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V. M. Simulik and I. Yu. Krivsky, Once more on the derivation of the Dirac equation, arXiv: 1309.0573v2 [math-ph] 22 Sep 2013.

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Quantum Particles as 3D Differentiable Manifolds

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

International Journal of Physics. 2018, Vol. 6 No. 5, 139-146
DOI: 10.12691/ijp-6-5-1
Copyright © 2018 Science and Education Publishing

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.

Correspondence to: Vu  B Ho, Advanced Study, 9 Adela Court, Mulgrave, Victoria 3170, Australia. Email:


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.