# What is electron? Is it a perfect point? What does it mean?

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

https://dl.dropboxusercontent.com/u/12405967/electron.png [Broken]

- 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/10.1007/s10701-008-9225-1 ).

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_constant#Running_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?

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?

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gracy

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ZapperZ
Staff Emeritus
Why electron is believed to be a point? What does it mean?
What would be expected if it wouldn't be?
I'm going to skip the rest of your post, because you're showing more of the basic properties of charges, than anything related to a "point charge". I will only address this part that I quoted.

IF the electron isn't a point charge, then it should have an internal "structure". It also means that when a strong-enough external E-field is applied, it should exhibit an electric dipole moment.

Within the past few years, there have been experiments looking for such dipole field, such as this: J.J. Hudson et al., Nature v.473, p.493 (2011). There's even a more accurate one done barely a couple of years ago!

Conclusion: they found NO such dipole effects. And note these are actual physics papers, not anonymously-written Wikipedia article.

Zz.

Sure, it doesn't have electric dipole moment - I completely agree, I don't see a need for such a hypothesis.

My question mainly regarded the E ~ 1/r^2 relation - can we be sure that it doesn't weaken for very small r?
For example E ~ q(r) / r^2 where q(r) is practically constant for r > r0, but drops to zero for very small r -> 0 ?
It would explain the reduction of effective charge in running coupling, and would allow for finite energy of electric field (integrate of E^2).

Can we exclude such reduction of effective charge?
How to repair the infinite energy problem? What does reduction of alpha in high energy collisions means?

ps. It is probably very related to vacuum polarization - what q(r) does it suggest?

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mfb
Mentor
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)
Physics also doesn't like attempts to use classical physics to describe elementary particles. It does not work.

My question mainly regarded the E ~ 1/r^2 relation - can we be sure that it doesn't weaken for very small r?
The directly tested range is ~100 GeV (LEP), indirect tests like the electron g-factor are often more precise (13 sigificant figures...) and reach higher energy scales.

I'm not aware of any model that predicts a composite electron and can reproduce the g-factor with the required precision. If you know a published one, feel free to link it, then we can discuss it. Otherwise the whole thread is too speculative.

So how QFT handles the issue of infinite energy of electric field of point charge?

So how QFT handles the issue of infinite energy of electric field of point charge?
Presently it doesn't. The "naked" electron still has a divergent energy in current theories.

QFT predicts that vacuum has energy, and "naive" computation, which assumes only electrons and photons exist (IOW: only QED), results in a very large number. This is inconsistent with observed Universe.

One of possible solutions is to declare that to get correct results, we must account for all forces and particles. They may cancel out so that final results (e.g. "naked" electron energy and such) will be finite, and vacuum energy will be small and consistent with observations. We still can't do these calculations. For example, QCD vacuum solution is out of reach yet.

Supersymmetry has these cancellations built-in (that's why so many people hope it will be detected - solves lots of problems).

Another train of thought is that unobservable quantities (such as "naked" electron energy and charge) don't need to be finite. They are not physical anyway.

jarekduda
Doesn't QFT suggest a finite size of particles:
- ultraviolet divergence/cutoff limits the maximal frequency and so minimal length,
- the sum over Feynman diagrams is usually divergent (asymptotic series) - we need to kind of limit the number of scenarios (diagrams) which can fit there.
Would a perfect point particle require such restrictions?

Alternative way to regularize the singularity is to treat particle as a soliton - the potential is activated to prevent infinite energy of electric field (e.g. lecture of prof. Faber).
One can finally quantize the soliton theory, but classical solitons would still require summation over scenarios (Feynman diagrams) while e.g. scattering.
However, their nonlinearity is very difficult to put into QFT, especially that it is crucial that the potential has topologically nontrivial minimum (like a circle in Higgs potential) - how to include such topology in QFT considerations?

A. Neumaier
2019 Award
So how QFT handles the issue of infinite energy of electric field of point charge?
By renormalization. This turns the bare electron into an extended object, as testified by its nontrivial form factors (which slightly deviate from those of a point particle, described by the Dirac equation). Therefore if one wants to be precise one only says that the electron is point-like, not a point particle.

The electron form factor is computed to 1 loop in every respectable book on QFT.

By the way, QED has no solitons but describes electrons perfectly at currently accessible energies.

mfb
Mentor
Supersymmetry has these cancellations built-in (that's why so many people hope it will be detected - solves lots of problems).
Supersymmetry (if it exists) is broken, so the cancellation is not perfect. Without gravity, this would be no problem - the absolute energy does not matter then. It becomes problematic if we try to plug this energy density into general relativity (or possible QFT formulations of gravity). It is unclear if that approach is meaningful.
- the sum over Feynman diagrams is usually divergent (asymptotic series) - we need to kind of limit the number of scenarios (diagrams) which can fit there.
Feynman diagrams are from perturbation theory, an approximation that works for weak couplings. We know that calculating Feynman diagrams is not the right thing (the full series doesn't converge), but it gives great approximations in many processes. The underlying equations describe fields, not Feynman diagrams.

By renormalization. This turns the bare electron into an extended object, as testified by its nontrivial form factors (which slightly deviate from those of a point particle, described by the Dirac equation). Therefore if one wants to be precise one only says that the electron is point-like, not a point particle.
If electron is not a perfect point, what do we know about its structure? Size limit?
We can be almost certain that it doesn't have electric dipole moment, so what should we expect from its nonzero radius?
Can it be seen in experiments? What finite-size corrections can/should we expect?
Can we interpret running coupling effect this way? - that reduction of alpha ~ q^2 in high energy scattering means e.g. that electric field is given by let say
E ~ q(r) / r^2, where q =~ e is practically constant above some radius, but q(r) -> 0 for r->0 ?
What do we know about electron's energy density (E^2 + ... ), such that it integrates to 511 keVs?

Is electron just a stable configuration of fields (soliton), among others: monopole of electric field, dipole of magnetic field ... or do we also need some additional entity for electron? (created in pairs while electron-positron creation)
What is the structure of its electric field? Magnetic field (this time 1/r^3 type singularity emphasizing a direction)?
How are they regularized to finite energy?

Why nature allows Gauss law only for integer charge (electron is energetically cheapest nonzero charge)? Why electron cannot split into e.g. two e/2 charges?
Can we interpret this quantization as having topological nature - electric charge as topological charge like in the model of prof. Faber? (Gauss-Bonnet theorem is topological analogue of Gauss law - https://dl.dropboxusercontent.com/u/12405967/soliton.pdf [Broken]).

What does electron's magnetic dipole mean?
Can electron be imagined as a tiny magnet?
Is it static magnetic dipole moment, or dynamical - just a response to external magnetic field and movement?

Finally - where its de Broglie's clock (E = mc^2 = hbar*omega), zitterbewegung come from - some internal oscillations.
Does this clock "always tick" or maybe these are just responses - resonant frequencies of its structure?

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A. Neumaier
2019 Award
If electron is not a perfect point, what do we know about its structure?
We know its form factors and its self-energy, to several loop orders.

The form factors describe the electromagnetic deviations from the behavior of a point-charge, and encode for example the slight deviation from the Dirac magnetic moment. See the exposition in Chapter B2: Photons and Electrons of my theoretical physics FAQ.
The self-energy describes the deviations from the dispersion relation of a point particle. It encodes the running coupling.

There are no solitons in QED. The electrons would however most likely be solitons in a bosonic field theory that only describes the uncharged sector of QED. States in this sector contain an indefinite number of photons and of electron-positron pairs. This sector should be describable by the Wightman axioms. However, nothing definite is known.

mfb and jarekduda