About the Different Approaches to Bohm's Quantum Potential in Non-Relativistic Quantum Mechanics

Margarete Mattes, M. Sorg

Pedro Luis Grande, Raul Carlos Fadanelli, Maarten Vos

Quantum mechanics is the most successful theory to describe microscopic phenomena. It was derived in different ways over the past 100 years by Heisenberg, Schrödinger, and Feynman. At the same time, other interpretations have been suggested, including the Bohm-De Broglie interpretation and the so-called Bohmian mechanics. Here, we show that Bohmian mechanics, which utilizes the concept of the Bohm quantum potential, can also serve as a starting point for quantizing classical non-relativistic systems. By incorporating the Bohm quantum potential into the Vlasov framework, we obtain a mean-field theory that captures the corpuscular nature of matter, in agreement with quantum mechanics within the Random Phase Approximation (RPA).

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.

H. van Haeringen

Davide Fiscaletti

Emilio Santos

It is proved that in non-relativistic quantum mechanics (without spin) the transition probability may be described in terms of particle paths, every path having a (positive) probability. This leads to a stochastic hidden variables theory providing an intuitive picture of particle motion. The change of velocity at every time has a probability that depends on the potential over a large region, at a difference with the local action of classical dynamics. Thus the hidden variables theory is non-local like Bohm's, but not deterministic.