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

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.