This is not repairable in the classical electrodynamics.

Moreover, the mentioned 511keV energy has nothing to do with any of electrostatic energies.

I disagree - treating electron as a stable (e.g. EM) field configuration - technically soliton, there are natural ways to regularize field for finite energy (e.g. below 511keVs).

For example modeling electric charge as topological charge: using Gauss-Bonnet theorem in place of Gauss law to get built in charge quantization. Here is a simple 2D vector field u with +1 topological charge and using V(u) = (|u|^2-1)^2 Higgs potential: which is usually zero (unit vectors), but allows to get to u->0 in the center of (topological) charge to prevent infinite energy of field - kind of electromagnetism deforming into other interactions (weak/strong) to prevent infinity (slides: https://www.dropbox.com/s/aj6tu93n04rcgra/soliton.pdf ).

Is it a proper way to repair Maxwell's equation for finite energy of charge (and lack of charge quantization)? Are there other known approaches?

You can treat any energy in virtually any way of course, but the 511keV is related to the electron mass which is no way connected to the charge :).

But nearly singular electric field of electron is something unavoidable for electron - energy conservation says that this energy has to be balanced e.g. when creating electron-positron pair from just EM field of two 511keV photons.

How large percentage of 511keVs is in energy of its field?

Dear Jarek,

welcome,

I am not an elementary particles physicist but an electrical engineer. My opinion in order to solve this challenge, one has to elaborate on the physical model of the electron. In the sense, one can not consider that the electron is a dimensionless point. As particle having mass, it must have physical volume. IF it is a sphere then this sphere must have a radius. Accordingly any other particle can not penetrate to its center where the potential energy will be infinity. I think the truncation of the potential energy comes from the nonzero volume of the electron.

Best wishes

Dear Abdelhalim,

Exactly, asking for structure of electric field of electron, we need to deform it on femtometer scale not to exceed the 511keV energy - technically it is no problem to do it, e.g. as I mentioned above.

The counterargument is that experiments exclude composite structure for electron - but it excludes parton structure (smaller particles building it), not regularization of its electric field - required not to exceed 511keVs for energy conservation.

Dear Jarek,

I could not understand clearly if my opinion may be applicable or not.

From the electrostatic point of view, Gauss law is applicable outside the particle volume it self. Also the Coulomb law has the same space validity.

Inside the volume of the particle one has to consider a distributed mass and charge. In this case the potential energy will be maximum at the surface of the particle. This what do seek. Is that right? This model may work and full fills your expectations.

Best wishes

Dear Abdelhalim,

Thinking in terms of charge distribution is misleading as nature has built-in charge quantization - they should appear in integer multiplicities of e (or e/3) - cannot be objectively smeared over nonzero volume (we have ~10^-18m experimental boundary for this radius).

Instead, the real question is about repairing electric field or electric potential near the center of such quantized charge - we need a regularization mechanism: deforming it on femtometer scale to get energy of field below 511keVs.

Hello esteemed colleagues,

Although not a particle physicist myself I have often felt the need for clarifications on physics models that somehow approach infinities. (whenever you see infinities in physics formulas they are usually generalizing, simplifying, or dead wrong). Whenever we get virtually no volume/negligible mass and a finite but much larger quantity associated with it we encounter impossible field concentrations or charge densities etc.The model has to be then understood as an approximation.

Returning to our beloved electrons, there is some talk about a possible sub-electronic split , a denaturation if you will, where in certain conditions the electron wave function is split and we get a fractionary charge pulse array- the so -called electron fractionalization, described loosely here:

https://phys.org/news/2015-05-electron.html

While not a proper particle split , this electron fractionalization suggest models that reveal some inner structure or at least suggest that there is some scale of phenomena that force the relativistic -wave-and-particle dual approach (although no large speeds are directly involved it is a question of scale).

The elementary particle is regarded as such when modelling interactions between particles, and between similarly sized and much larger atoms, molecules or various EM fields.

But as the scale of the problem descends BELOW the electron radius you hit one of physics' most interesting general problems: the characteristic length problem. In numeric modelling but also in general , physics modelling- i.e. when you describe a natural phenomenon by a formula- you constrain your observation and description of the phenomena to a scale according to the simplest observable interaction at that scale.

For example, if one was to model the movement of a pendulum, the "scale" of the problem is the trajectory arc length - a few cm, a few m, etc. One will surely look at you funny if you started modelling quantum interactions of particles in atoms in the pendulum with the environment- although they are there. But adjust the scale of the pendulum sufficiently enough, or think of those interesting light pressure rotors in vacuum where one side painted white reflects light and the opposite ,black, absorbs it- and the very light rotor starts to turn when exposed to light- and interactions at that level become significant.

In summary, I think you have hit the limit of the scale of the classical electronic model. Go relativistic henceforth and you will find a way to describe it.

Our physics teacher had this example to familiarize us with the concept of modelling scale problems : think of a scale model of, say, a plane. It is not too big, say it has 10 cm and the real plane is about 25 m long. If the model is simple, it does not show much detail. A more complex model will include representations of rivets, panels, even cockpit instrumentation, but now I ask you, does the model altimeter work? It is supposed to be a scaled model of an aircraft- not an exact copy. It approximates by simply ignoring finer structures than a certain scale it can conveniently represent. That is just the case with physics modelling.

To conclude- the problem of what happens below an electron radius is simply not solvable with classical electricity laws.We are boldly stepping into the murky waters of quantum physics. I personally stay away- I am more of a radio-RC- power amplifier -kind of guy..not a particle physicist.

I hope my lengthy post did not bore you, all the best,

Marius.

I think the quantization does no forbid the distribution. Quantization means that the quantized quantities assumes discrete values rather than continuous values. The space occupied by the mass of the electron may be quantized but it is not zero.

Best wishes

Dear Abdelhalim,

If charge quanization itself does not forbid smearing of charge of electron, it seems forbidden by experiments, claiming e.g. 10^-22m bound for radius of this charge ( http://iopscience.iop.org/article/10.1088/0031-8949/1988/T22/016/pdf ).

However, this experiment does not restrict shape of electric field, which needs to be deformed so that its energy does not exceed 511keVs.

From experimental perspective, there is required running coupling effect: that alpha increases from ~1/137 e.g. to ~1/127 for 90GeV collisions. This kind of effect is also observed in soliton models of electron like above - potential of two particle system drops with 1/r like in Coulomb, however, it turns out to drop slightly faster for huge energies - increasing coupling - like in plot in fig. 4 in http://iopscience.iop.org/article/10.1088/1742-6596/361/1/012022/pdf

What other experimental effects should be expected for point electron, with EM field regularized from 1/r^2 to a finite energy (below 511keV)?

Dear Alexandru,

Indeed physics doesn't like infinities, and there are plenty of them especially around QFT e.g. with ~120 orders of magnitude overestimated vacuum energy. We shouldn't just sweep them under blanket of QM mysteries.

While we often need QFT for calculation in this scale, there still remains a basic question: what is configuration of electric field (or electric potential) around electron?

For example we use electric potential in Schrodinger equation - how should its correction look like when such femtometer scale becomes crucial, e.g. in postronium, or while interaction with nucleus: in electron capture or internal conversion?

I have spent the last few days searching literature and Internet for experimental evidence for "well known fact that electron is a point" ... and usually there is only pointed Dehmelt's 1988 paper ( http://iopscience.iop.org/article/10.1088/0031-8949/1988/T22/016/pdf ), which fits parabola to two points: g-factor for proton and triton (both built from 3 fermions), concluding that electron would need to have nearly zero radius to be "composed of 3 smaller fermions" - here are graphs: https://physics.stackexchange.com/questions/397022/experimental-evidence-for-lack-of-femtometer-scale-structure-of-electron

Looking at cross section for electron-positron collisions (in link above), the plot clearly shows that beside resonances we have a line sigma ~ 1/gamma2, corresponding to Lorentz contraction. Extrapolating it to resting: gamma=1, we get ~2fm "radius of electron".

To get energy of electric field not exceeding 511keV, what is required by energy conservation, we would need to integrate E~1/r2 from ~1.4fm - there is required deformation on femtometer scale to regularize electric field to finite energy, below 511keVs.

Is there any experimental evidence for electron having no structure on femtometer scale?

Dear Mr Jarek, daer Mr. Abdelhalim,

Do you consider that at femtometer scale we approach the vicinity of scale of electron diameters?Just like a millimetric structure is relevant at the scale of meters so is the case here. At this point, the observations above about the characteristic length of the problem mean that we cease to regard the electron as a "point" source. This necessarily means we must consider the possibilities of some kind of inner structure for the electron- although from what I know there is no currently universally accepted model and some just reject the hypothesis. But it has long been discussed- what if the electron itself has an inner structure- and the "minimum field radius" is just that- a distance from the common charge center or something that takes into account the charge distribution caused by this inner structure?

I confess I am not very familiar with the field- but watching from the sidelines it appears that field shaping/deforming to make it fit a certain value or distribution can be justified if we take into account a model for a non-trivial inner structure for the electron. (i.e. not a "ball" of infinitely small radius, also known as a point source). At this scale the "point source" becomes a complicated/convoluted system of ..maybe sub-particles or vibrating strings or.. something else entirely (wild guess here) but this presumptive inner structure will cause a field to be limited to finite values and also to have a non-uniform distribution once one reaches below the electron radius.

In molecular chemistry also some properties of molecules are clear only after we take into account the types of atoms, and their inner structure (electronic orbitals etc). Some types of bonds are only explainable after understanding the atom's inner structure and the polar and non-polar aspect of some molecules- with their associated electric field -again, explainable only after taking into account sub-atomic structures.

Again we can face a similar problem- there MUST be something inside the electron- something that limits charge density, something that distributes and shapes the field and limits its value to something other than infinity.

From dancing with infinities- if we condense any finite value in a zero-size 4D space what do we get? Infinity or nonexistence? Never upon a time.. all of this was nowhere...

Dear Mr Alexandru,

Indeed from one side it seems there is no experimental evidence for lack of femtometer-scale structure of electron (?), from the other some deformation is required not to have infinite energy of electric field for E~1/r^2.

We should search for this structure. Assuming some "rigid ball" seems non-physical, what would this ball be made of? Instead, there is no problem to deform electric field to finite energy - seeing electron as a stable field configuration: soliton, like in Faber's model: http://iopscience.iop.org/article/10.1088/1742-6596/361/1/012022/pdf

Definitely agreed. As things stand now, we can say we have strong reasons to suspect a finer structure of the electron -and the reason for the field behavior /need for the mathematical artifice is that we observe it without considering (or indeed knowing ) this finer structure or its properties. Although the beach-ball model for the electron is as wrong as it can be, for simplicity s sake we can use the known model for larger scale interactions-where the so-called near field is no object- and look for accurate representations to describe theory. That is the paradox I often see in academia - once the theory reaches its limits, many who really know it inside out so to speak still are amazed or put off by the need to find a new, more complex model. Models in physics as in other disciplines ,really- from art to mathematics and engineering- are just representations of the things as they look at a particular level of interaction. Models can be as simple as point sources, infinite planes or lines, or as complex as atomic interaction in a stable molecule.

The point is I don't like the phrase "the theory is wrong" when talking about merely adding to it- particularizing and observing finer structures and their laws. Newtonian mechanics, the one that gave us planes, trains and automobiles, is not going away even if we know today to use relativistic terms for timing in GPS position computation (that's the most mundane relativistic mechanics application I could think of -it's in your phone, and maybe in your watch; your laptop could have it and your tablet surely does).The old theory is ok when dealing with large scale interactions. For interactions at the level of the diameter of the electron knowing its inner structure is crucial.Otherwise we can improvise until we do- in the way that you suggested- use the generic theory and impose some apparently unexplained conditions- and attach a blame ticket for the yet undiscovered inner structure of the electron and its works, and to justify the values -use the available experimental data that says so. And there, we have a provisional extended model. One that cannot yet predict all sub-electronic interactions yet, but they can be factored in once we see them-and now we're just following in the steps of giants like Maxwell who modeled the famous equations from the experiment- and did so by arranging some terms- historically as data became available to him, he is known to have adjusted/reformulated some terms . He did stop and publish once his theoretical apparatus could describe accurately what was seen at the time. Once new evidence showed up, he found confirmation. Contrary bits of evidence showed an initial modelling error and he corrected it, re-verified it and...voila, we have classical electrodynamics for almost two centuries now, and we started looking for high-energy, high-velocity particle interactions. His models no longer apply but we must not forget what he was really doing when he started: describing magnetic fields obtained by rotating or oscillating (often manually!) a bar magnet around/in a large hand-wound solenoid coil and other things like that- Oersted's experiments level stuff.And he got it right- an induction motor, a DC motor, an electromagnet, simple circuit theory, the basics of electric propulsion and so many other things we owe to this gentleman and his collaborators.Yet in the beginning, he was one mathematician trying to capture into some usable formulae the practical observations of his peers and friends who experimented with this new thing at the time, electro-magnetism and its applications.

Indeed, the current approach is just fitting constants and adding new terms while disagreement ... ignoring problems at the bottom, starting with 120 orders of magnitude overestimated vacuum energy, or infinite energy of field of charge.

I thought I have found a safe way to try to discuss this issue with mainstream people ignoring it: imagine single electron in universe, we know it contains 511keVs of energy. So let's take radius r sphere around it and define e(r) as energy in this sphere, so e(r) ~ 511 keVs for large r.

However, e(r) has to reduce with r, subtracting energy of electric field on intermediate distances. The main question is behavior of e(r) for r->0?

But trying to discuss it with mainstream person, electron has no position, is defined as result of electron creation operator - end of discussion ... https://www.reddit.com/r/AskPhysics/comments/89z0mp/experimental_boundaries_for_size_of_electron/dwuwytc/

Dear Jarek,

Please understand that the statement "electron contains 511 keV of energy" is a non-sense. You cannot extract or use or change any part of it, apart from total changes like e.g. annihilating the electron with a positron or making some other elementary particle reactions.

Please note that these reactions - including either creation or annihilation of an electron - deal with interactions of a totally different nature than the Coulomb law you're only discussing.

The value of ~511 keV is an equivalent of energy you need to have to create one real isolated electron - though it's not possible without involvement of other particles and corresponding additional energies, either before or after the creation event.

Your attempts to find a match between the 511keV energy value and the hypothetical electron radius is equivalent to accounting the consumed home electricity in the number of electrons: just divide some 200 kW*h by 511 keV and learn how many electrons were eaten.

Please don't consider my comment as an attack of any kind!

I really appreciate all attempts to investigate the open questions and missing points, and the infinite electrostatic self-energy of an electron is exactly one of them. At it doesn't seem to be explained just by some finite size of the particle.

You can also further think of a neutron as a totally neutral particle and still possessing a noticeable mass. Does this "mass energy" correspond to some electrostatic energy? Even if we think that neutron=(proton+electron+something else), how would you arrange this field distribution?..

Dear Victor,

In pair creation two 511keV EM waves are converted into electron+positron pair, including energy of their electric field - energy conservation does not allow it to exceed the initial energy.

Dear Jarek ~

Euclid defined a point as “that which has no part”. A point is a mathematical abstraction, not an aspect of physical reality. At the level of quantum theory the electrostatic field around an electron is a particular instance of the interaction between electrons and photons. In QED these basic interactions are represented by the Feyman diagrams on which calculations are based. Anomalous infinities arise because the interactions are conceived of as taking place at a point. Those infinities are eliminated by “renormalization” - ad-hoc procedures whose only justification is that they work and provide correct finite answers. Clearly, a deeper understanding is required that doesn’t start out from the mistaken idea that there is anything in Nature corresponding to Euclid’s “points”.

Dear Eric,

Beside renormalization, perturbatvie QFT itself has more indications that using perfect points is only idealization (mathematically difficult to avoid): ultraviolet divergence requiring lower bound for distance, and divergence of series: bounding size of scenarios (Faynman diagrams) fitting there.

Fortunately we have also alternative mathematical tool to work with varying number of particles, convenient for understanding non-point particles: soliton models - asking for stable field configurations building particles, which effective description also requires pertubative QFT: considering ensembles of scenarios (Feynman diagrams).

There are such quite successful models of nuclei (e.g. http://eprints.whiterose.ac.uk/85454/9/aloof10_published.pdf ), we need to get down to electron if wanting to really understand charge.

Jarek ≪How to repair the problem of infinite energy of electric field of point electron?≫

It Quantum Electrodynamics this problem is much less severe than in Classical Electrodynamics, because the electron is a fermion. This was first calculated by Vicky Weisskopf,

Article Über die Selbstenergie des Elektrons

Article On the Self-Energy and the Electromagnetic Field of the Electron

In classical electrodynamics, one has to attribute a linear size equal to the classical electron radius re to the electron, in order equate all its mass-energy mec2 with its electrostatic field energy. The corresponding linear size in QED (estimated, to first order in the fine structure constant α) is much smaller, of order (re/α) × exp(-1/α). Since 1/α ≈ 137, this is such a small distance that it can for all practical purposes be equated to zero. The situation is more severe (even worse than in classical electrodynamics) for charged bosonic particles, like the W± bosons.

The favoured general recipe to get rid of all (I believe) infinites is to turn to superstring theory, which so far has had limited phenomenological success. All infinites may also be removable in some quantum field theories with a very high degree of supersymmetry.

For classical electrodynamics, there is not only the problem that a finite electron radius is needed; there is also a 4/3 mismatch (according to special relativity) between the electromagnetic contributions to energy and momentum for moving charged particles, indicating that some additional contributions to the energy-momentum tensor are needed. Such contributions would anyway be needed to keep the (continuous) charge distribution localised, and to have a conserved energy-momentum tensor. Here is a short discussion of these issues by Julian Schwinger:

Article Electromagnetic mass revisited

It has been speculated that the necessary extra contributions could be explained by the Casimir effect. However, explicit calculations of this effect for a spherical shell came out with the wrong sign (which is also happens to be a counter-intuitive sign).

Dear Jarek,

Your question is quite sophisticated and not simple to solve properly without using complex calculations in QED. You need to calculate the loop corrections to the photon propagator due to the vacuum polarization Feynman diagram. After some tricks you can solve the integrals for obtaining a finite value which you multiply by the initical electric charge given in the Lagrangian (bare coupling constant) and that it is the renormalized charge (physical) that is measured in experiments.

Now let me give you a more intuitive physical image of this process. Imagine you have a photon associated to the electromagnetic field which sorrounds the electric charge of the electron. When you decrease the distance to the center of the electric charge of the electron, it obtains a lot of energy. But this has a threshold in QED because the photon can transform in electron-positron pairs, which created an electric polarization P on the bare charge and this gives you a finite value of the charge having the vacuum polarized electrically. Obviously the finite value of these amount of particles needs to use the previous techniques of the renomalization group in QED. One good book which has developed all these calculations is the

C.Itzykson and J.B Zuber, Quantum Field Theory, Dover Ed. 2006.

I hope that this can help you

It is necessary to recognize first that electrodynamics is never classical, should never be an extension of the classical Newtonian model, as taught at lower levels, even recently. This may change how Jarek can see the problem, and comments here.

It needs to use special relativity from the start, may not even need the Maxwell's equations, but can -- they are covariant with Lorentz transformations, as proved by Voigt and Lorentz. See https://www.researchgate.net/post/Should_college_students_learn_about_electromagnetism_before_mechanics#view=5b30fdf4565fbaeae90aaf2f

Victor V Solovyev is right

511keVs = 8.19E-14J

From the equation E=mc^2 we get mas=9.10940197475E-31 kg

and the electron mass is 9.1093837E-31

That is nihilated mass only.

What has happened to the electric field energy?

“How to repair the problem of infinite energy of electric field of point electron?”

- - to answer to this question it is necessary before to understand – what are “energy”? “electric field”? ], “electron”? – and what are “fundamental Nature force’s fields” and “particles” at all?, though.

All these points are fundamentally transcendent in mainstream physics, and so the thread problem, and quite not only, exist in the mainstream, in both – in classical theories and QFT.

Rigorous scientific definitions of the points above can be, and are, obtained only in framework of the Shevchenko-Tokarevsky’s informational physical model 3 main papers are

https://www.researchgate.net/publication/354418793_The_Informational_Conception_and_the_Base_of_Physics,

https://www.researchgate.net/publication/355361749_The_informational_physical_model_and_fundamental_problems_in_physics

https://www.researchgate.net/publication/369357747_The_informational_model_-Nuclear_Force

In the whole model it is rigorously shown that particles fundamentally aren’t “points”, though interact as points [briefly see, say, recent SS post in https://www.researchgate.net/post/What_underlying_rules_might_have_lead_to_dual_nature_of_light].

More concretely in this thread case see Shevchenko-Tokarevsky’s 2007 initial model of Gravity and Electric Forces in

https://www.researchgate.net/publication/365437307_The_informational_model_-_Gravity_and_Electric_Forces ,

- where, including, it is rigorously explained what are Gravity and Electric Forces charges and fields, including that electric charge isn’t point-like, while both Forces fields fundamentally don’t contain energy at least at statics, so the thread’s problem really doesn’t exist. That is true, though, in Nuclear [that binds nucleons in nuclei] Force [see the 3-rd link], Matter is rather simple logical system and so at least these 3 Forces act by the same scheme.

Cheers

Dear Jarek Duda

"We would get 511keVs from electric field if integrating from r~1.4fm instead of zero - there is needed some deformation of electric field on femtometer scale not to exceed rest energy (mass) of electron."

What should r be when using 511/2 KeV?

Best Regards, Chian Fan

“…"We would get 511keVs from electric field if integrating from r~1.4fm instead of zero - there is needed some deformation of electric field on femtometer scale not to exceed rest energy (mass) of electron."

What should r be when using 511/2 KeV?……”

Classical [in classic ED] electron radius r=λ×α = 2.817 940 3262 fm, λ is electron [reduced] Compton length, , i.e. real, more than first approximation, electron radius, α is fine structure constant. In Jarek Duda’s the “We would get 511keVs” seems means 2x511 keV energy for creation e± pair, for energy 511/2 keV r should be 5.63 fm.

Though all that really is senseless – again – see the SS post above – Coulomb field energy is equal to zero at any radius, just provided this the energy conservation isn’t violated in systems of electrically [and at least gravitationally and by Nuclear Force coupled] coupled systems of particles, bodies, etc.

More see the SS post above and links in the post.

Cheers

Daniel Baldomir Kåre Olaussen

Are you aware of any estimates what this finite energy of the electron field might be? I am anxious to see any approximations.

I have recently proposed a conjecture that this energy is equal to the Stoney Mass rest energy as per Einstein's equation.

The reason for this is that there is no real entity "Stoney Mass" (Ms).

When you eliminate kg and C=As units in favor of [m^3/s^2] and the mass and charge become commensurable, there is no place for Stoney Mass -- becoming identical to the electron charge.

The Stoney "Mass" energy can be calculated Es=ec^2. Although e is in different dimensions of [m^3/s^2], that energy is the same physical energy as the SI Stoney Mass energy Msc^2 1.6709746158593E+08J which is astronomical comparing to electron mass rest energy in the famous ratio of:

√(4.165 61 × 10^42)

My article that includes all necessary details has been accepted by Journal of Physics Communications and the peer reviewed preprint already published.

Preprint From Newton to universal Planck natural units - disentanglin...

DOI 10.1088/2399-6528/ad0090

I would greatly appreciate any comments on the electron field energy estimates resulting from any known theory.