The Clifford Algebra Approach to Quantum Mechanics B: The Dirac Particle and its relation to the Bohm Approach

TL;DR

This paper develops a complete relativistic Bohm model for the Dirac particle using Clifford algebras, showing it can be derived without classical mechanics or wave functions, unifying quantum mechanics within a single mathematical framework.

Contribution

It presents the first fully relativistic Bohm model for the Dirac particle based on Clifford algebras, avoiding classical mechanics and wave functions, and unifies quantum mechanics in a single structure.

Findings

Derived exact relativistic energy-momentum density expressions

Established a relativistic quantum Hamilton-Jacobi equation including quantum potential

Demonstrated reduction to non-relativistic cases for Pauli and Schrödinger particles

Abstract

In this paper we present for the first time a complete description of the Bohm model of the Dirac particle. This result demonstrates again that the common perception that it is not possible to construct a fully relativistic version of the Bohm approach is incorrect. We obtain the fully relativistic version by using an approach based on Clifford algebras outlined in two earlier papers by Hiley and by Hiley and Callaghan. The relativistic model is different from the one originally proposed by Bohm and Hiley and by Doran and Lasenby. We obtain exact expressions for the Bohm energy-momentum density, a relativistic quantum Hamilton-Jacobi for the conservation of energy which includes an expression for the quantum potential and a relativistic time development equation for the spin vectors of the particle. We then show that these reduce to the corresponding non-relativistic expressions for the Pauli particle which have already been derived by Bohm, Schiller and Tiomno and in more general form by Hiley and Callaghan. In contrast to the original presentations, there is no need to appeal to classical mechanics at any stage of the development of the formalism. All the results for the Dirac, Pauli and Schroedinger cases are shown to emerge respectively from the hierarchy of Clifford algebras C(13),C(30), C(01) taken over the reals as Hestenes has already argued. Thus quantum mechanics is emerging from one mathematical structure with no need to appeal to an external Hilbert space with wave functions.