Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

JOURNAL OF ADVANCES IN PHYSICS

Introduction to Quantum Mechanics in Four-Dimensional Space Based on the Theory of Space

Dariusz Stanisław Sobolewski

Published Dec 21, 2024• 0 citations
Publisher

Abstract

The article, based on the structure of elementary particles in the form of spatial vortices revealed in the Theory of Space, the four-dimensional structure of space, and the essence of electromagnetic interactions, formulates Hamilton and Hamiltonian-Jacobi functions describing the free motion of an elementary particle and the system of elementary particles in a hydrogen atom, approximating elementary particles with a single spatial vortex. It is shown that the complex nutation and precession of a spatial vortex identified with an elementary particle can be identified with matter waves described by transverse de Broglie plane waves of classical quantum mechanics on the boundary hypersurface βϰβ, which also determines the limit of applicability of classical quantum mechanics, which reduces the complex trajectory of spatial vortices to the analysis of deformations on one of the boundary hypersurfaces.

DOI
10.24297/jap.v22i.9695

Related papers

Analytical Quantum Dynamics in Infinite Phase Space

Maxim Raykin

We develop a dynamical theory, based on a system of ordinary differential equations describing the motion of particles which reproduces the results of quantum mechanics. The system generalizes the Hamilton equations of classical mechanics to the quantum domain, and turns into them in the classical limit $\hbar\rightarrow 0$. The particles' motions are completely determined by the initial conditions. In this theory, the wave function $ψ$ of quantum mechanics is equal to the exponent of an action function, obtained by integrating some Lagrangian function along particle trajectories, described by equations of motion. Consequently, the equation for the logarithm of a wave function is related to the equations of motion in the same way as the Hamilton-Jacobi equation is related to the Hamilton equations in classical mechanics. We demonstrate that the probability density of particles, moving according to these equations, should be given by a standard quantum-mechanical relation, $ρ=|ψ|^2$. The theory of quantum measurements is presented, and the mechanism of nonlocal correlations between results of distant measurements with entangled particles is revealed. In the last section, we extend the theory to particles with nonzero spin.

arXiv:1204.1540•Apr 6, 2012•0 citations

Quantum mechanics justifies the existence of a fourth space dimension

Ratikanta Das

In this short article, we have tried to generalize Born’s rule on the product of wave functions and applied it to the free particle solution of the Schrödinger equation. The interpretation of the result of this operation reveals that the electron has a four-dimensional Gaussian structure capable of moving as a stable wave pulse. Confirmation of the validity of this interpretation will assert that our universe has an extra space dimension, and the physics of this four-dimensional space is consistent with the quantum theory of three-dimensional space. Again, to have an understanding about the motion of the electron as a four-dimensional wave pulse, we attempt to rectify a longstanding misconception about the phase velocity of the matter wave and subsequently derive the de Broglie equation in a unique manner.

Physics Essays•Mar 5, 2026•0 citations

Lectures on Quantum Mechanics (nonlinear PDE point of view)

Alexander Komech

We expose the Schrödinger quantum mechanics with traditional applications to Hydrogen atom. We discuss carefully the experimental and theoretical background for the introduction of the Schrödinger, Pauli and Dirac equations, as well as for the Maxwell equations. We explain in detail all basic theoretical concepts. We explain all details of the calculations and mathematical tools: Lagrangian and Hamiltonian formalism for the systems with finite degree of freedom and for fields, Geometric Optics, the Hamilton-Jacobi equation and WKB approximation, Noether theory of invariants including the theorem on currents, four conservation laws (energy, momentum, angular momentum and charge), Lie algebra of angular momentum and spherical functions, scattering theory (limiting amplitude principle and limiting absorption principle), the Lienard-Wiechert formulas, Lorentz group and Lorentz formulas, Pauli theorem and relativistic covariance of the Dirac equation, etc. We give a detailed oveview of the conceptual development of the quantum mechanics, and expose main achievements of the ``old quantum mechanics'' in the form of exercises. One of our basic aim in writing this book, is an open and concrete discussion of the problem of a mathematical description of the following two fundamental quantum phenomena: i) Bohr's quantum transitions and ii) de Broglie's wave-particle duality. Both phenomena cannot be described by autonomous linear dynamical equations, and we give them a new mathematical treatment related with recent progress in the theory of global attractors of nonlinear hyperbolic PDEs.

arXiv:math-ph/0505059•May 21, 2005•5 citations

The wave function in Quantum Mechanics

Kiyoung Kim

Through a new interpretation of Special Theory of Relativity and with a model given for physical space, we can find a way to understand the basic principles of Quantum Mechanics consistently from Classical Theory. It is supposed that natural phenomena have a connection with intangible reality which cannot be measured directly. Furthermore, the intangible reality is supposed as vacuum particles -- stationary vacuum electrons as a model. In addition, 4-Dimensional Complex Space is introduced, in which each dimension has an internal complex space.

arXiv:quant-ph/9701023•Jan 19, 1997•1 citation

Are elementary particles vortices in a 'vacuum' fluid?

Daniël Bischoff van Heemskerck

Quantum theory and relativity exhibit several formal analogies with fluid mechanics. This paper examines under which conditions a classical fluid model may reproduce the simplest formulations of both theories. By identifying the angular momentum of irrotational vortices in an inviscid, barotropic, isothermal fluid with sound speed c with the reduced Planck constant, and incorporating Korteweg capillary stress, a complex wave equation describing the momentum and continuity fields of a vortex is obtained. When uniform convection is introduced, the weak field approximation of this wave equation is equivalent to Schrödinger’s equation. The model is shown to yield classical analogues for de Broglie wavelength, the Einstein–Planck relation, the Born rule and the uncertainty principle. Accounting for the retarded propagation of the wavefield of a vortex in convection yields the Lorentz transformation and the Klein–Gordon equation, with Schrödinger’s equation appearing as the low-Mach-number limit. These results demonstrate that, under explicit assumptions, a classical continuum can reproduce quantum and relativistic theory in their simplest form, without assuming the postulates principal to those theories.

•Dec 20, 2025•0 citations

Modified de Broglie approach applied to the Schrödinger and Klein-Gordon equations

James Baker‐Jarvis, Pavel Kaboš

The goal of this paper is an extension of the de Broglie wave mechanics model for a single spinless particle in an electromagnetic field. The analysis indicates that the motion of a particle separates naturally into particle dynamics through the classical Hamilton-Jacobi equation and quantum wave behavior through a pilot or interaction wave equation. The interaction wave equation travels at the classical particle velocity. We study gauge invariance and interpret it in the light of the interaction wave. The Heisenberg uncertainty relations are shown to be implicit in the interaction wave. We also develop a complex quantum-mechanical, relativistic energy-momentum conservation expression using a complex quantum-mechanical four-vector.

Physical Review A•Oct 21, 2003•19 citations
The article, based on the structure of elementary particles in the form of spatial vortices revealed in the Theory of Space, the four-dimensional structure of space, and the essence of electromagnetic interactions, formulates Hamilton and Hamiltonian-Jacobi functions describing the free motion of an elementary particle and the system of elementary particles in a hydrogen atom, approximating elementary particles with a single spatial vortex. It is shown that the complex nutation and precession of a spatial vortex identified with an elementary particle can be identified with matter waves described by transverse de Broglie plane waves of classical quantum mechanics on the boundary hypersurface βϰβ, which also determines the limit of applicability of classical quantum mechanics, which reduces the complex trajectory of spatial vortices to the analysis of deformations on one of the boundary hypersurfaces.
We develop a dynamical theory, based on a system of ordinary differential equations describing the motion of particles which reproduces the results of quantum mechanics. The system generalizes the Hamilton equations of classical mechanics to the quantum domain, and turns into them in the classical limit ℏ→0\hbar\rightarrow 0ℏ→0. The particles' motions are completely determined by the initial conditions. In this theory, the wave function ψψψ of quantum mechanics is equal to the exponent of an action function, obtained by integrating some Lagrangian function along particle trajectories, described by equations of motion. Consequently, the equation for the logarithm of a wave function is related to the equations of motion in the same way as the Hamilton-Jacobi equation is related to the Hamilton equations in classical mechanics. We demonstrate that the probability density of particles, moving according to these equations, should be given by a standard quantum-mechanical relation, ρ=∣ψ∣2ρ=|ψ|^2ρ=∣ψ∣2. The theory of quantum measurements is presented, and the mechanism of nonlocal correlations between results of distant measurements with entangled particles is revealed. In the last section, we extend the theory to particles with nonzero spin.
In this short article, we have tried to generalize Born’s rule on the product of wave functions and applied it to the free particle solution of the Schrödinger equation. The interpretation of the result of this operation reveals that the electron has a four-dimensional Gaussian structure capable of moving as a stable wave pulse. Confirmation of the validity of this interpretation will assert that our universe has an extra space dimension, and the physics of this four-dimensional space is consistent with the quantum theory of three-dimensional space. Again, to have an understanding about the motion of the electron as a four-dimensional wave pulse, we attempt to rectify a longstanding misconception about the phase velocity of the matter wave and subsequently derive the de Broglie equation in a unique manner.
We expose the Schrödinger quantum mechanics with traditional applications to Hydrogen atom. We discuss carefully the experimental and theoretical background for the introduction of the Schrödinger, Pauli and Dirac equations, as well as for the Maxwell equations. We explain in detail all basic theoretical concepts. We explain all details of the calculations and mathematical tools: Lagrangian and Hamiltonian formalism for the systems with finite degree of freedom and for fields, Geometric Optics, the Hamilton-Jacobi equation and WKB approximation, Noether theory of invariants including the theorem on currents, four conservation laws (energy, momentum, angular momentum and charge), Lie algebra of angular momentum and spherical functions, scattering theory (limiting amplitude principle and limiting absorption principle), the Lienard-Wiechert formulas, Lorentz group and Lorentz formulas, Pauli theorem and relativistic covariance of the Dirac equation, etc. We give a detailed oveview of the conceptual development of the quantum mechanics, and expose main achievements of the ``old quantum mechanics'' in the form of exercises. One of our basic aim in writing this book, is an open and concrete discussion of the problem of a mathematical description of the following two fundamental quantum phenomena: i) Bohr's quantum transitions and ii) de Broglie's wave-particle duality. Both phenomena cannot be described by autonomous linear dynamical equations, and we give them a new mathematical treatment related with recent progress in the theory of global attractors of nonlinear hyperbolic PDEs.
Through a new interpretation of Special Theory of Relativity and with a model given for physical space, we can find a way to understand the basic principles of Quantum Mechanics consistently from Classical Theory. It is supposed that natural phenomena have a connection with intangible reality which cannot be measured directly. Furthermore, the intangible reality is supposed as vacuum particles -- stationary vacuum electrons as a model. In addition, 4-Dimensional Complex Space is introduced, in which each dimension has an internal complex space.
Quantum theory and relativity exhibit several formal analogies with fluid mechanics. This paper examines under which conditions a classical fluid model may reproduce the simplest formulations of both theories. By identifying the angular momentum of irrotational vortices in an inviscid, barotropic, isothermal fluid with sound speed c with the reduced Planck constant, and incorporating Korteweg capillary stress, a complex wave equation describing the momentum and continuity fields of a vortex is obtained. When uniform convection is introduced, the weak field approximation of this wave equation is equivalent to Schrödinger’s equation. The model is shown to yield classical analogues for de Broglie wavelength, the Einstein–Planck relation, the Born rule and the uncertainty principle. Accounting for the retarded propagation of the wavefield of a vortex in convection yields the Lorentz transformation and the Klein–Gordon equation, with Schrödinger’s equation appearing as the low-Mach-number limit. These results demonstrate that, under explicit assumptions, a classical continuum can reproduce quantum and relativistic theory in their simplest form, without assuming the postulates principal to those theories.
The goal of this paper is an extension of the de Broglie wave mechanics model for a single spinless particle in an electromagnetic field. The analysis indicates that the motion of a particle separates naturally into particle dynamics through the classical Hamilton-Jacobi equation and quantum wave behavior through a pilot or interaction wave equation. The interaction wave equation travels at the classical particle velocity. We study gauge invariance and interpret it in the light of the interaction wave. The Heisenberg uncertainty relations are shown to be implicit in the interaction wave. We also develop a complex quantum-mechanical, relativistic energy-momentum conservation expression using a complex quantum-mechanical four-vector.