Energy and momentum conservation in Bohm's model for quantum mechanics

Roderick I. Sutherland

A Lagrangian description is presented here that can be used in conjunction with a particle interpretation of quantum mechanics. The advantages of this description are that the field equations and the particle equations of motion can both be deduced from a single Lagrangian density expression and that conservation of energy and momentum are assured. This Lagrangian formulation is then applied to the special case of the well-known Bohm model as an example. It is thereby demonstrated that such a formulation is compatible with the predictions of quantum mechanics.

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.

Dmitri Martila

The Templeton Prize winner wrote: „In the second half of the eighteenth century, the deterministic character of Newton’s equations encouraged many people to see the physical world in strictly mechanical terms, as if the universe is a gigantic piece of cosmic clockwork.” Such wrong view comes from the believe, that the energy and momentum are always conserve.

Sean M. Carroll, Jackie Lodman