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Progress of Theoretical Physics

Vortex, Spin and Triad for Quantum Mechanics of Spinning Particle.I: General Theory

Takehiko Takabayasi

Published Jul 1, 1983• 59 citations
Publisher

Abstract

It is shown that a natural extension of the hydrodynamical formalism of quantum mechanics for a Schrödinger particle to include vortical flows leads to the hydrodynamical formalism of quantum mechanics for a spinning particle. This latter formalism is then analysed in regard to its characteristic features, especially the subsidiary condition connecting the vorticity of flow with the inhomogeneity of spin field and the existence of spin stress. Also the formalism is brought to completion by establishing the global condition that quantizes circulation around a singular vortex line. The geometro-hydrodynamical formalism which is equivalent to the above hydrodynamical formalism but introduces a triad structure underlying the classical spin is reconstructed on its own footing. The geometrical property of the triad implies the invariance of theory with respect to the rotation of each triad around its symmetry axis by an arbitrary angle, and this necessitates the introduction of the electromagnetic potential, providing the geometrical interpretation of local gauge invariance. Applications of theory to various special cases and typical examples are deferred to Part II.

DOI
10.1143/ptp.70.1

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It is shown that a natural extension of the hydrodynamical formalism of quantum mechanics for a Schrödinger particle to include vortical flows leads to the hydrodynamical formalism of quantum mechanics for a spinning particle. This latter formalism is then analysed in regard to its characteristic features, especially the subsidiary condition connecting the vorticity of flow with the inhomogeneity of spin field and the existence of spin stress. Also the formalism is brought to completion by establishing the global condition that quantizes circulation around a singular vortex line. The geometro-hydrodynamical formalism which is equivalent to the above hydrodynamical formalism but introduces a triad structure underlying the classical spin is reconstructed on its own footing. The geometrical property of the triad implies the invariance of theory with respect to the rotation of each triad around its symmetry axis by an arbitrary angle, and this necessitates the introduction of the electromagnetic potential, providing the geometrical interpretation of local gauge invariance. Applications of theory to various special cases and typical examples are deferred to Part II.
Hydrodynamical formalism of quantum mechanics is revisited for a particle under external electromagnetic field, with detailed explanations as to its subtle aspects such as the role of singular vortex lines with quantized circulation. The theory is applied to several problems where a charged particle moves under certain new configurations of external magnetic field, which are suitable to provide examples of Aharonov·Bohm effect in different situations. A remarkable consequence is given about the flow in the case of string magnetic field and is employed to obtain solutions explicitly for the movement under the field of a circular magnetic string.
We present the development of a realistic geometro-hydrodynamical formalism of quantum mechanics for the spinning particle, that involves the vortical flows and is based on the idea that the spinor wave represents a new type of physical field, propagating in space-time and influencing a corpuscle embedded in the wave. We assume that this new field is created by the intrinsic spin of the particle and can produce new intrinsic forces and torques affecting the particle with spin. We derive and discuss the physical effects of this field.
In quantum theory, vortex lines arise in the hydrodynamic interpretation of the wave equation. In this interpretation, which is originally due to Madelung, the flow of the probability density for a single particle is described in terms of the hydrodynamic variables. For the sake of simplicity, the standard time-dependent Schrödinger equation, and the related vortex lines embedded in the probability fluid of the quantum particle, are considered here. A vortex line in this case is simply the curve defined by equating the wave function to zero. The linearity of the Schrödinger equation enables us to obtain a large family of exact time-dependent analytic solutions for the wave functions with vortex lines. Moreover, the method is general enough to allow for various initial configurations of the vortex lines. Although the equation of motion of the quantum mechanical probability fluid is different in its literal form from the equations describing the real physical fluid, we believe that the evolution of the vorticity in the quantum and in the real fluid share the same qualitative features that can be described in terms of the topology of the vortex lines configurations. The general phenomena such as the switch-over, creation and annihilation of vortices can be observed in the quantum mechanical fluid. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Using the hydrodynamical formalism of quantum mechanics for a Schrodinger spinning particle, developed by T. Takabayashi, J. P. Vigier and followers, that involves vortical flows, we propose the new geometrical interpretation of the wave-pilot theory. The spinor wave in this interpretation represents an objectively real field and the evolution of a material particle controlled by the wave is a manifestation of the geometry of space. We assume this field to have a geometrical nature, basing on the idea that the intrinsic angular momentum, the spin, modifies the geometry of the space, which becomes a manifold, that is represented as a vector bundle with a base formed by the translational coordinates and time, and the fiber of the bundle, specified at each point by the field of an tetrad eμae^a_μeμa​, forms from the bilinear combinations of spinor wave function. It was shown, that the spin vector rotates following the geodesic of the space with torsion and the particle moves according to the geometrized guidance equation. This fact explains the self-action of the spinning particle. We show that the curvature and torsion of the spin vector line is determined by the space torsion of the absolute parallelism geometry.
Abstract Using the hydrodynamical formalism of quantum mechanics for a Schrödinger spinning particle developed by Takabayashi, Vigier, and followers, which involves vortical flows, we propose a new geometrical interpretation of the pilot wave theory. The spinor wave in this interpretation represents an objectively real field, and the evolution of a material particle controlled by the wave is a manifestation of the geometry of space. We assume this field to have a geometrical nature, basing on the idea that the intrinsic angular momentum, the spin, modifies the geometry of the space, which becomes a manifold, represented as a vector bundle with a base formed by the translational coordinates and time, and the fiber of the bundle, specified at each point by the field of a tetrad eμae^a_{\mu}eμa​, forms from bilinear combinations of the spinor wave function. It has been shown that the spin vector rotates following the geodesic of the space with torsion, and the particle moves according to the geometrized guidance equation. This fact explains the self-action of the spinning particle. We show that the curvature and torsion of the spin vector line is determined by the space torsion of the absolute parallelism geometry.
The formulation of quantum mechanics in terms of the picture of trajectory ensemble or of the hydrodynamical picture, developed in a previous article, is further supplemented for some points. Individual trajectories for some typical non-stationary state are explicitly obtained, illustrating that they generally show complicated fluctuations due to the action of quantum potential. Next, emphasis is placed on the fact that the Schrödinger field admits a mechanical model. Also the Klein-Gordon field is represented with a quasi-irrotational flow under some additional scalar field. This representation of Klein-Gordon field suggests its possible generalization to include the vortical motion. Such treatments of interacting Klein-Gordon and electromagnetic fields involve the formalism of Dirac's recent classical electrodynamics as a limiting case. Some remarks are given on this Dirac's theory. Finally, the picture of trajectory ensemble is generalized so as to represent wave equations with sources. In an appendix criticism is given on the attempt to interpret quantum mechanics in terms of Markoff process.