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arXiv:0802.2269

On possibility of topological interpretation of quantum mechanics

Oleg A. Olkhov

Published Feb 15, 2008• 0 citations
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Abstract

Geometrical model for quantum objects is suggested. It is shown that equations for free material Dirac field and for Maxwell electromagnetic field can be considered as relations describing propagation of the space topological defects. This interpretation explains irrational properties of quantum objects such as wave-corpuscular duality, stochastic behavior, instantaneous nonlocal correlation in EPR-paradox, the light velocity invariance and so on. It is shown also that Dirac equation for hydrogen atom can be also considered as relation describing the space topological defect. Electromagnetic potentials appears within this approach as connectivities of the defect universal covering space and gauge invariance of electromagnetic field happens to be a natural consequence of topological interpretation. Proposed approach can be also considered as a nonlocal model with hidden variables. Preliminary results were published by parts early, and here they are presented completely.

DOI
10.48550/arxiv.0802.2269

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Geometrical model for quantum objects is suggested. It is shown that equations for free material Dirac field and for Maxwell electromagnetic field can be considered as relations describing propagation of the space topological defects. This interpretation explains irrational properties of quantum objects such as wave-corpuscular duality, stochastic behavior, instantaneous nonlocal correlation in EPR-paradox, the light velocity invariance and so on. It is shown also that Dirac equation for hydrogen atom can be also considered as relation describing the space topological defect. Electromagnetic potentials appears within this approach as connectivities of the defect universal covering space and gauge invariance of electromagnetic field happens to be a natural consequence of topological interpretation. Proposed approach can be also considered as a nonlocal model with hidden variables. Preliminary results were published by parts early, and here they are presented completely.
Geometrical model for material Dirac wave field and for Maxwell electromagnetic field is suggested where above fields are considered as propagating regions of the space itself with distorted euclidean geometry. It is shown that equations for these fields can be considered as relations describing the space topological defects. These defects, being closed topological manifolds, are embedded in the outer five-dimensional space, and observable objects appear to be intersections of above defects with the physical space. This interpretation explains irrational properties of quantum objects such as wave-corpuscular duality, stochastic behavior, instantaneous nonlocal correlation in EPR-paradox, the light velocity invariance and so on. Wave-corpuscular properties arise as a result of the defect periodical movement in the outer space relative to its intersection with the physical space, and just this periodical movement attributes phase to the propagating object. Appearance of probabilities within the formalism is a consequence of uncertainty of the closed topological manifold shape, and ensemble of all possible shapes for the same object can be considered as an ensemble of hidden variables that leads to probabilistic description. Embedded in the outer space topological defects provide channels for nonlocal correlations between their intersections-- noninteracting particles in EPR-experiments, and this means that the proposed approach can be considered as a nonlocal model with hidden variables.
It is shown that the Dirac equation for free particle can be considered as a description of specific distortion of the space euclidean geometry (space topological defect). This approach is based on possibility of interpretation of the wave function as vector realizing representation of the fundamental group of the closed topological space-time 4-manifold. Mass and spin appear to be topological invariants. Such concept explains all so called “strange” properties of quantum formalism: probabilities, wave-particle duality, nonlocal instantaneous correlation between noninteracting particles (EPR-paradox) and so on. Acceptance of suggested geometrical concept means rejection of atomistic concept where all matter is considered as consisting of more and more small elementary particles. There is no any particles a priori, before measurement: the notions of particles appear as a result of classical interpretation of the contact of the region of the curved space with a device.
The hypothesis was earlier suggested by the author where all micro-objects are considered as specific distortions of the physical space-time pseudo-Euclidean geometry, namely, as closed topological 4-manifolds. The foundation of the hypothesis is a geometrical interpretation of the basic equation of quantum mechanics for classical (not quantized) wave fields -- the Dirac equation for free particle. Such hypothesis does not contradict to any physical laws and experimental facts and gives firstly an opportunity to explain qualitatively within classical notions (geometrical) the so called “paradoxical” properties of quantum particles such as wave-corpuscular duality, appearance of probabilities in the quantum mechanics formalism, spin, EPR-paradox.To demonstrate prospects for suggested geometrical approach the author early attempted to find new dynamic equations other than known quantum-mechanical ones for atomic spectra calculations. In this work above investigation is being continued on a more rigorous basis, representing a new geometrical interpretation of the equation for hydrogen atoms. Results of calculations of ionization potentials for helium atom are in agreement with experimental data.
Interpretation of wave function for free particle is suggested as a description of microscopic distortion of the space-time geometry, namely, as some closed topological 4-manifold. Such geometrical object looks in three-dimensional Euclidean space as its topological defect having stochastic and wave-corpuscular properties of quantum particle. All possible deformations (homeomorphisms) of closed topological manifold play the role of “hidden variables”, responsible for statistical character of the theory.
The quantum mechanics description of a physical object stretched in space and stable in time from the relativistic space-time properties point of view, introduced in special theory of relativity, is considered and analysed. The mathematical model of physical objects is proposed. This model gives a possibility to unite a description of corpuscular and wave properties of real physical objects, i.e. fields and particles. There are substantiated an approach and a mathematical pattern which give a possibility to describe physical object not only in causal, but also in absolute remote fields of the Minkowski space. Applying the proposed approach to the microcosm description, one can get the equations that in passage to the limit transfer to such quantum mechanics equations as Schrodinger, Klein-Gordon-Fock and in particular case - the wave equation. The event nature of the received equations is discussed. It is shown that all mentioned equations reflect the space-time relativistic properties during the description of the invariant and non-invariant physics object characteristics.