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

摘要

In this paper we present for the first time a complete description of theBohm model of the Dirac particle. This result demonstrates again that thecommon perception that it is not possible to construct a fully relativisticversion of the Bohm approach is incorrect. We obtain the fully relativisticversion by using an approach based on Clifford algebras outlined in two earlierpapers by Hiley and by Hiley and Callaghan. The relativistic model is differentfrom the one originally proposed by Bohm and Hiley and by Doran and Lasenby. Weobtain exact expressions for the Bohm energy-momentum density, a relativisticquantum Hamilton-Jacobi for the conservation of energy which includes anexpression for the quantum potential and a relativistic time developmentequation for the spin vectors of the particle. We then show that these reduceto the corresponding non-relativistic expressions for the Pauli particle whichhave already been derived by Bohm, Schiller and Tiomno and in more general formby Hiley and Callaghan. In contrast to the original presentations, there is noneed to appeal to classical mechanics at any stage of the development of theformalism. All the results for the Dirac, Pauli and Schroedinger cases areshown to emerge respectively from the hierarchy of Clifford algebrasC(13),C(30), C(01) taken over the reals as Hestenes has already argued. Thusquantum mechanics is emerging from one mathematical structure with no need toappeal to an external Hilbert space with wave functions.