Electric charge can be calculated through Gauss law by integrating electric field over closed surface - what has also topological analogue if interpreting spatial curvature of field as electric field.
In similar way they experimentally get Coulomb-like interaction in liquid crystals: https://www.nature.com/articles/s41598-017-16200-z
Temporal change fits better magnetic field, e.g. in Faber's model: https://inspirehep.net/files/f…21cb8020c10f84ca4fdf0ff68
This way he get Coulomb potential if calculating energy of pairs of charges in various distances ... with slight deformation for very small distances - as in running coupling effect.
Electric charge can be calculated through Gauss law by integrating electric field over closed surface
Unluckily this does not work for the near field of electron/proton. It's perfect for the far field or a large set of charges. It's does not fit particles charges. Also the Coulomb law breaks down for the e/p near field (just gives you 3-4 digits out of 10). So for basic physics you must find an other approach.
This is one reason why the standard model is a total fail. Charge is not a point!
Indeed electron being perfect point makes no sense - e.g. would have infinite energy of electric field, wouldn't have running coupling effect (as experiment and Faber have).
Here are gathered experimental arguments for size of electron: https://physics.stackexchange.…ries-for-size-of-electron
And so what I and Faber emphasize is regularization - deformation of this field to prevent infinite energy, what can be easily realized using Higgs-like potential: preferring e.g. unitary vectors, but also allowing to deform them to prevent infinity, discontinuity.
See e.g. https://arxiv.org/abs/2108.07896
or just top above:
And so what I and Faber emphasize is regularization - deformation of this field to prevent infinite energy, what can be easily realized using Higgs-like potential:
In math you can do all you like. But in physics a potential needs a physical generator. There is none for Higgs.
I would no longer waste my time with the standard model math as obviously the solutions for fields using a 3:1 metric are fringe physics. There are no physical generators for 3D fields! The only allowed actions in physics are 1:1 and 2:1
what leads to 1/r and 1/r2 force fields. The unification with 1/r fields in 3D is not possible! So only a magnetic "field" (flux) solution will help.
I did it in SO(4) 6D! what works pretty well and introduces the needed golden ratio flux quantization.
The approach you did follow is similar to what Kransohovelet did try on the sub-matter = instanton level.
Higgs potential e.g. (|n|2 - 1)^2, quite popular in physics, here just means nontrivial vacuum - e.g. "director field" in liquid crystals, leading to electromagnetism as Goldstone bosons ... potential allows to deform to finite energy to prevent infinite energy singularities.
The 1/r2 (Coulomb) force is proportional to field curvature around topological charges e.g. hedgehog configuration - hence we interpret curvature as electric field, in Gauss law getting charge quantization as topological.
Here they experimentally get even stronger 1/r force in liquid crystals: https://pubs.rsc.org/en/conten…m/c9sm01710k#!divAbstract
Biaxial nematic has SO(3) vacuum in 3D space ... which naturally extends to SO(4) in 4D spacetime - adding 4th axis which tiny perturbations are governed by second set of Maxwell equations - popular GEM approximation of general relativity ( https://en.wikipedia.org/wiki/Gravitoelectromagnetism )
Indeed in physics and our models we have asymptotically 1/r2 Coulomb force for topological/electric charges.
For very low distances it is deformed - so called running coupling ( https://en.wikipedia.org/wiki/…constant#Running_coupling ) seen experimentally and in our models - see distance energy dependence diagram in #742 above.
Update: Prepared slides with lots of materials about this resemblance between liquid crystal topological defects and particle physics:liquid crystal particles.pdfShared with Dropboxwww.dropbox.com