Clifford Algebras and the Dirac-Bohm Quantum Hamilton-Jacobi Equation

Basil J. Hiley, R. E. Callaghan

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.

A. K. Kwaśniewski, W. Bajguz, I. Jaroszewski

Wadyr A. Rodrigues, Samuel A. Wainer

Pedro Amao, Hernán Castillo

T. Djama

For one dimensional systems, we derive from the Dirac spinor equation (DSE) the relativistic quantum stationary Hamilton–Jacobi equation (RQSHJES) for particles of spin . We establish that the RQSHJES ±1/2 is constituted of two equations, each one corresponding to the positive and negative energies (particle and antiparticle). The solution of the RQSHJES ±1/2 is well established. We also investigate the Dirac–Klein–Gordon equation and set its corresponding Hamilton–Jacobi equation.

Antonio S. de Castro, ALVARO DE SOUZA DUTRA