. The Lorentz covariance of the trajectory equations is established by invoking the relativity of the trajectory label. We show how these results extend to the inclusion of external potentials. We further show that the angular version of the Dirac equation implies continuity equations for currents with non-negative densities, for which the Dirac current defines the mean flow. This provides an alternative trajectory construction of free evolution. Finally, we examine the polar representation of the Dirac equation, which also implies a non-negative conserved density but does not map into a stand-alone trajectory theory. It reveals how the quantum potential is tacit in the Dirac equation.