The Dirac-Hestenes Equation and its Relation with the Relativistic de Broglie-Bohm Theory

TL;DR

This paper derives conservation laws from the Dirac-Hestenes equation using Clifford algebra, clarifies misconceptions in relativistic quantum potential, and relates it to the de Broglie-Bohm theory without Lagrangian formalism.

Contribution

It provides a novel derivation of conservation laws from the Dirac-Hestenes equation and clarifies the connection between relativistic quantum potential and de Broglie-Bohm theory.

Findings

Energy-momentum and angular momentum are not conserved in null electromagnetic fields.

Correct relativistic quantum potential is obtained within the de Broglie-Bohm framework.

Critiques misconceptions in existing literature with rigorous mathematical analysis.

Abstract

In this paper we provide using the Clifford and spin-Clifford formalism and some few results of the extensor calculus a derivation of the conservation laws that follow directly from the Dirac-Hestenes equation (DHE) describing a Dirac-Hestenes spinor field (DHSF) in interaction with an external electromagnetic field without using the Lagrangian formalism. In particular, we show that the energy-momentum and total angular momentum extensors of a DHSF is not conserved in spacetime regions permitting the existence of a null electromagnetic field F but a non null electromagnetic potential A. These results have been used together with some others recently obtained (e.g., that the classical relativistic Hamilton-Jacobi equation is equivalent to a DHE satisfied by a particular class of DHSF) to obtain the correct relativistic quantum potential when the Dirac theory is interpreted as a de Broglie-Bohm theory. Some results appearing in the literature on this issue are criticized and the origin of some misconceptions is detailed with a rigorous mathematical analysis.