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Motion of a charged particle in its own field: avoiding the ultraviolet divergence

This paper proposes that the Current Magnetisation Hypothesis (CMH) resolves the ultraviolet divergence and infinite self-energy of classical point charges by introducing an effective magnetisation potential that vanishes at the source worldline, thereby yielding a finite self-field and zero self-force without requiring mass renormalisation, while also offering a method to eliminate infrared divergences in extended sources.

Original authors: Sherif Tawfik

Published 2026-09-09

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CC BY 4.0

Original authors: Sherif Tawfik

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the world of classical physics, a single electric charge is a simple thing: a tiny point that creates an electric field stretching out into space. When that charge moves, it also creates a magnetic field. For over a century, physicists have tried to calculate exactly how a charged particle interacts with its own field. The problem is that if you treat the particle as a true mathematical point with no size, the math breaks down. As you get closer and closer to the particle, the energy of its own field grows without limit, becoming infinite. This "ultraviolet divergence" has forced scientists to use complicated workarounds, such as pretending the particle has a tiny but finite size, or subtracting infinite numbers from other infinite numbers to get a finite result. It is a persistent headache in the foundations of electromagnetism, suggesting that our standard picture of a point charge might be missing something fundamental about how it behaves in its own immediate vicinity.

A researcher at Deakin University, Sherif Tawfik, has proposed a different way to look at this problem by shifting the focus from a single isolated charge to the collective behavior of charges in a conductor. Instead of asking how one point charge interacts with itself, the study examines a concept called the "current magnetisation hypothesis." This idea suggests that magnetic fields in tiny conductors arise not from a single moving charge, but from the correlated motion of many charges interacting with one another. By building a mathematical model based on these interactions, Tawfik found that the troublesome infinite energy disappears entirely without needing any artificial fixes or hidden assumptions.

The core of the discovery lies in how the new model handles the space right next to the particle. In the traditional view, the potential energy of a particle's own field explodes as you approach it, like a tower of blocks growing infinitely tall as you get closer to the base. In Tawfik's new model, the behavior is the exact opposite. As the observation point gets closer to the particle's path, the potential does not grow; it shrinks. It vanishes smoothly and linearly, reaching exactly zero right at the particle's location. Because the potential drops to zero instead of shooting up to infinity, the magnetic field generated by the particle remains finite and well-behaved. The energy stored in this field near the particle is no longer infinite; it is a specific, calculable number. This result holds true even when the particle is moving at high speeds, and it requires no arbitrary cutoffs or adjustments to the mass of the particle to make the math work.

This finding also resolves a long-standing issue regarding the force a particle exerts on itself. In the standard theory, the infinite self-energy leads to a self-force that causes unphysical behaviors, such as particles accelerating before a force is even applied or accelerating forever without limit. In this new framework, because the self-potential is exactly zero at the particle's location, the self-force is also exactly zero. The particle does not push against itself in a way that creates these paradoxes. The model achieves this naturally through its mathematical structure, which treats the magnetic field as an effective result of charge correlations rather than a direct property of a single point. This means the equations describing the motion of the particle are stable and do not require the complex, often unstable corrections found in older theories.

However, the study also clarifies what this new model does and does not solve. While the infinite energy problem near the particle is gone for the magnetisation field, the model explicitly does not remove the usual electrostatic Coulomb self-energy of a bare point charge. The findings are specific to this magnetisation sector. Furthermore, the model still predicts that the magnetic field extends far out into space, fading away slowly. If you sum up the fields from many particles in a large structure, this long-range tail can still cause a different kind of mathematical problem at very large distances, known as an infrared divergence. The paper shows that this can be fixed by a simple adjustment: removing the net magnetic moment from the description of each individual carrier. This adjustment does not change the fact that the energy near the particle is finite; it simply cleans up the behavior at a distance. When applied to structures with specific symmetries, like a ring of current, this adjustment reduces the far-field energy to the correct, expected level, matching what is observed in real-world systems like the electron currents in a benzene molecule.

The researchers confirmed these theoretical results through detailed numerical experiments. They simulated a particle moving in a circle and tracked the field values as they approached the particle's path. The data showed that the new model's potential dropped to zero exactly as predicted, while the traditional model's potential grew infinitely large. The energy density in the new model remained flat and finite, whereas the traditional model's energy density spiked wildly. These simulations also verified that the model satisfies the fundamental laws of electromagnetism and that the adjustments made to fix the long-range behavior did not spoil the short-range results.

Ultimately, this work offers a fresh perspective on an old problem. It suggests that the infinite self-energy of a point charge might be an artifact of looking at the charge in isolation. By viewing the magnetic field as a product of interactions between charges, the model naturally avoids the infinities that have plagued physics for decades. It provides a way to describe the motion of charged particles in their own fields that is mathematically clean, free of infinite values, and consistent with the laws of relativity, without needing to invent new particles or modify the basic laws of physics. The findings are specific to this magnetisation field and do not change the electrostatic energy of a single charge, but they offer a robust, divergence-free description of how magnetic fields behave at the smallest scales.

Technical Summary: Motion of a Charged Particle in its Own Field: Avoiding the Ultraviolet Divergence

Problem Statement
The paper addresses the long-standing difficulty in classical electrodynamics regarding the self-interaction of a point charge. In the standard Liénard–Wiechert formalism, the vector potential $A^\mu$ and the associated field tensor $F^{\mu\nu}$ diverge as the observation point approaches the particle's worldline ( $R \to 0$ ). This singularity leads to an ultraviolet-divergent self-energy ( $U_{self} \sim 1/r_0$ as the cutoff $r_0 \to 0$ ) and an infinite self-force. Historically, resolving this has required external prescriptions such as mass renormalization (Dirac), direct-action theories with absorber conditions (Wheeler–Feynman), or the subtraction of singular terms to derive the Lorentz–Abraham–Dirac (LAD) equation, which itself suffers from runaway and pre-acceleration pathologies.

Methodology
The author employs the Current Magnetisation Hypothesis (CMH), a framework that models magnetic fields in nanoscale conductors not as smoothed current densities, but as arising from microscopic charge–charge correlations.

The CMH Potential: The paper utilizes a covariant four-potential $A^\mu_{CMH}$ derived from an axial source four-vector $S^\mu_i$ carried by each charge carrier. This potential is constructed using a screened radial kernel $B(\ell) = \kappa/(\ell + \kappa)$ and the projected field-source separation $\Delta^\mu$ .
Analytical Approach: The study analyzes the behavior of $A^\mu_{CMH}$ and the resulting field tensor $F^{\mu\nu}$ in the limit where the observation point approaches the source worldline ( $\ell \to 0$ ). It compares the scaling behavior of the CMH potential against the Liénard–Wiechert potential.
Ensemble Treatment: For systems of multiple carriers, the paper investigates the far-field behavior. It proposes a "residual construction" where the net weighted axial-source moment is subtracted from every carrier's source term to eliminate the leading-order long-range tail.
Numerical Verification: The theoretical claims are tested via numerical experiments simulating a single carrier on a circular worldline and an ensemble of static carriers, verifying the scaling of potentials, field tensors, and energy densities.

Key Contributions and Results

Linear Vanishing of Self-Potential:
Unlike the Liénard–Wiechert potential, which diverges as $1/\ell$ , the CMH self-potential vanishes linearly ( $A^\mu_{CMH} \sim \ell$ ) as the observation point approaches the worldline. This is a direct consequence of the potential's algebraic structure, which includes an explicit factor of the projected separation $\Delta^\mu_i$ .
Bounded Self-Field and Finite Self-Energy:
Because the potential vanishes linearly, the field tensor $F^{\mu\nu}$ remains bounded (though generally direction-dependent) at the worldline. Consequently, the electromagnetic stress-energy tensor $T^{\mu\nu}$ is bounded, and the integrated field energy in any neighborhood of the worldline is finite. This result is achieved without any ultraviolet cutoff, subtraction, or mass renormalization.
Vanishing Self-Force:
The self-coupling action $S_{self} = -\int d\tau u_\mu A^\mu_{CMH}$ is identically zero because $A^\mu_{CMH}$ vanishes on the worldline. Consequently, the variational derivative (the self-force) is identically zero. This eliminates the need for the LAD equation and avoids its associated runaway and pre-acceleration solutions within the magnetization sector.
Resolution of Infrared Divergence in Ensembles:
While a single carrier's field is ultraviolet-regular, it retains a standard $1/r$ magnetic tail. When summed over an extended structure with a non-zero net moment, this leads to an infrared divergence in the total field energy. The paper demonstrates that subtracting the net moment from every carrier's source (the residual construction) removes this leading tail term. For sources with rotational symmetry (e.g., a bound ring current), this reduces the far-field decay to the correct magnetic-dipole order ( $1/r^3$ ), rendering the total field energy finite at both short and long distances.
Lorenz Gauge Identity:
The CMH potential satisfies the Lorenz gauge condition ( $\partial_\mu A^\mu = 0$ ) identically, derived from the antisymmetry of the projector and the Levi-Civita tensor structure, without requiring additional constraints.

Significance and Claims
The paper claims to provide an ultraviolet-regular effective magnetization field that resolves the self-energy divergence of the magnetization sector of a classical point charge without ad-hoc regularization or renormalization.

Scope: The results apply specifically to the magnetization sector of the CMH framework, which describes magnetic fields arising from charge–charge correlations. The paper explicitly states that this construction does not remove the electrostatic Coulomb self-energy of a bare point charge; it addresses only the magnetic self-interaction.
Physical Interpretation: The CMH potential is treated as an effective field generated by correlations, not as the Liénard–Wiechert field of a single boosted charge. In this view, a lone moving charge with no correlated source partners possesses no magnetic field, and thus no magnetic self-force.
Practical Implication: In the context of nanoscale calculations (where carrier drift speeds are non-relativistic), the bounded nature of the self-field implies that self-interaction corrections are unnecessary when summing contributions from multiple carriers, provided the net moment is handled correctly to ensure infrared convergence.

The paper concludes that the CMH framework offers a mathematically consistent description of magnetic self-interaction where the self-force vanishes and the self-energy is finite, avoiding the pathologies of traditional point-charge electrodynamics within its defined domain.

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