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arXiv:2606.05955

Magnetic flux as a quantized Lorentz pseudoscalar and its relation to electric charge quantization

Cyril Belardinelli

quant-phSubmitted Jun 4, 2026• Announced Jun 5, 2026• 0 citations
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Abstract

In this paper, we re-examine the well-known question of why electric charges exist only in quantized portions. In this context, we revisit the motion of a charged particle in a field-free region around a current-carrying solenoid. Solving the corresponding Schrödinger equation leads to a simultaneous quantization condition for the magnetic flux and the electric charge. We also demonstrate the Lorentz invariance of this condition by showing that the magnetic flux behaves like a pseudoscalar under Lorentz transformations.

Comments
9 pages, 2 figures

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In this paper, we re-examine the well-known question of why electric charges exist only in quantized portions. In this context, we revisit the motion of a charged particle in a field-free region around a current-carrying solenoid. Solving the corresponding Schrödinger equation leads to a simultaneous quantization condition for the magnetic flux and the electric charge. We also demonstrate the Lorentz invariance of this condition by showing that the magnetic flux behaves like a pseudoscalar under Lorentz transformations.
It is noted that the elementary electric charge is equal to e , from which fact the well-known principle of quantization of the electric charge definitely follows, namely, the electric charge is quantized and a quantum is the charge of the electron e (or positron). Or any change in the charge is equal to an integer number of electrons (or positrons). The formally identical transformation of the principle of quantization of the electric charge allows formulating the principle of quantization of the magnetic flux, namely, the quantity inverse to the magnetic flux is quantized and a quantum is the inverse of the quantum of the F. London magnetic flux. Or any change in the reciprocal of the magnetic flux is equal to an integer number of quantities inverse to the quantum of the F. London magnetic flux. The reciprocal of the magnetic flux quantum is equal to the sum of two quantities inverse to the quantum of the F. London magnetic flux. The validity of the principle of quantization of the magnetic flux with respect to the hydrogen atom is illustrated by Theorem 1: quantization of the energy of the hydrogen atom is a consequence of the principle of quantization of the magnetic flux. Theorem 2 is proved: the magnetic flux of a hydrogen atom in the ground state is equal to the quantum of the F. London magnetic flux. Theorem 3 is proved: the quantum of the magnetic flux is not the minimum possible for a nonzero magnetic flux. It is established that the quantum of the magnetic flux is not a quantum in the sense of a portion (like the F. London quantum). A quantum is the inverse of the quantum of the F. London magnetic flux. It is established that the magnitude of the magnetic flux quantum is not the minimum possible for a nonzero magnetic flux. It is established that the magnetic flux of a hydrogen atom in the ground state is equal to the quantum of the F. London magnetic flux. A discrete set of energies of the hydrogen atom is noted to be a consequence of the solution of the Schrödinger equation, which, in turn, is phenomenological. The reverse discourse used in the proof of Theorem 1 can show that the Schrödinger equation is a consequence of the principle of quantization of the magnetic flux.
The problem of the relation between the Ahronov-Bohm effect and traditional postulates of electrodynamics, which claim that only electric and magnetic fields are observable, is resolved by denial of the statement about validity of the Maxwell equations for microscopic fields. We proceed from the idea that the Maxwell equations, as the generalization of experimental data, are valid only for averaged values. We show that microscopic electrodynamics should be based on postulation of the d'Alembert equations for four-vector of the electromagnetic field potential. The Lorentz condition is valid only for the averages and provides the implementation of the Maxwell equations for averages. This concept eliminates the problem of electromagnetic field quantization and provides the correctness of all known results of quantum electrodynamics. Therefore, the "virtuality" of the longitudinal and scalar photons has a formal mathematical character, conditioned by the Maxwell equations for averaged fields. The longitudinal and scalar photons provide not only the Coulomb interaction of charged particles, but also allow the electrical Aharonov-Bohm effect.
In this paper one deals with the quantization of mesoscopic LC-circuits under the influence of an external time dependent voltage. The canonically conjugated variables, such as given by the electric charge and the magnetic flux, get established by resorting to the Hamiltonian equations of motion provided by both Faraday and Kirchhoff laws. This time the discretization of the electric charge is accounted for, so that magnetic flux operators one looks for should proceed in terms of discrete derivatives. However, the flux operators one deals with are not Hermitian, which means that subsequent symmetrizations are in order. The eigenvalues characterizing such operators can be readily established in terms of twisted boundary conditions. Besides the discrete Schrödinger equation with nearest-neighbor hopping, a nontrivial next nearest neighbor generalization has also been established. Such issues open the way to the derivation of persistent currents in terms of effective k-dependent Hamiltonians. Handling the time dependent voltage within the nearest neighbor description leads to the derivation of dynamic localization effects in L-ring configurations, such as discussed before by Dunlap and Kenkre. The onset of the magnetic flux quantum has also been discussed in some more detail.
In some textbooks on quantum mechanics, the description of flux quantization in a superconductor ring based on the Aharonov-Bohm effect may lead some readers to a (wrong) conclusion that flux quantization occurs as well for a long solenoid with the same quantization condition in which the charge of cooper pair 2e2e2e is replaced by the charge of one electron eee. It is shown how this confusion arises and how can one avoid it.
The motion of a charged particle moving on a flat surface is studied through the constants of motion associated with the system, given the magnetic gauge. The usual Landau’ solution and the non-separable solution for the Landau’s gauge are found, and a new non-separable solution for the symmetric gauge is given. As a consequence of this, the quantization of the magnetic flux results from the invariance of the solutions under the unitary transformations which arise from the operator’s constants of motion.
We summarize our results about the quantization of a charged particle motion without spin inside a flat box under a static electromagnetic field with Landau’s gauge for the magnetic field, where Fourier’s transformation was used to analyze the problem, to point out that there exists a wave function which is different to that one given by Landau with the same Landau’s levels. The quantization of the magnetic flux is deduced differently to previous one, and a new solution is presented for the case of symmetric gauge of the magnetic field, and having the same Landau’s levels.