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?