Is electron a perfect point? Why charge is quantized?

[Jarek]

Electron is usually imagined as a simple point charge, but in fact it is a very complex entity:



- being electric charge itself means singular(-like?) configuration of electric field - E behaves like 1/r^2,
- it is also magnetic dipole moment - singular(-like?) configuration of magnetic field - B behaves like 1/r^3,
- it acts like a tiny gyroscope: attaching a force leads to response perpendicular to this force and to direction of gyroscopic moment, for example in Larmor precession,
- it has some internal oscillations - seen as zitterbewegung in Dirac theory, or as de Broglie's clock (E = mc^2 = hbar*omega), which can be directly observed (e.g. http://link.springer.com/article/10.100 ... 008-9225-1 ).

These properties suggest that electron is quite a complex entity - how to fit them into a perfect point? (physics doesn't like discontinuities as they would have infinite energy)
The electric field itself says that a single electron would affect the entire universe ... we could even say that this singular field defines the charge (can't we?), that electron is a configuration of the fields (what more is it?) - that it is a soliton of, among others, electromagnetic field.

So what does the popular claim that electron is a point means?
I understand it that the central singularity is perfect E ~ 1/r^2 ... however, calculating energy of such point singularity we get integrate of E^2*r^2 ~ 1/r^2, what is divergent in r = 0.
In other words - point charge would require infinite energy of electric field only - what is a nonsense as we know that 2 x 511keVs is sufficient to create electron-positron pair.

Another argument against point charge is running coupling - from https://en.wikipedia.org/wiki/Coupling_ ... g_coupling
"In particular, at low energies, α ≈ 1/137, whereas at the scale of the Z boson, about 90 GeV, one measures α ≈ 1/127."
So alpha ~ charge^2 decreases for high energy collisions - the effective charge is reduced while particles are very close together.
Doesn't it suggest that the E ~ 1/r^2 is weakened very close to the center - additionally allowing to repair the problem with infinite energy of 1/r^2 electric field?

A connected question is of charge quantization - why electron cannot decay e.g. into two half-charges?
In other words - why Gauss law can return only integer charges?
A natural explanation is topological charge as electric charge, e.g.

where Gauss-Bonnet theorem acts as Gauss law, but counting topological charge - which can be only integer.
Prof. Faber has shown that the simplest topological charges already recreate electromagnetism in 3D - behave accordingly to Maxwell's equations - video lecture, paper, slides, essay about expansion to further particles.

Why electron is believed to be a point? What does it mean?
What would be expected if it wouldn't be?
Doesn't running coupling and infinite energy of point charge suggest that E ~ 1/r^2 is weakened for very small r?
Why electric charge is quantized? Are there different explanations than that is a topological charge?
What is the structure of (e.g. EM) fields of electron?

[FrediFizzx]

...Why electron is believed to be a point? What does it mean?...

I think right now it means simply that it has no constituent parts. At least none have been found so far. However given all the properties of an electron, it doesn't necessarily have to be a point. If you invoke the quantum "vacuum" as a relativistic medium, then for sure it doesn't have to be a point. But upon interactions like in experiments to find constituent parts, the interaction "collapses" to a point. That is probably the best way I can put it.

A question: What is the torsion energy of an electron? Quantitatively?

[Jarek]

Indeed, the question of electron being a point is polluted by the perturbative field theory way of thinking - requiring "a corpuscular part" as a charge or interaction carrier (boson) ... but this picture is idealized - for example imagine antenna producing spherical EM waves - as intensity drops like 1/r^2, it is hard to imagine spherical waves as made of corpuscular photons. We can be nearly certain that there are no additional corpuscular entities inside electron.
From the other side, it is not a question of field being point-like, as single charge affects electric field of the entire Universe in 1/r^2 way.

The question is asking about the structure of fields of electron - for example if E ~ 1/r^2 in the center, what leads to a nonsense: infinite energy of electric field of a charge.
One interpretation of not being a point charge is assuming that
E ~ q(r) / r^2
where q(r) ~ e for r > r0, but q(r) -> 0 for r->0 to get a finite electric field of a charge.
Such reduction of effective charge is suggested by the observed running coupling - for high energy collisions, alpha ~ q^2 is reduced, for example "In particular, at low energies, α ≈ 1/137, whereas at the scale of the Z boson, about 90 GeV, one measures α ≈ 1/127" from https://en.wikipedia.org/wiki/Coupling_ ... andau_pole

This is also a natural type of regularization appearing in soliton particle models (e.g. Faber): for example assume a field of unitary vectors - they can take hedgehog-like configuration v(x) = x/|x| (it has topological charge +1) ... however there is a problem in the center of such hedgehog - we would get discontinuity, and so infinite energy like for electric field of point charge.
The solution is to use Higgs-like potential to constrain e.g. to unitary vectors, like
V(v) = (|v|^2 - 1)^2
it makes that field prefers unitary vectors, but allows to regularize in the center of hedgehog/charge - the field goes to zero there, like q(r) -> 0 for r->0, at cost of nonzero Higgs potential - giving particle mass (rest energy e.g. released while annihilation).
The vacuum dynamics (V~0) are rotations of the field and it recreates electromagnetism (Goldstone bosons of Higgs potential).
The Higgs potential is activated only near singularities-particles (to prevent infinities) - electromagnetism is kind of being deformed into other interactions (weak/strong).

Regarding "torsion" of electron, its gyroscopic moment and magnetic dipole moment doesn't seem to be well explained by spinning - torsion.
The topological charge explanation suggests that its not a dynamical spinning, but a structural one - of configuration like in charge picture above, or equivalently as fluxon/Arbrikosov vortex: quant of magnetic field in superconductor. Take a loop around a singularity and count how many times field (e.g. quantum phase) rotates along this loop.

[Ben6993]

Jarek wrote:
Doesn't it suggest that the E ~ 1/r^2 is weakened very close to the center - additionally allowing to repair the problem with infinite energy of 1/r^2 electric field?

Yes, I agree (for what that is worth as I am not an expert). Likewise, the strong force is weaker at the centre.
Also, the force law is an aggregate law and doesn't seem to me to be fundamental. The force is fundamentally carried out by exchange of bosons which need a finite time and distance to operate in.
I am not saying that the following applies here as a cause of weakening, but it seems to be relevant that if a particle is squeezed for time, say causing it to have a failed interaction attempt, it [or the particle which fails to form properly] can have a reduced or off-shell mass.

[FrediFizzx]

...Regarding "torsion" of electron, its gyroscopic moment and magnetic dipole moment doesn't seem to be well explained by spinning - torsion. The topological charge explanation suggests that its not a dynamical spinning, but a structural one - of configuration like in charge picture above, or equivalently as fluxon/Arbrikosov vortex: quant of magnetic field in superconductor. Take a loop around a singularity and count how many times field (e.g. quantum phase) rotates along this loop.

Torsion is not "spinning". It is a twisting. I am wondering what might happen in Faber's soliton model if he used torsion in S^3 instead of curvature? A reason for that is that in Joy Christian local-realistic model that explains the correlations of EPR, he uses a parallelized 3-sphere topology; zero curvature so flat and non-zero torsion. Perhaps that could solve some of the problems Faber has with his soliton model? Do a search for "torsion energy" and ignore the Russian stuff that has been discredited.

[Jarek]

Ben6993, weak/strong interaction act only very close to particle (singularity) - it is well explained in soliton models:
Pure electromagnetism would lead to infinite energy of electric field of a charge - to prevent it, soliton models use Higgs-like potential (see my previous post).
This potential is usually nearly zero (so called vacuum state, e.g. field of unitary vectors), the potential is activated to prevent infinities caused by topological constraints (e.g. hedgehog configuration) - the Higgs potential is activated to regularize charge (e.g. vector field goes to zero vectors in the center of charge).
In other words, electromagnetic interaction is kind of deformed into weak/strong interaction - only near singularity/particle.

There can be different types of such deformations - corresponding to weak or strong interaction.
In a model I consider (essay), weak interaction corresponds to direct deformation caused by singularity, like grayness in below spin +1/2, -1/2 pair:

Strong interaction corresponds to more distant relations, for example holding nuclei together:

It provides simple explanation of why proton is lighter than neutron (or deuteron than p + n): baryons themselves require some charge (not necessarily complete): let say +2/3. So neutron has to compensate it with two -1/3, making it wider - what costs energy. In contrast, proton can be narrower - it has just +1 charge. In deuteron proton and neutron share the same charge.

FrediFizzx, the simplest charge configuration seems to be hedgehog-like: v(x) = x/|x|, which has topological charge 1.
We need to get 1/r^2 electric field for it - curvature is a natural choice, Faber shows that we naturally get the entire electromagnetism this way - Maxwell equations to describe evolution.
Can you get it using torsion instead?

[FrediFizzx]

...FrediFizzx, the simplest charge configuration seems to be hedgehog-like: v(x) = x/|x|, which has topological charge 1.
We need to get 1/r^2 electric field for it - curvature is a natural choice, Faber shows that we naturally get the entire electromagnetism this way - Maxwell equations to describe evolution.
Can you get it using torsion instead?

That is basically what I was asking you. Do you think torsion might work instead of curvature in Faber's soliton model? Torsion is actually a form of curvature though different from "normal" curvature. Maybe this will help?

https://en.wikipedia.org/wiki/Torsion_tensor

[FrediFizzx]

...FrediFizzx, the simplest charge configuration seems to be hedgehog-like: v(x) = x/|x|, which has topological charge 1.
We need to get 1/r^2 electric field for it - curvature is a natural choice, Faber shows that we naturally get the entire electromagnetism this way - Maxwell equations to describe evolution.
Can you get it using torsion instead?

That is basically what I was asking you. Do you think torsion might work instead of curvature in Faber's soliton model? Torsion is actually a form of curvature though different from "normal" curvature. Maybe this will help?

https://en.wikipedia.org/wiki/Torsion_tensor

This paper seems to indicate that it would work. And seems to also require that an electron is not a point particle.

http://arxiv.org/abs/0910.1181
"Nonsingular Dirac particles in spacetime with torsion"

[Jarek]

There is probably a large freedom to choose the kinetic term to get 1/r^2 Coulomb interaction for topological charges - start with 1/r^2 electric field for hedgehog.
Having Coulomb interaction and Lorentz invariance, you obtain the entire electromagnetism: https://en.wikipedia.org/wiki/Relativis ... omagnetism
This way of thinking for example extends Newton's law to gravitomagnetism, which is seen as approximation of GRT and its corrections were tested by Gravity Probe B: https://en.wikipedia.org/wiki/Gravitoelectromagnetism

Indeed the choice of both kinetic and potential term is a big question - ellipsoid field I consider seems to nicely recreate our particle menagerie in qualitative way, however the quantitative part overwhelms me.
It is a natural expansion of Faber's picture: with unitary vectors in vacuum (far from singularities): we additionally distinguish rotation around this vector - there is a single additional degree of freedom, which corresponds to quantum phase, rotated by de Broglie's clock of particle, its evolution leads to interference.
Instead of a single vector in each point, we have emphasized three orthogonal distinguishable directions - axes of ellipsoid in this point:
- it can perform 3 types of hedgehog - with one of 3 axes - getting three leptons because of living in 3D,
- performing a hedgehog with one axis, topology forbids us to smoothly align the second axis on a sphere ( https://en.wikipedia.org/wiki/Hairy_ball_theorem ) - explaining why the simplest charge (electron) need to also have magnetic singularity (magnetic dipole moment),
- it also get meson and baryon-like structure, e.g. explaining why proton is lighter than neutron, what bind nucleus,
- going to 4D spacetime, we get 4 axes - the 4th usually aligns toward time direction and leads to gravitomagnetism.