Whatever we assume the electron radius to be is a pretty much speculative matter; to me once we talk about radius, the only sensible way of looking at it is how we can observe it -- yes, experimentally, or at least using a theory that aims to describe a feasible experiment. The most straightforward kind of experiment would be to observe/measure a scattering and respective cross-section by any other particle (preferably neutral). And the most universal "particles" would be photons.

Now. for low photon energies, hbar omega m_0 c^2, where the cross-section slowly goes down as energy encreases due to generation-annihilations of electron-positron pairs; to the first order in alpha, it is described by a powerful Klein-Nishina theory. In the limit eps >> m_0 c^2, where eps=hbar omega,   we have approximately sigma_KN / sigma_0 ~ (3/ (8 eps)) [ln(2 eps) + 1/2]. All this of course have nothing to do with Planck radius...

Hello John,

Nice question , I have explained structure of electron and positron in attached paper and conference link.

Electron radius should be in order of Compton wavelength.

Regards,

Bhushan Poojary

https://www.youtube.com/watch?v=WtA6W7KkEp0

Thesis Fine Structure Decoded

Different definitions of the radius must not be mixed up.  The Dirac equation starts from an electron of radius 0.  The compton wave length correspond to its radius after renormalization, that is, in quantum field theory. This radius includes the virtual particle-antiparticle pairs in the vicinity of the point electron.  It is approximately equal to the so called classical radius.

Spin is correctly described by the Dirac equation.  It isn't an orbital angular momentum, but the consequence of the multi-components wave function, that transforms like a spinor.

Whatever we assume the electron radius to be is a pretty much speculative matter; to me once we talk about radius, the only sensible way of looking at it is how we can observe it -- yes, experimentally, or at least using a theory that aims to describe a feasible experiment. The most straightforward kind of experiment would be to observe/measure a scattering and respective cross-section by any other particle (preferably neutral). And the most universal "particles" would be photons.

Now. for low photon energies, hbar omega m_0 c^2, where the cross-section slowly goes down as energy encreases due to generation-annihilations of electron-positron pairs; to the first order in alpha, it is described by a powerful Klein-Nishina theory. In the limit eps >> m_0 c^2, where eps=hbar omega,   we have approximately sigma_KN / sigma_0 ~ (3/ (8 eps)) [ln(2 eps) + 1/2]. All this of course have nothing to do with Planck radius...

In the context where the notion of an electron makes sense, experiments indicate it is point-like, i.e. its radius can be taken to be equal to zero-and the theoretical description of the experiments is consistent with these findings, qualitatively and quantitatively. The notion of an electron as a charged sphere is classical and not appropriate to describe its quantum properties. These energies are much less than the Planck scale, which means that the length scales, where the notion of a point-like electron makes sense, are much larger than the Planck length. What is the appropriate description of the electron at length scales comparable to the Planck length,i.e. when gravitational interactions become comparable to the electroweak interactions that affect the electron, isn't known yet.

And what about the radius of the

Planck length only makes sense when time is assumed to be one-dimensional.  In causal set theory, where all dimensions are formed by discrete time, any quantum can be assigned unit length ("length" is duration in causal set theory) and all others assigned relative duration with respect to the chosen unit.  That's how quanta have their variation in wavelength/frequency values.  In the modeling of Bohr's formula by causal sets, when we take the DeBroglie values of an electron to be the chosen unit of frequency and wavelength, the main Bohr lines have their frequency values in whole-number ratios, reflecting the structure of the underlying cloud activity.  There are higher and lower quantized frequencies than that of the electron, but the modeling achieved for Bohr's formula is well-established without regard to the higher frequencies and shorter wavelengths of the nuclear structure and whatever higher frequencies exist beyond that.

Causal sets form inherent frequency ratios, useful for defining energy ratios in accord with E=hf.  The causal link, or discrete step of time, is the unit of the energy ratios, and thus it is identified as the quantum of energy.  The common particles are easily constructed from causal links using the arrow diagrams of causal set theory.  The 4-D manifold is likewise constructible,  The manifold and the particles propagate together as a "causal web," to use a term from a few decades ago.

http://temporalsuccession.wordpress.com/

So far this question has received 90 views but no one has been motivated to defend the string theory model of an electron. It is my understanding that the string theory model of an electron is a one dimensional vibrating closed loop that is about Planck length in diameter. If there are supporters of string theory reading this, I have two questions that I have always wanted to ask. 1) What is the frequency of vibration for an electron? 2) Is there a calculation which connects the vibrational frequency, loop length, etc. to arrive at the electron’s energy?

One new approach to understanding the size and energy structure of an electron is that it is a circulating charged photon. See   https://www.researchgate.net/publication/266392325_The_Electron_Is_a_Charged_Photon . The relativistic energy-momentum equation for an electron is consistent with the electron's linear momentum being the longitudinal component of the circulating momentum of a helically moving charged photon with the same energy as the electron, while the transverse component of the momentum of this circulating charged photon is an invariant mc, which gives rise to the resting electron's spin hbar/2 .

John,

I think that the String Theory model is just that.  It has no real meaning in a 4 dimensional space time and only comes to light in the 11 dimensions of the theory.  In my humble opinion the theory has some merit in the way it makes people look at the electron and for that mater all particles.  If instead of looking at the electron as a string that is a closed loop that is vibrating in open space we may take the premise of a string instead of a single point particle.  This allows us to look at the electron differently.  

In this way we can build a new model of the atom combining the concept of a string as an electron being a string that has a charge of negative at each end of the string and a proton as being a string that is made up of three quark strings that when combines are a string with a positive charge at each end and the neutron as being a string that is made up of 3 quark strings that has a positive charge at one end and a negative charge at the other end being neutral in charge over all.  This configuration of the atom would not need to have a hypothetical particle of a graviton as the charge differentials in the atom would allow for the attractive nature of a string dipole to be held into the center via the string.  This would also explain the reason that as we have larger amounts of neutrons in an atom it becomes less stable and will tend to radiate or at least want to be smaller.  

But this is a theory that I have been working on for the last several years.  Most do not what to combine the theories as they would both have to be modified to fit.  The advantage of this theory is that both parts of the two theories that have the most promise are kept and the parts (11 dimensions of space) that can not be seen can be disregarded.  It explains a lot about the atom that we can not see with our eyes.  I am of course disregarding the part of string theory that takes us on the trip through  Calabi–Yau manifolds and much more.

There is more to this theory that I have not been able to put in a talk like this.  The Neutron for instance is not just three quarks of the up and down variety but is a proton that has Neutrinos as strings involved.  You get my point.   We can go even deeper into the string of an electron, photon, and neutrino as being the component parts of the quarks as well.  It sets up a detailed explanation of the problems of how many different particles there are and why the super colliders are always going to be coming up with new particles.  As the strings can be strong together to make even the Higgs particle and eventually bigger particles that are just as unstable and only last for a billionth of a second or so. 

This theory is a work in progress but you may be able to glean the light at the end of the tunnel that I have seen from this new way of looking at the atom.  All this work came from my working in the lab and realizing that the old model did not fit everything that was happening.  Also making it more complex did not solve the problems.     

My work as a child in science had left me with questions that this new model addresses in full. 

The electron becomes more point-like as energy scale increases. If you consider the field to be part of it’s structure, then it becomes a localized entity or excitation of Planck-scale structure. For example, the Dirac equation is actually tracking two points in the case of an orbiting electron; the classical position of the electron (uncertainty principle) and a (family of) point(s) along its spin plane. At the radius of the reduced Compton wavelength, the magnetic dipole moment becomes twice the Bohr magneton before relativistic corrections. Thus the application of mass in the Dirac equation or QED is related to this extra component of spin rather than something more fundamental like an invariant scalar field (e.g. Higgs field) or the origin of mass.

Think in terms of the double slit experiment and wave-particle duality. As the energy of a particle increases, its final distribution becomes more concentrated. For a massive particle with v -> c the diffraction pattern would become point-like, although this is not achievable in nature (also note that at v = c, the particle’s field would no longer be spinning).

Dear John, as other participants claimed in details, the answer to your question is negative. But your motivation seems to be string theory, which although has a nice starting point, it has been trapped to the concept of spacetime manifolds and this is, according to my opinion, the origin of it s failure after 30 years of working. BTW, don't you agree that 30 years are too much for a theory without even one experimental connection? Anyway, there exist so many theories of electron that we have to agree on the principles of acceptance before examining each one of them.

Recently it was found that the muon, also supposed to be a point particle, when used as a probe of proton radius gave a different result than the electron does in the same experiment. I don't understand how it can be supposed the electron and muon have the same radius.

John Macken:

> 1) What is the frequency of vibration for an electron?

Divide the speed of light by the string scale (~Planckian). "Vibrations" are string eigenmodes.

> 2) Is there a calculation which connects the vibrational frequency, loop length, etc. to arrive at the electron’s energy?

You mean the electron mass? No, as of today.

At superstring unification scale electrons (as well as quarks, gauge bosons etc) are massless. There are also massive states, too massive (these correspond to higher excited states of string "vibration") from the point of view of currently accessible energies -- for these one too can relate their masses at this superstring unfication scale to their excitation state (vibration modes excited); but things get much harder once one wants to figure these masses at human energy scales. All these electrons, quarks etc acquire mass as you go through all the CY compactification, supersymmetry breaking etc, the exact details of which are not known as of today (even the dimensionality of our spacetime (d=4, above string scales) is not understood, probably our world is just one of the numerous solutions, tehre exist others with different dimensionality on large scales, and completely different particle masses, coupling constants etc). 

I think the difference is that most of the particles are not point particles at all but we just look at them that way.  They are all real and are somewhat different in many aspects.

James: if the electron had a finite size (larger than the muon), it could explain why it gives a larger result for the proton radius. However, the experimental limits on the intrinsic size of the electron are much, much! smaller than what would be needed to explain the difference in electron and muon measurements of the proton charge radius.

Yes I know that. So the experimental results don't reconcile.

John and James,

I am going to repeat parts of your discussion to set up my point. John said, "If the electron had a finite size (larger than the muon), it could explain why it gives a larger result for the proton radius." James answered, "Yes I know that. So the experimental results don't reconcile." The reason that this is so interesting to me is that in the preprint of the paper cited below (to be published in December) I propose a fundamental particle model (electron, muon etc.) that is a rotating "dipole" wave in spacetime. This is a unit of quantized angular momentum existing in the energetic vacuum which has a physical size equal to the particle's reduced Compton wavelength. A collision momentarily adds energy which contracts its radius to preserve angular momentum. A calculation shows that this model always would appear to be a point particle in collision experiments. It also explains why an electron appears to be a point particle in calculations of an electron’s g-factor.

This paper also shows that this particle model generates the correct weak gravity curvature of spacetime and the correct gravitational force between particles. According to this model, the rotating distortion of spacetime that forms an electron has a radius of 3.86×10-13 m and a muon’s radius is about 1.87×10-15 m. It would be a very important finding if a calculation showed that this difference explained the observed difference between the electron and muon experiments. I ask someone with expertise in this area to make this calculation.

http://onlyspacetime.com/QM-Foundation.pdf

Dear John, I think you are on right way! The small electron (about 10^-18 m) does not explain anything, if put there elementary charge then we get unbelievable huge energy-mass that cannot be (as you notice!) Its property of "dancing - jumping" is sadly fiction only, it changes nothing essential. I will read your reference, the wave model of particles really gives many actual parameters of particles! You can find something interesting in below link:

http://vixra.org/author/george_kirakosyan

Dear John, I completely agree with you when you say:

"This work Makes the Case That everything in the universe (all particles, fields and forces) is derived from the single building block of 4 dimensional spacetime."

According to Smolin, the universe is formed by Atoms of space and of time whose diameter is what we call Plank length.

That is, the universe is made up of four-dimensional space spheres. Mass is the rotational energy of the sphere, is what we call rest mass or frequency.

An electron is a particle of Planck rotating speed w (c = wr) in 3D space and speed w / 2 in the fourth dimension. Where r is the Compton wavelength.

An electron is like a flat disk. The radius is the Compton wavelength and the height is the radius of Planck. This consists of approximately 1060 spheres Planck

This allows to deduce and calculate electron properties, such as:

Mass, charge, spin, particle, wave, size, shape, external structure (no internal structure), in the free state velocity, magnetic moment, etc.

All of the above is applicable to other particles, such as quarks, proton, neutron, muon, pion, etc

Everything is reduced to spacetime and movement.

Dear Sanjay, A good question. Electron models don't have to flatten like pancakes at speeds approaching the speed of light. An electron that is modeled as a helically circulating charged photon actually decreases its helical radius as 1/gamma^2 .

Dear Richard, Accordingly, the electron is modeled as a photon helically circulating charged photon. Now I just need to know: What is a Photon ?. What is electric charge? What is mass? What is spin? and How the photon with spin +1 becomes spin +1/2 in the electron? Easy questions to answer and calculate with Planck’s spheres model

@Michael Peck

Thus the application of mass in the Dirac equation or QED is related to this extra component of spin rather than something more fundamental like an invariant scalar field (e.g. Higgs field) or the origin of mass.

The De Broglie frequency of the electron is determined by the spinorial relation of dirac and the two spinors are related by a coupling coefficient which is determined by the HIGGS field. In such interaction of the spinors the mass is created by resonance.

Hello Jose, Thanks for your questions. You can find some answers in my article at https://www.researchgate.net/publication/267922850_The_Electron_Is_a_Charged_Photon .  No one knows what electric charge is, but it may be related to the twisting of space (in my opinion). Mass is the inertia of localized circulating energy (in my opinion) and this is the case in my electron/charged photon model. Spin is the angular momentum of the circulating charged photon in my electron/charged photon model. The photon has spin z-component plus or minus hbar/2 in my model because it is charged and is moving in a clockwise or counterclockwise helical path with a transverse momentum mc which combines with the electron/charged photon model's Dirac radius of the resting electron/charged photon hbar/2mc to give hbar/2 for its longitudinal component of momentum, i.e. the charged photon's z-component of spin, for a slow moving electron.

Stefano,

Including the Higgs field into the discussion is beyond the plain Dirac equation. Simply replacing m with gØ is not constrained by the Dirac equation itself. In fact, the mass of the Higgs boson cannot be predicted, but must be measured. It does not explain what mass is itself; for example, what gives the higgs boson/field mass?

Electrons are not just point-like objects, they have extended electromagnetic fields. You can conceptualize an electron field as spinning on an axis, where each point of the electric field (E) is traveling at the speed of light (c) to create a dipole moment. The Dirac equation encodes this by following a single point at a fixed radius from the classical position. By doing so, the motion of a field becomes the motion of a single point upon the field. Furthermore, the velocity of this point must be balanced with particle velocity (i.e. the total is always c), so that a relativistic electron would be rotating relatively slower than a stationary one.

If we include the field(s) of electrons into what an “electron” is, it will have infinite length; it interacts with distant particles instantaneously.

The electron diameter may be smaller than Planck length, explaining why our science is not able to measure it. Spin and magnetic moments indicate geometry and differentiation. It probably has parts that could eventually be inferred from experiments. A flexible size in collisions is probably ruled out by scattering angles, which differ for soft and hard objects.

Considering that ZPE amplitude is of Planck length and the ZPE oscillators have smaller components, these smaller components must reside in a media of some sort that allows distance between the ZPE components. That media may be made of even smaller objects, and there is no way at present to predict how many generations of smaller objects may exist, but on the smallest scale there must be degeneracy in which all the energy states are filled, and there is no distance between one object and its neighbors. There is much data from particle experiments which might be used to infer some properties of the scales smaller than Planck.

I forgot to mention earlier that the electron is quick to reveal its quantum structure in the reconstruction of physics from causal sets. It, and its sister particle the electron-neutrino, are the simplest particles after the photon.

What evidence is there that the electron has any finite size at all?

It's just a point.

The reconstruction of physics from quanta-- from the causal links of causal set theory-- in which every quantum, and anything constructed from quanta, has finite extent, including the electron, is incontrovertible evidence of the finite extent of the electron. The construction of the common particles with their DeBroglie values and mass-ratios to one another, together with the other fundamentals of physics-- all from a temporal successor relation-- is no accident by any stretch of the imagination. Such proof is nowhere else to be found in the history of theoretical physics.

Dear Carey: Sad about your answer - all empty words and no reality. Math is a possible description of simple situations that needs expositions on how the symbols are detected as existents in reality. SET theory is bogus from its very foundations as a set cannot even be defined, so anything is possible, as in medieval theology.

The theory is best interpreted in terms of the logic and ontology of relations rather than set theory. The interpretation of the formal constructions owes to Russell and Whitehead, who were scientific structural realists, motivated to treat the mind-body problem by reformulating dual-parameter space-time as a single-parameter causal web. Their solution to the mind-body problem is part of the bargain in the reduction of physics to Whitehead's discrete version of the event ontology, which has been re-invented for physics today as "causal set theory" without knowledge of Whitehead's interpretation for the elements and causal links that compose causal sets. Russell and Whitehead were quite aware of the distinction between mathematics and the causal structure of contingent facts.

"The electron model proposed by the Authors consists in a current ring generated by a massless charge that rotates at speed of light along a circumference whose length is equal to the Compton wavelength of the electron: about 2.4 x 10^-12 meters. The charge is not point-like but distributed on the surface of a sphere, whose radius is equal to the classic radius of the electron: about 2.8 x 10^-15 meters."

"The rotating charge momentum has – when expressed in natural units – the dimension of an energy and a value equal to electron mass. Using natural units, the electron mass is also equal to the angular frequency of the charge rotation, and is equal to the inverse of the ring radius. But the mass value can also be obtained by integrating the square of the electric and magnetic field or by integrating the product of the current density for the vector potential. Starting from such model, it is thus possible to obtain the electron rest mass in six different ways."

from

http://www.ecat-thenewfire.com/blog/electromagnetic-electron-new-fire/

see also:

Article The Electron and Occam's Razor

We know electron contains 511keVs of energy, e.g. released while annihilation. From the other side, point charge would have infinite energy of electric field - to get only 511keVs we would have to integrate electric field from r0 ~ 1.4fm. So pure energy conservation requires some mechanism reducing electric field in femtometer scale.

I am most familiar with topological soliton model of prof. Faber for this purpose (e.g. http://iopscience.iop.org/article/10.1088/1742-6596/361/1/012022/pdf ) - using Gauss-Bonnet theorem in place of Gauss law. It says that integrating curvature over a closed surface gives topological charge inside - which has to be integer. So interpreting curvature of some more fundamental field (v) as electric field, and using standard Lagrangian for it, he recreates electromagnetism with additionally build in charge quantization as topological.

Regularization to finite energy (511keVs) is made by using Higg's like potential, e.g. (|v|^2-1)^2. It prefers unitary vectors which evolution gives electromagnetism, but allows for v->0 in the center of (topological) charge like hedgehog to prevent infinite energy of the field. Intuitively, electromagnetism is deformed into other interactions (strong/weak) in femtometer scale to prevent infinity.

Slides: https://www.dropbox.com/s/aj6tu93n04rcgra/soliton.pdf

Dear Jarek:

I am a physicist and your "answer" is meaningless to me - all math, no reality.

Dear Herb,

Could you elaborate? Electron having electric charge means it is (nearly singular) stable configuration of fields (at least ~1/r^2 electric) - technically is a soliton, we would like to understand its physical structure.

Annihilation turns electron + positron into 2x511keV electromagnetic radiation. Most physicists believe in energy conservation - requiring the same energy before annihilation, and electric field itself would exceed it if assuming it is unchanged 1/r^2 also in femtometer scale.

Some other arguments against point electron are ultraviolet divergence: requiring lower bound for distance to consider, and divergence of QFT perturbative series - like there is physically no place for too large scenarios: Feynman diagrams.

Herb,

All concede set theory is circular reasoning with no Set of all sets; however, that does not preclude that math is not without terra firma. See my posts for the tangible physical foundation of mathematics in to the fact and so certainty that: There is Being, which may therefore be equated to the fundamental concept or meaning of 1 and the fact that each number is so many ones or beings, followed by asking what in nature or the universe or in Being would constitute the Being of all beings, or if you will Set of all sets.

That would be light, since linear in a vacuum it is the fastest of all beings/phenomena/entities and therefore all else would be relative to it (the standard timekeeper by which all else may be measured/predicted, as speed=distance/time and thus time=speed/distance). Also, since light is the fastest, physics does know by the least action/time principle that light travels the shortest path of an ellipse or in 3D, a geodesic, ultimately I contend and Grigori Perelman's Poincare conjecture solution would agree, two cones facing each other, a rugby ball shape. Now as relatively linear light travels this ellipse or shortest path, it must slow to round the 'end' curves and in slowing physics also does know, its energy is stored or becomes matter (aka gravity, that heavier feeling we experience braking a car before rounding a curve) conically (from minimally the smallest length, Planck, to maximally largest expanse of less than 180 degrees or linearity), such as, an oblate planet, helical dna, quanta, etc., depending on the radii. The very first entry point of this sharper end curve constitutes the Planck length, magnitudes smaller before this slowed light continues on and forms an electron and self-similarly a brain, etc. -- all of this Being of relative beings, conically simultaneous that an observer might measure with self-same instruments as always closely distributed according to Ulam's spiral.