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Global Journal of Human Social Science

A CLASSIFICATION OF QUANTUM PARTICLES

Vu B Ho

Published Jan 1, 2018• 1 citation
Publisher

Abstract

In this work, by summarising our recent works on the differential geometric and topological structures of quantum particles and spacetime manifolds, we discuss the possibility to classify quantum particles according to their intrinsic geometric structures associated with differentiable manifolds that are solutions to wave equations of two and three dimensions. We show that fermions of half-integer spin can be identified with differentiable manifolds which are solutions to a general two-dimensional wave equation, in particular, a two dimensional wave equation that can be derived from Dirac equation. On the other hand, bosons of integer spin can be identified with differentiable manifolds which are solutions to a general three-dimensional wave equation, in particular, a three-dimensional wave equation that can be derived from Maxwell field equations of electromagnetism. We also discuss the possibility that being restricted to three-dimensional spatial dimensions we may not be able to observe the whole geometric structure of a quantum particle but rather only the cross-section of the manifold that represents the quantum particle and the space in which we are confined. Even though not in the same context, such view of physical existence may comply with the Copenhagen interpretation of quantum mechanics which states that the properties of a physical system are not definite but can only be determined by observations.

DOI
10.13140/rg.2.2.23239.96165

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In this work, by summarising our recent works on the differential geometric and topological structures of quantum particles and spacetime manifolds, we discuss the possibility to classify quantum particles according to their intrinsic geometric structures associated with differentiable manifolds that are solutions to wave equations of two and three dimensions. We show that fermions of half-integer spin can be identified with differentiable manifolds which are solutions to a general two-dimensional wave equation, in particular, a two dimensional wave equation that can be derived from Dirac equation. On the other hand, bosons of integer spin can be identified with differentiable manifolds which are solutions to a general three-dimensional wave equation, in particular, a three-dimensional wave equation that can be derived from Maxwell field equations of electromagnetism. We also discuss the possibility that being restricted to three-dimensional spatial dimensions we may not be able to observe the whole geometric structure of a quantum particle but rather only the cross-section of the manifold that represents the quantum particle and the space in which we are confined. Even though not in the same context, such view of physical existence may comply with the Copenhagen interpretation of quantum mechanics which states that the properties of a physical system are not definite but can only be determined by observations.
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In our previous works, we suggest that quantum particles are composite physical objects endowed with the geometric and topological structures of their corresponding differentiable manifolds that would allow them to imitate and adapt to physical environments. In this work, we show that Dirac equation in fact describes quantum particles as composite structures that are in a fluid state in which the components of the wavefunction can be identified with the stream function and the velocity potential of a potential flow formulated in the theory of classical fluids. We also show that Dirac quantum particles can manifest as standing waves which are the result of the superposition of two fluid flows moving in opposite directions. However, for a steady motion a Dirac quantum particle does not exhibit a wave motion even though it has the potential to establish a wave within its physical structure, therefore, without an external disturbance a Dirac quantum particle may be considered as a classical particle defined in classical physics. And furthermore, from the fact that there are two identical fluid flows in opposite directions within their physical structures, the fluid state model of Dirac quantum particles can be used to explain why fermions are spin-half particles.
In this work we discuss the topological transformation of quantum dynamics by showing the wave dynamics of a quantum particle on different types of topological structures in various dimensions from the fundamental polygons of the corresponding universal covering spaces. This is not the view from different perspectives of an observer who simply uses different coordinate systems to describe the same physical phenomenon but rather possible geometric and topological structures that quantum particles are endowed with when they are identified with differentiable manifolds that are embedded or immersed in Euclidean spaces of higher dimension. We present our discussions in the form of Bohr model in one, two and three dimensions using linear wave equations. In one dimension, the fundamental polygon is an interval and the universal covering space is the straight line and in this case the standing wave on a finite string is transformed into the standing wave on a circle which can be applied into the Bohr model of the hydrogen atom. In two dimensions, the fundamental polygon is a square and the universal covering space is the plane and in this case the standing wave on the square is transformed into the standing wave on different surfaces that can be formed by gluing opposite sides of the square, which include a 2-sphere, a 2-torus, a Klein bottle and a projective plane. In three dimensions, the fundamental polygon is a cube and the universal covering space is the three-dimensional Euclidean space. It is shown that a 3-torus and the manifold K×S^1 defined as the product of a Klein bottle and a circle can be constructed by gluing opposite faces of a cube therefore in three-dimensions the standing wave on a cube is transformed into the standing wave on a 3-torus or on the manifold K×S^1. We also suggest that the mathematical degeneracy may play an important role in quantum dynamics and be associated with the concept of wavefunction collapse in quantum mechanics.
In this work we discuss the nature of matter wave of quantum particles whose dynamics are described by Dirac equation. Since we have shown that both Dirac equation and Maxwell field equations can be derived from a general system of linear first order partial differential equations, it is reasonable to suggest that matter wave may have similar physical formation to that of the electromagnetic field in the sense that matter wave is also the result of a coupling of two physical fields, such as the electric field and the magnetic field in electromagnetism. In particular, we show that when Dirac equation is reformulated as a system of real equations, like Maxwell field equations, then Dirac equation describes a quantum particle as a string-like object whose cross-section vibrates as a membrane.