International Journal of Physics
ISSN (Print): 2333-4568 ISSN (Online): 2333-4576 Website: Editor-in-chief: B.D. Indu
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International Journal of Physics. 2018, 6(5), 139-146
DOI: 10.12691/ijp-6-5-1
Open AccessArticle

Quantum Particles as 3D Differentiable Manifolds

Vu B Ho1,

1Advanced Study, 9 Adela Court, Mulgrave, Victoria 3170, Australia

Pub. Date: September 26, 2018

Cite this paper:
Vu B Ho. Quantum Particles as 3D Differentiable Manifolds. International Journal of Physics. 2018; 6(5):139-146. doi: 10.12691/ijp-6-5-1


As shown in our work on spacetime structures of quantum particles, Schrödinger wavefunctions in quantum mechanics can be utilised to construct the geometric structures of quantum particles which are considered to be three-dimensional differentiable manifolds. In this work we will extend this kind of geometric formulation of quantum particles by showing that wavefunctions that are normally used to describe wave phenomena in classical physics can in fact also be utilised to represent three-dimensional differentiable manifolds which in turns are identified with quantum particles. We show that such identification can be achieved by using a three-dimensional wave equation to construct three-dimensional differentiable manifolds that are embedded in a four-dimensional Euclidean space. In particular, the dual character that is resulted from the identification of a wavefunction with a three-dimensional differentiable manifold may provide a classical basis to interpret the wave-particle duality in quantum mechanics.

quantum particle wave equation 3D differentiable manifold general relativity wave mechanics wave-particle duality

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[1]  Erwin Schrödinger, Collected Papers on Wave Mechanics (AMS Chelsea Publishing, New York, 1982).
[2]  A. Einstein, The Principle of Relativity (Dover Publications, New York, 1952).
[3]  Vu B Ho, Spacetime Structures of Quantum Particles (Preprint, ResearchGate, 2017), viXra 1708.0192v2, Int. J. Phys. vol 6, no 4 (2018): 105-115.
[4]  R. Haberman, Elementary Applied Partial Differential Equations, Prentice-Hall Inc, Sydney, 1987.
[5]  E. Kreyszig, Introduction to Differential Geometry and Riemannian Geometry (University of Toronto Press, 1975).
[6]  Walter A. Strauss, Partial Differential Equation (John Wiley & Sons, Inc., New York, 1992).
[7]  Ray D’Inverno, Introducing Einstein’s Relativity (Clarendon Press, Oxford, 1992).
[8]  Richard S. Hamilton, Three-Manifolds with Positive Ricci Curvature, J. Diff. Geo., 17 (1982) 255-306.
[9]  Huai-Dong Cao and Xi-Ping Zhu, A Complete Proof of the Poincaré and Geometrization Conjectures-Application of the Hamilton-Perelman Theory of the Ricci Flow, Asian J. Math., Vol 10, No. 2, 165-492, June 2006.
[10]  B. H. Bransden and C. J. Joachain, Introduction to Quantum Mechanics (Longman Scientific & Technical, New York, 1989).
[11]  Vu B Ho, Formulation of Maxwell Field Equations from a General System of Linear First Order Partial Differential Equations (Preprint, ResearchGate, 2018), viXra 1802.0055v1.
[12]  Vu B Ho, Formulation of Dirac Equation for an Arbitrary Field from a System of Linear First Order Partial Differential Equations (Preprint, ResearchGate, 2018), viXra 1803.0645v1.
[13]  Vu B Ho, On Dirac Negative Mass and Magnetic Monopole (Preprint, ResearchGate, 2018), viXra 1806.0319v1.
[14]  S. V. Melshko, Methods for Constructing Exact Solutions of Partial Differential Equations, Springer Science & Business Media, Inc, 2005.
[15]  V. M. Simulik and I. Yu. Krivsky, Once more on the derivation of the Dirac equation, arXiv: 1309.0573v2 [math-ph] 22 Sep 2013.
[16]  Vu B Ho, A Classification of Geometric Interactions (Preprint, ResearchGate, 2018), viXra 1805.0329v1.
[17]  Allen Hatcher, Algebraic Topology, 2001.
[18]  K. Yasuno, T. Koike and M. Siino, Thurston’s Geometrization Conjecture and cosmological models, arXiv:gr-qc/0010002v1, 2000.
[19]  Allen Hatcher and William Thurston, Moduli Spaces of Circle Packings, 2015.
[20]  Lewis Ryder, Introduction to General Relativity (Cambridge University Press, Melbourne, 2009).
[21]  Vu B Ho, Euclidean Relativity (Preprint, ResearchGate, 2017), viXra 1710.0302v1.
[22]  Vu B Ho, Temporal Geometric Interactions (Preprint, Research Gate, 2018), viXra 1807.0134v1.
[23]  Vu B Ho, On the Geometric Structure of the Spatiotemporal Manifold (Preprint, ResearchGate, 2018), viXra 1808.0144v1.