[Paper Review] A defect of the electromagnetism and it serves as a hidden variable for quantum mechanics

H. Y. Cui|ArXiv.org|Mar 12, 2006

Experimental and Theoretical Physics Studies3 references3 citations

TL;DR

This paper argues that Maxwell's electromagnetism contains a hidden, undefined component in its 4-vector force formulation due to incomplete orthogonality with 4-velocity, reducing the electromagnetic force to only two independent components in 4D spacetime instead of the expected three. The missing component is proposed as a candidate for a hidden variable in quantum mechanics, with a relativistic wave equation derived in Euclidean 4D space-time (using $ict$) that reproduces the fine structure of the hydrogen atom, confirming consistency with quantum mechanics and indirectly supporting the existence of this undefined component.

ABSTRACT

According to the theory of relativity, the 4-vector force acting on a particle is orthogonal to the 4-vector velocity of the particle, but we found that the electromagnetic force of the Maxwell's theory can not completely satisfy this orthogonality, this incompletion leads us to find that the electromagnetic force has two independent components in the 4 dimensional space time. A 4-vector force has 4 components, because of the orthogonality of 4-vector force and 4-vector velocity, the number of independent components of the force reduces to 3, while the electromagnetic force can merely provide 2 independent components, this situation means that there is an undefined component accompanying the electromagnetic force. This missing undefined component may serve as a hidden variable which quantum mechanics is eager to find for a long time. The primary purpose of this paper is to strictly proof that the electromagnetic force has two independent components in the 4 dimensional space time. We also briefly discuss the possibility that the undefined component of the electromagnetic force serves as a hidden variable for quantum mechanics.

Motivation & Objective

• To identify a fundamental defect in Maxwell's electromagnetism related to the orthogonality of 4-force and 4-velocity.
• To demonstrate that the electromagnetic 4-force provides only two independent components in 4D spacetime, despite the theoretical possibility of three due to orthogonality constraints.
• To propose that the missing, undefined component of the electromagnetic force may serve as a hidden variable for quantum mechanics.
• To validate the existence of this undefined component by deriving the fine structure of the hydrogen atom energy levels using a relativistic wave equation in Euclidean 4D spacetime.

Proposed method

• Formulating the 4-vector force in a 4D spacetime with $x_4 = ict$, ensuring orthogonality between force and velocity via $u_\mu f^\mu = 0$.
• Expressing the electromagnetic 4-force as $f_\mu = q F_{\mu\nu} u^\nu$, where $F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$, with $A_\mu = \frac{k q' u'_\mu}{c^2 r}$.
• Using a 4D Euclidean coordinate system to derive a relativistic wave equation for the hydrogen atom, treating $x_4 = ict$ as a spatial dimension.
• Separating variables in spherical coordinates and solving the radial, angular, and azimuthal parts of the wave equation to obtain energy eigenstates.
• Applying boundary conditions requiring periodicity and standing wave solutions to derive quantized energy levels.
• Matching the derived energy levels $E = m_e c^2 \left[1 + \frac{\alpha^2}{(\sqrt{j^2 - \alpha^2} + s)^2}\right]^{-1/2}$ with the fine structure of the hydrogen atom, confirming consistency with Dirac theory.

Experimental results

Research questions

• RQ1 Does the electromagnetic 4-force in Maxwell’s theory fully satisfy the relativistic orthogonality condition with 4-velocity?
• RQ2 Why does the electromagnetic 4-force only provide two independent components in 4D spacetime, despite the theoretical constraint reducing the number of independent components to three?
• RQ3 Could the missing, undefined component of the electromagnetic 4-force serve as a hidden variable in quantum mechanics?
• RQ4 Can a relativistic wave equation formulated in 4D Euclidean spacetime reproduce the fine structure of the hydrogen atom energy levels?
• RQ5 Is the existence of this undefined component indirectly confirmed by agreement with established quantum mechanical results?

Key findings

• The electromagnetic 4-force in Maxwell’s theory fails to fully satisfy the relativistic orthogonality condition $u_\mu f^\mu = 0$, implying a missing component in its 4D formulation.
• The electromagnetic 4-force provides only two independent components in 4D spacetime, while the orthogonality constraint allows for three, indicating a missing, undefined component.
• The missing component is proposed as a candidate for a hidden variable in quantum mechanics, potentially resolving long-standing foundational issues.
• The derived relativistic wave equation in 4D Euclidean spacetime yields energy levels that exactly match the fine structure of the hydrogen atom as predicted by the Dirac equation.
• The agreement with the fine structure constant $\alpha = e^2/\hbar c$ and the quantized quantum numbers $j = k + |m|$ confirms the physical consistency of the model and indirectly supports the existence of the undefined component.
• The solution of the wave equation requires a standing wave condition, leading to the quantization rule $\frac{1}{\hbar c} \int_0^\infty \sqrt{(E + e^2/r)^2 - m_e^2 c^4 - \frac{j^2 \hbar^2 c^2}{r^2}} dr = \pi s$, which yields the correct energy spectrum.