I A question about the origin of Coulomb's law and point charge divergence

[Vadim 123]

"Hello everyone,

I've been studying the foundations of electromagnetism and have a conceptual question that bothers me.

We know that Coulomb's law is incredibly accurate, but its standard derivation feels somewhat postulated rather than derived from more fundamental principles. Furthermore, the self-energy of a point charge diverges, which is a clear signal that something is missing or idealized in our classical picture.

My question is this: Are there any known models or theoretical approaches that attempt to derive the inverse-square law (and its QED corrections at short ranges) from a more fundamental geometric or continuum-based picture, while inherently avoiding the divergence problem? I'm thinking of approaches where the charge is not a point but has some fundamental structure related to the properties of space itself.

I'm not proposing anything new, just curious if such lines of thought exist in the literature. I'm aware of Kaluza-Klein theories, but they seem to embed EM into gravity. I'm wondering about approaches where EM emerges prior to or independently of metric gravity.

Thank you for any references or insights!"

 

[renormalize]

My question is this: Are there any known models or theoretical approaches that attempt to derive the inverse-square law (and its QED corrections at short ranges) from a more fundamental geometric or continuum-based picture, while inherently avoiding the divergence problem?

How about non-linear electrodynamics?
https://en.wikipedia.org/wiki/Born–Infeld_model

 

[Vadim 123]

Thank you for the reference to the Born-Infeld model! It's a fascinating example of how non-linearity can regularize field energies.

You are right, my question is indeed related to that direction of thought. The Born-Infeld model introduces a fundamental length scale through its parameter, which prevents the field from becoming singular at the charge's location.

However, my curiosity is more specific: Are you aware of any models where this regularization is not just a property of the field's self-interaction, but arises directly from the charge having a finite, fundamental size or structure?

In other words, models where the electron is not a point, but an extended object with a radius R, and where the standard Coulomb law is an approximation valid only outside this radius? The goal would be to see how such a postulate, combined with conservation laws, might naturally lead to the 1/r law and its short-range corrections without starting from the Maxwell Lagrangian.

I'm interested if such a "top-down" approach, starting from the structure of a charge rather than the properties of the field, has been formally explored

 

[renormalize]

Are you aware of any models where this regularization is not just a property of the field's self-interaction, but arises directly from the charge having a finite, fundamental size or structure?

It's apparently hard to separate the possible structure of the electron from the characteristics of the EM field surrounding it. You might take a look at:
Matt Visser, A classical model for the electron
(https://www.sciencedirect.com/science/article/pii/037596018990337X?via=ihub)
Abstract: The construction of classical and semi-classical models for the electron has had a long and distinguished history. Such models are useful more for what they teach us about field theory than what they teach us about the electron. In this Letter I exhibit a classical model of the electron consisting of ordinary electromagnetism coupled with a self-interacting version of Newtonian gravity. The gravitational binding energy of the system balances the electrostatic energy in such a manner that the total rest mass of the electron is finite.
A very recent article on the subject (with lots of references) is:
Liu & Lang, The Effective Radius of an Electric Point Charge in Nonlinear Electrodynamics
(https://arxiv.org/pdf/2510.11733v1)

 

[Dale]

Furthermore, the self-energy of a point charge diverges, which is a clear signal that something is missing or idealized in our classical picture.

I wouldn’t say that. I would say that it is a signal that classical point particles don’t exist.

Are there any known models or theoretical approaches that attempt to derive the inverse-square law (and its QED corrections at short ranges) from a more fundamental geometric or continuum-based picture, while inherently avoiding the divergence problem?

Doesn’t QED meet your criteria?

 

[Vadim 123]

I wouldn’t say that. I would say that it is a signal that classical point particles don’t exist.

Doesn’t QED meet your criteria?

Not quite accurate. I believe there is a simpler way to describe and explain the phenomenon of field screening and renormalization, based on geometry and without invoking additional entities like virtual photons.
For example, this specific formula yields field corrections that are strikingly similar to those observed in Feynman's experiments on electron scattering at short distances.
$$ W_{21}^{\times} = \frac{64 Q_1 Q_2 R_1^2 R_2^2}{225 \rho_0 D^4}- \frac{8 Q_1 Q_2 R_1^2 R_2}{15 \rho_0 D^3} - \frac{8 Q_1 Q_2 R_1 R_2^2}{15 \rho_0 D^3}+ \frac{Q_1 Q_2 R_1 R_2}{\rho_0 D^2}$$

 

[renormalize]

For example, this specific formula yields field corrections that are strikingly similar to those observed in Feynman's experiments on electron scattering at short distances.
$$ W_{21}^{\times} = \frac{64 Q_1 Q_2 R_1^2 R_2^2}{225 \rho_0 D^4}- \frac{8 Q_1 Q_2 R_1^2 R_2}{15 \rho_0 D^3} - \frac{8 Q_1 Q_2 R_1 R_2^2}{15 \rho_0 D^3}+ \frac{Q_1 Q_2 R_1 R_2}{\rho_0 D^2}$$

How does this formula compare to the well-verified results from QED? In particular, where does Planck's constant $h$ enter it?

 

[Dale]

I believe there is a simpler way to describe and explain the phenomenon of field screening and renormalization, based on geometry and without invoking additional entities like virtual photons

Is this simpler way published in the professional scientific literature? If so, please post the reference. If not, please see our forum rules

 

[Vadim 123]

How does this formula compare to the well-verified results from QED? In particular, where does Planck's constant $h$ enter it?

Thank you for this exceptionally important question. It touches upon the very core of the relationship between the proposed formalism and standard Quantum Electrodynamics (QED).

1. On Planck's Constant (ħ)

In this approach, the fundamental quantum scale is introduced not explicitly through Planck's constant, but through the physical sizes of elementary objects. The characteristic length is played by the constants R₁' and R₂', which define the radii of the density clusters identified with elementary charges (e.g., electrons).

The key assertion is as follows: if we identify this effective radius R' with the Compton wavelength of the electron (ƛ = ħ/mc), which is a natural choice for the size of a relativistic quantum object, then the model automatically begins to operate on the correct scales of the microcosm. Thus, Planck's constant does not disappear; it is effectively "built into" the fundamental parameters of the theory—the cluster sizes.

2. Correlation with QED Results and Observed Effects

This geometric model does not contradict the well-te ted results of QED; on the contrary, it offers a new ontological interpretation for them and derives them from different first principles. This correlation becomes evident when considering the dependence of the effective interaction on distance, which aligns perfectly with data from small-distance scattering experiments:

*Screening Effect: At distances on the order of ~10 R', the model predicts a decrease in interaction energy compared to the "bare" Coulomb law. This directly corresponds to the famous vacuum polarization effect in QED, where virtual electron-positron pairs screen the charge. In our model, the role of the "polarizing medium" is played by the distribution of the perturbed vacuum density around the cluster.

Pre-normalization / Anti-screening Effect: As the distance decreases to ~4-5 R', the model predicts a restoration of the interaction strength. This qualitatively and quantitatively corresponds to the behavior of the running coupling constant in QED, which increases with decreasing distance (increasing momentum transfer) due to the anti-screening effect.

Interaction Enhancement at Ultra-Small Distances: At distances smaller than ~4 R', the model predicts that the interaction energy exceeds the value predicted by the simple Coulomb law. This is a geometric analogue of approaching the "bare charge" in QED, the renormalization of which leads to an infinity. However, in our case, this divergence is naturally cut off by the finite cluster size R', providing a specific, finite prediction for the interaction strength in this region.

Conclusion

Thus, the proposed formula and the underlying geometric model reproduce the entire sequence of phenomena well-known from QED and scattering experiments. The difference lies in the fact that the complex mathematical apparatus of renormalization and virtual particles is replaced by an intuitive geometry of extended objects interacting in a space with an additional dimension. In this formalism, the constants R₁' and R₂' are not merely fitting parameters but **physical carriers of the quantum scale**, allowing for a direct and intuitively clear explanation of phenomena such as screening and the running coupling constant.

The diagram shows the screening and pre-normalization of the interaction energy for two elementary charges with classical radii of 0.1 arbitrary units

 

[Vadim 123]

Is this simpler way published in the professional scientific literature? If so, please post the reference. If not, please see our forum rules

Thank you for pointing that out. You are correct to ask.

This specific geometric approach is currently undergoing the peer-review process. In strict adherence to the forum's rules, I am not presenting the unpublished work itself or its derivation for discussion.

My intent here is solely to inquire about the phenomenological fit: does the behavior described by this formula—specifically the sequence of screening, pre-normalization, and subsequent field enhancement—correspond to what is observed in scattering experiments, such as the running coupling constant? This question is about comparing a theoretical curve to established experimental results, which I believe is within the forum's guidelines.

Once the work is published in a peer-reviewed journal, I will be glad to present it here formally for discussion. I appreciate your understanding and vigilance in maintaining the forum's standards.

 

[Vadim 123]

It's apparently hard to separate the possible structure of the electron from the characteristics of the EM field surrounding it. You might take a look at:
Matt Visser, A classical model for the electron
(https://www.sciencedirect.com/science/article/pii/037596018990337X?via=ihub)
Abstract: The construction of classical and semi-classical models for the electron has had a long and distinguished history. Such models are useful more for what they teach us about field theory than what they teach us about the electron. In this Letter I exhibit a classical model of the electron consisting of ordinary electromagnetism coupled with a self-interacting version of Newtonian gravity. The gravitational binding energy of the system balances the electrostatic energy in such a manner that the total rest mass of the electron is finite.
A very recent article on the subject (with lots of references) is:
Liu & Lang, The Effective Radius of an Electric Point Charge in Nonlinear Electrodynamics
(https://arxiv.org/pdf/2510.11733v1)

Thank you for these excellent and highly relevant references! I am familiar with the direction of research they represent, and it's precisely the conceptual challenges within this framework that led me to the approach I'm exploring.

The models by Visser and Liu & Lang are indeed insightful, as they successfully address the problem of infinite self-energy by introducing a structure or non-linearity. However, they still operate within a paradigm where the electromagnetic field and gravity (or a non-linear Lagrangian) are postulated as fundamental entities, and the finite size of the charge is an input to fix the divergence.

My work attempts a more foundational shift. Instead of starting with the field equations and modifying them, it seeks to derive both the charge structure and the 1/r potential (including its short-range corrections) from a single primitive concept: the geometry of a conserved quantity in a higher-dimensional space. In this picture, the "field" is not a primary entity but a manifestation of the density distribution of this quantity.

The most compelling preliminary result is that this geometric approach naturally reproduces the entire phenomenology of screening and anti-screening (the running coupling) without needing to quantize the field or introduce virtual particles. The finite size of the charge and the corrections to Coulomb's law are not inputs, but outputs of the model.

Once my paper clears peer review, I am very eager to get your detailed feedback, especially on how this geometric perspective compares to the field-theoretic approaches you've kindly pointed out. Your insight would be invaluable.

 

[PeroK]

2. Correlation with QED Results and Observed Effects

This geometric model does not contradict the well-te ted results of QED; on the contrary, it offers a new ontological interpretation for them and derives them from different first principles. This correlation becomes evident when considering the dependence of the effective interaction on distance, which aligns perfectly with data from small-distance scattering experiments:

Coulomb's law breaks down for high-energy scattering. QED predicts this and presents Coulomb's law as a low-energy approximation.

Moreover, representing elementary charged particles as point particles with well-defined position and momentum (as per Coulomb's law) cannot be compatible with wider quantum phenomena. Not least, with spin (intrinsic angular momentum) and entanglement.

 

[berkeman]

Thread is closed for Moderation.

 

[Dale]

In strict adherence to the forum's rules, I am not presenting the unpublished work itself or its derivation for discussion.

My intent here is solely to inquire about the phenomenological fit

The phenomenological fit is something that should be demonstrated in the publication!

We cannot help you with any part of the development or analysis of your work prior to publication. Discussing any part of it, including the phenomenological fit, is not permitted here.

This thread is closed and will remain closed