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arXiv:0806.4050

On the Quantization Procedure in Classical Mechanics and Problem of Hidden Variables in Bohmian Mechanics

V. D. Rusov, D.S. Vlasenko

Published Jun 25, 2008• 0 citations
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Abstract

Basing on the Chetaev's theorem on stable trajectories in dynamics in the presence of dissipative forces we obtain a generalized stability condition for Hamiltonian systems that has the form of the Schrodinger equation. We show that the energy of the dissipative forces generating generalized Chetaev's stability condition exactly coincides with Bohm's "quantum" potential. Using the principle of least action of perturbation we prove that the Bohmian quantum mechanics complemented with the Chetaev's generalized theorem does not have hidden variables.

DOI
10.48550/arxiv.0806.4050

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Basing on the Chetaev's theorem on stable trajectories in dynamics in the presence of dissipative forces we obtain a generalized stability condition for Hamiltonian systems that has the form of the Schrodinger equation. We show that the energy of the dissipative forces generating generalized Chetaev's stability condition exactly coincides with Bohm's "quantum" potential. Using the principle of least action of perturbation we prove that the Bohmian quantum mechanics complemented with the Chetaev's generalized theorem does not have hidden variables.
Using Chetaev's theorem on stable trajectories in dynamics in the presence of perturbation forces we obtain a generalized stability condition for Hamiltonian systems that has the form of the Schrodinger equation. We show that the energy of the perturbation forces generating generalized Chetaev's stability condition is of electromagnetic nature and exactly coincides with Bohm's "quantum" potential. We stress that not taking into account the reality of Bohm's electromagnetic Psi-field turns out to be the fundamental reason of violation of the famous Bell inequality, in the recent most precise direct experiments, as well as in theoretical calculations based on the simple Wigner model. We discuss the possibility of direct experiments, alternative to Bell's ideology and related, e.g. with observation of nuclear stochastic resonance during the alpha-decay process, which cannot occur in the framework of the probabilistic interpretation of quantum mechanics, but has to occur in the alternative Bohm interpretation.
Based on the Chetaev theorem on stable dynamical trajectories in the presence of dissipative forces, we obtain the generalized condition for stability of relativistic classical Hamiltonian systems (with an invariant evolution parameter) in the form of the Stückelberg equation. As is known, this equation is the basis of a competing paradigm known as parametrized relativistic quantum mechanics (pRQM). It is shown that the energy of dissipative forces, which generate the Chetaev generalized condition of stability, coincides exactly with Bohmian relativistic quantum potential. Within the framework of Bohmian RQM supplemented by the generalized Chetaev theorem and on the basis of the principle of least action for dissipative forces, we show that the squared amplitude of a wave function in the Stückelberg equation is equivalent to the probability density function for the number of particle trajectories, relative to which the velocity and the position of the particle are not hidden parameters. The conditions for reasonableness of trajectory interpretation of pRQM are discussed. Based on analysis of a general formalism for vacuum-flavor mixing of neutrino within the context of the standard and pRQM models we show that the corresponding expressions for the probability of transition from one neutrino flavour to another differ appreciably, but they are experimentally testable: the estimations of absolute value for neutrino mass based on modern experimental data for solar and atmospheric neutrinos show that the pRQM results have a preference. It is noted that the selection criterion of mass solutions relies on proximity between the average size of condensed neutrino clouds, which is described by the Muraki formula (29th ICRC, 2005) and depends on the neutrino mass, and the average size of typical observed void structure (dark matter + hydrogen gas), which plays the role of characteristic dimension of large-scale structure of the Universe.
In the hidden variable theory, Bohm proved a connection between the Schrodinger and Hamilton-Jacobi equations and showed the existence of classical paths, for which the generalized Bohr quantization condition is valid. In this paper we prove similar properties, starting from the equivalence between the Schrodinger and wave equations in the case of the conservative bound systems. Our approach is based on the equations and postulates of quantum mechanics without using any additional postulate. Like in the hidden variable theory, the above properties are proven without using the approximation of geometrical optics or the semiclassical approximation. Since the classical paths have only a mathematical significance in our analysis, our approach is consistent with the postulates of quantum mechanics.
Abstract This work presents a further development of causal interpretation of quantum mechanics. Existence of ‘non-local hidden variables’ like a physical field is taken as an assumption. Under this assumption, on the basis of the causal approach it is proven that Schrödinger equation is a necessary condition for stability of the motion of a particle. For the case of a hydrogen atom this approach gives a mathematical base with which to suggest that (1) the electron’s spin in an atom is precessing; (2) the energy of the precessional motion on Bohr orbits satisfies the Rydberg formula, and (3) structures are formed in the physical vacuum which stabilize the motion of an electron on Bohr orbits.
In a recent paper [B. Poirier, Chem. Phys. 370, 4 (2010)], a formulation of quantum mechanics was presented for which the usual wavefunction and Schrödinger equation are replaced with an ensemble of real-valued trajectories satisfying a principle of least action. It was found that the resultant quantum trajectories are those of Bohmian mechanics. In this paper, analogous ideas are applied to Bohmian Mechanics with Complex Action (BOMCA). The standard BOMCA trajectories as previously defined are found not to satisfy an action principle. However, an alternate set of complex equations of motion is derived that does exhibit this desirable property, and an approximate numerical implementation is presented. Exact analytical results are also presented, for Gaussian wavepacket propagation under quadratic potentials.
Schrodinger’s equation is a basic mathematical expression for a single particle in quantum mechanics. However, its physical meaning has never been assigned in any standard quantum mechanics text book. Here, a detailed complete re-derivation of hidden variable equation in David Bohm’s 1952 papers is given to assign some physical meanings to Schrodinger’s equation. This work shows some required detailed transformation of Schrodinger’s Equation into a form of Newtonian second law of motion with definite trajectory with some initial conditions given. All the quantum effects come from the single localized term called the quantum potential.