Authors: Vu B Ho
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. A rigorous approach would be a complete formulation of wave dynamics on two and three-dimensional geometries that are classified according to the uniformisation theorem of Riemannian surfaces and the Thurston geometrisation conjecture on three-dimensional differentiable manifolds. However, for the purpose of physical illustration, we will follow a modest approach in which we will 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. The wave dynamics on a circle can also be described in terms of projective elliptic geometry. Since a circle is a 1-sphere which is also a 1-torus therefore the Bohr model of the hydrogen atom can also be viewed as a standing wave on a 1-torus. 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 particular, we show that when the wave dynamics on a projective plane is described in terms of projective elliptic geometry then it is identical to the wave dynamics on a 2-sphere. 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 discuss a transformation of a stationary wave on the fundamental cube into a stationary wave on a 3-sphere despite it still remains unknown whether a 3-sphere can be constructed directly from a cube by gluing its opposite faces. In spite of this uncertainty, however, we speculate that mathematical degeneracy in which an element of a class of objects degenerates into an element of a different but simpler class may play an important role in quantum dynamics. For example, a 2-sphere is a degenerate 2-torus when the axis of revolution passes through the centre of the generating circle. Therefore, it seems reasonable to assume that if an n-torus degenerates into an n-sphere then wavefunctions on an n-torus may also be degenerated into wavefunctions on an n-sphere. Furthermore, since an n-sphere can degenerate itself into a single point, therefore the mathematical degeneracy may be related to the concept of wavefunction collapse in quantum mechanics where the classical observables such as position and momentum can only be obtained from the collapse of the associated wavefunctions for physical measurements. This consideration suggests that quantum particles associated with differentiable manifolds may possess the more stable mathematical structures of an n-torus rather than those of an n-sphere.
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[v1] 2018-10-20 04:55:25
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