What is the physical makeup of an electron?

[CF.Gauss]

HI,
Firstly I'd like to open with I know what an electron is and I know all about its charge and the role it plays in electricity, current, free electron model etc etc.
My question is what is an electron 'made' out of? My reasoning is that it can't be made out of anything physical as its charge would distribute evenly throughout its-self and would fly apart as every part of the electron would repel every other part of the electron.
In physics the electron is thought of as a mathematical point particle but in a 3-spacial dimensional universe a 1-d object can't physically exist so that rules that out.
If i could magically enlarge an electron to the size of a car what would i physically see?
or is there even any credence to asking a question like that?

 

[phyzguy]

I'd like to know, too. We don't have a model of elementary particles which gives them any structure. Many people have tried to build such a model (Lorentz, Poincare, Feynman ...), but no one has succeeded. Modern Quantum Field Theory assumes that elementary particles are pointlike entities with no internal structure. Whether this is true or whether this is only an approximation is an open question. "The Feynman Lectures on Physics" Vol 2, Chapter 28 gives a very readable history of these attempts.

 

[Vanadium 50]

What exactly is a lion? If I pointed at one and said "that's a lion", wouldn't that be an acceptable answer?

What is unacceptable about pointing at an electron and saying "that's an electron?" Until you've answered that question, it will be difficult to write an answer that will satisfy you.

 

[vanhees71]

To conclude: To the best of our knowledge today (i.e., in this case the standard model of particle physics) the electron is an elementary spin-1/2 Dirac particle with one negative elementary charge and a mass of about 511 \; \mathrm{keV}/c^2. It's a lepton, i.e., participates only in the electroweak interaction (let alone gravitation, which acts universally on anything that has energy and momentum).

 

[Drakkith]

When you ask what something is, the most accurate description is detailing the physical properties of it, such as mass, charge, etc. Asking what it "really" is simply doesn't make any sense, as there is no more available information. Any answer is simply speculation.

 

[A. Neumaier]

My question is what is an electron 'made' out of? [...]
In physics the electron is thought of as a mathematical point particle [...]
If i could magically enlarge an electron to the size of a car what would i physically see?
or is there even any credence to asking a question like that?

In enlarging quantum objects one makes their quantum properties disappear. Macroscopic objects behave classically.

The electron is an elementary particle, hence not composed of anything but itself. But it is not a point - only pointlike (which means, the formal, unobservable, bare electron in the defining action is a point). Due to radiative corrections stemming from the renormalization procedure for relativistic quantum field theories, an observable, renormalized electron has a positive charge radius (though far too small to be probed experimentally with current methods).

 

[Bill_K]

What is the charge radius of the electron predicted to be? Order of magnitude.

 

[nitsuj]

CF.Gauss why do electrons not look like sparks/lightening?

 

[Khashishi]

CF.Gauss why do electrons not look like sparks/lightening?

I think the lightning you see is actually emission from partially ionized nitrogen and oxygen plasma.

 

[nitsuj]

Yes, and when I "see" anything else what am I "seeing"? Say fire for example, am I seeing fire or what.

Simular to what Vanadium 50 said "What exactly is a lion? If I pointed at one and said "that's a lion", wouldn't that be an acceptable answer?"

an electron looks like a bzzt, and feels like a bzzt, so it must be a bzzt.

 

[Fastman99]

This question is similar to asking what is a photon? Photons and electrons and other elementary particles are not actually little billiard balls that are flying around high speeds. They are both quantum excitations of their respective fields.

The entire universe is filled with a photon field, and it's mostly empty. You can think of it as an empty EM field as well. At every point in space there is a quantum harmonic oscillator for each possible spatial frequency, and thing about quantum harmonic oscillators is that only allowed energy levels come in steps of hw. The minimum energy of the oscillator is 3/2hw in 3 dimensions, and then it goes up to 5/2hw, then 7/2 hw, etc. One step above the zero-point level is considered one photon at that spatial frequency. The photon could have a range of frequencies, and be localized in some way, or be more spread out and less localized.

Just think of it of a field as an infinite set of harmonic oscillators at every point in space, and think of the particles as quantum vibrations of this field.

In a similar way, there is an electron field that fills of space with a zero-point energy, and it has certain linearly quantized energy levels above the zero level that indicate the number of electrons. This explains why every electron has exactly the same mass, charge, spin, and g-factor. Saying an electron is the same thing as saying a quantum vibration of the electron field, but the latter is too wordy. The electron vibration can be localized, as in a vibration around an atom, or more spread out like a free particle, or an electron in a double slit experiment.

The big difference between the electron field and the photon field is that with electron vibrations, they can't stack directly on top each other. This is described as the Pauli Exclusion rule. The electron field is a fermion field, described by the Dirac equation. Two electron vibrations can be in almost the same state very close to each other, but they can never occupy the same exact state.

I like to visual all quantum particles, whether they are photons or electrons, as 3 dimensional fuzz balls, and those fuzz balls oscillate and move around and sometimes disappear according the probabilistic laws of QFT. It's the sudden collapse of the fuzz balls that's most shocking to me, (wavefunction collapse is mysterious).

 

[Drakkith]

Fastman, while your explanations seems to make sense, I am hesitant to really accept it, as I've never heard of "photon fields" or "electron fields" and the like. What model is this from?

 

[Fastman99]

You've heard of fermionic fields though right? I just sort of made it up the terms "photon field" and "electron field" on the spot.
http://en.wikipedia.org/wiki/Fermionic_field
The electron field is just a type of fermionic field governed by the Dirac equation. That's my definition anyway. It's what helps me envision quantum field theory better.

The most disappointing aspect of the field theory is that it predicts a large zero-point energy. It's been dismissed before, but now that dark energy is around, we need some explanation for why there is a negative energy field permeating the entire universe and causing cosmic acceleration. The zero-point energy of the QFs were a candidate, but the calculations were done and it's 120 orders of magnitude larger than the measured value! That's a terrible model error.

http://en.wikipedia.org/wiki/Zero-point_energy#Gravitation_and_cosmology

 

[Drakkith]

You've heard of fermionic fields though right? I just sort of made it up the terms "photon field" and "electron field" on the spot.

Actually no. My knowledge of QFT is severely lacking. Thanks for the links by the way!

 

[A. Neumaier]

What is the charge radius of the electron predicted to be? Order of magnitude.

10^{-16}cm according to
http://adsabs.harvard.edu/cgi-bin/nph-bib_query?1992PhDT...130L

 

[Vanadium 50]

False.

There is no prediction for the charge radius of the electron. There are experimental limits suggesting that any charge radius must be smaller than some number, but the number that A. Neumaier posted is neither a prediction nor a measurement.

 

[derek101]

If you could magnify an electron to the size of a car,you would have to slow it down as well ,so as to observe the individual oscillations.

 

[A. Neumaier]

There is no prediction for the charge radius of the electron. There are experimental limits suggesting that any charge radius must be smaller than some number, but the number that A. Neumaier posted is neither a prediction nor a measurement.

I haven't seen the paper, but according to the abstract: ''indicate the validity of the SM down to the distance of order ~10^{-17} cm and the electron charge radius of ~ 10^{-16} cm.''
Thus there seems to have been a comparison between experiment and a prediction, though the number given is maybe only a bound.

I haven't found a calculation that I could have checked, But in principle, a prediction is possible: In his book

S. Weinberg,
The quantum theory of fields, Vol. I,
Cambridge University Press, 1995,

Weinberg defines and explicitly computes in (11.3.33) a formula for the
charge radius of a physical electron. But his formula is not
fully satisfying since it is not fully renormalized (infrared
divergence: the expression contains a fictitious photon mass,
and diverges if this goes to zero, as infrared corrections from soft
photons are not accounted for). See also Section V in the
review article

M.I. Eides, H. Grotch, and V.A. Shelyuto,
Theory of Light Hydrogenlike Atoms,
Phys. Rep. 342 (2001) 63-261.
http://arxiv.org/pdf/hep-ph/0002158

where the authors says:
''According to QED an electron continuously emits and absorbs virtual
photons (see the leading order diagram in Fig. 8) and as a result its
electric charge is spread over a finite volume instead of being
pointlike''. Then they give without proof the explicit formula (28)
for the charge radius, depending logarithmically on the charge of the
central field in which the electron moves.

But according to (7.12) in Phys. Rev. D 62, 113012 (2000),
the charge radius of neutrinos, another pointlike particle, computed
from the standard model to 1 loop order, is in the range of
4...6 10^-14 cm for the three neutrino species.

 

[Vanadium 50]

I haven't seen the paper

Then maybe you should. You brought it up.

 

[A. Neumaier]

Then maybe you should. You brought it up.

I don't know how to get it.

 

[Vanadium 50]

Normally when someone references a book or article that they haven't read, that pretty much ends the discussion. However, I don't want this thread to end on such a misleading note.

How does one quantify substructure? One can model an electron as a uniformly charged sphere of radius r. The sensitivity of an experiment is R if it can distinguish between a sphere of radius R and one of radius r, with r << R, but not any smaller than R. (In the interest of full disclosure, hard spheres have technical difficulties, so

Today we know that for an electron r < 10^-18 or a few 10^-19 meters. These numbers come from both precision measurements at low energy and a search for deviations from a point-like geometry in high energy scattering. Note that this isn't saying that an electron is a point; it's saying that it appears pointlike, but that we cannot resolve a distance smaller than about 10^-18 meters.

So where does this 10^-16 number come from? That's the distance at which vacuum polarization becomes important and thus where the electric field starts to depart from a ~1/r potential. This happens for all charged objects: electrons, muons, quarks It would be profoundly misleading to attribute this to electron substructure because a) it is a property of every single charged particle, not just the electron and b) happens whether the electron is fundamental or composite.

 

[A. Neumaier]

Normally when someone references a book or article that they haven't read, that pretty much ends the discussion. However, I don't want this thread to end on such a misleading note.

I had given references that I read and understood (Weinberg Chapter 11.3, with an explicit formula for the renormalized charge radius, though in terms of a tiny photon mass, because this kind of computations have IR divergences). Only in response to the query for an explicit value, I referred to something I couldn't access but seemed to give such a value.

How does one quantify substructure? One can model an electron as a uniformly charged sphere of radius r. The sensitivity of an experiment is R if it can distinguish between a sphere of radius R and one of radius r, with r << R, but not any smaller than R.

I wasn't speaking of substructure in the sense of being composite, but of not having the properties of a point.

The electron is an elementary particle, hence not composed of anything but itself. But it is not a point - only pointlike (which means, the formal, unobservable, bare electron in the defining action is a point). Due to radiative corrections stemming from the renormalization procedure for relativistic quantum field theories, an observable, renormalized electron has a positive charge radius (though far too small to be probed experimentally with current methods).

This is the case for purely theoretical reasons, a direct consequence of QED, which is generally acknowledged to be reliable in this regime. (Corrections from the standard model would be tiny and not alter the general fact.)

So where does this 10^-16 number come from? That's the distance at which vacuum polarization becomes important and thus where the electric field starts to depart from a ~1/r potential. This happens for all charged objects: electrons, muons, quarks It would be profoundly misleading to attribute this to electron substructure because a) it is a property of every single charged particle, not just the electron and b) happens whether the electron is fundamental or composite.

I didn't attribute it to electron substructure but to Weinberg's discussion in his QFT book. (One might attribute it to virtual substructure the physical electron being a composite of a bare electron and a cloud of bare virtual photons, but I don't like this sort of imagery.)

 

[A. Neumaier]

One can model an electron as a uniformly charged sphere of radius r. The sensitivity of an experiment is R if it can distinguish between a sphere of radius R and one of radius r, with r << R, but not any smaller than R. (In the interest of full disclosure, hard spheres have technical difficulties, so

Today we know that for an electron r < 10^-18 or a few 10^-19 meters. These numbers come from both precision measurements at low energy and a search for deviations from a point-like geometry in high energy scattering. Note that this isn't saying that an electron is a point; it's saying that it appears pointlike, but that we cannot resolve a distance smaller than about 10^-18 meters.

Let me complement your experimental view with the theoretical side of the matter.

The deviations from pointlikeness are usually described by means of
form factors that would be constant for a point particle but become
momentum-dependent for particles in general.

The form factors contain everything that can be observed
about single particles in an electromagnetic field.

http://en.wikipedia.org/wiki/Electric_form_factor :
''The electric form factor is the Fourier transform of electric
charge distribution in space.''

http://en.wikipedia.org/wiki/Magnetic_form_factor :
''a magnetic form factor is the Fourier transform of an electric
current distribution in space.''
In particular, the charge radius is defined as the number r such that
the electric form factor has an expansion of the form

F_1(q^2) = 1-(r^2/6) q^2 if r^2q^2&lt;&lt;1.

(Units are such that c=1 and hbar=1.) This definition is motivated
by the fact that the average over exp(i q dot x) over a spherical shell
of radius r has this asymptotic behavior.
See Formula (11.3.32) in

S. Weinberg,
The quantum theory of fields, Vol. I,
Cambridge University Press, 1995.

QED (which treats the electron as pointlike in the usual sense of the
word - that it appears as a fundamental field in the Lagrangian) imply
a positive value for the charge radius of the electron. Indeed, this
is Weinberg's conclusion from his calculations in Section 11.3,
together with an estimate of infrared effects taken from (14.3.1).

 

[Orion1]

What exactly is an electron?

The electron is a elementary subatomic particle with a negative elementary electric charge. It has no known components or substructure.

Observation of a single electron in a Penning trap shows the upper limit of the particle's radius is 10^−22 meters.
r_e \leq 10^{-22} \; \text{m}

I request to make a recommendation for the Physics Forums Science Advisers to simply cite and reference a Wikipedia webpage for layman original poster (OP) subject questions, instead of referencing high level physics science papers, in order to avoid a lot of confusion and hyperbole.
[/Color]
Reference:
Electron - Wikipedia

 

[lordsandman]

electron is electron only. we are still investigating that elementary "thing". we know some of it's properties, ie. particle property, wave property, charge, mass etc. we don't know what exactly it is, only can measure some of it's characteristics, that's all

 

[ailog]

Well said lordsandman. If we ever come to understand what an electron really is then we won't need high energy machines anymore.

 

[Drakkith]

Well said lordsandman. If we ever come to understand what an electron really is then we won't need high energy machines anymore.

How do you know we don't understand what an electron really is?

 

[mfb]

How do you define "what an electron really is", without just describing its measured properties?
And even worse: How do you test this, if you cannot use any measurement by construction?

 

[ailog]

To Drakkith

Well, it depends on what understand means to those studying the issue. I suppose the words "understand" and "really" should banned from the world of physics.

 

[Drakkith]

To Drakkith

Well, it depends on what understand means to those studying the issue. I suppose the words "understand" and "really" should banned from the world of physics.

I think people should understand that we can't even know if our knowledge about something is complete. So making statements like "when we really understand" is 100% meaningless.

 

[jtbell]

If we ever come to understand what an electron really is

How will we know when we have reached that point?

 

[ailog]

I think people should understand that we can't even know if our knowledge about something is complete. So making statements like "when we really understand" is 100% meaningless.

I am not sure if you are quoting me but what I said was "If we ever come to understand what an electron really is". There's a big giant if in the statement. I don't believe we ever will and not just for the electron. The universe is too complex for our order of intelligence.

 

[ailog]

How will we know when we have reached that point?

If we are smart enough we will admit that there will always be something to learn and we will never reach that point. But I left the possibility open with an if statement. Now if I can figure out a method to create an electron out of a vacuum at my garage workbench then I'm getting closer.

 

[Intrastellar]

If the universe is governed by a number of laws, we can learn these laws, no matter how complex they are. Being so distant doesn't mean we will never reach.

About the OP's question, I believe he is asking about it's volume and shape, and other physical properties usually observed by eyes.

 

[wizwom]

The electron is just a static wave-form that self-reflects. The energy needed for this wave to form is the rest mass of the electron.

 

[Karimspencer]

An electron is a negatively charged sub atomic particle in an atom that is around the nucleus of the atom in the electron cloud. The electron is a very light mass subatomic particle and you can take away an electron from an atom or give electrons to an atom which will result in an ion(cation or anion).

 

[Bob S]

This question is similar to asking what is a photon? Photons and electrons and other elementary particles are not actually little billiard balls that are flying around high speeds. They are both quantum excitations of their respective fields.

The entire universe is filled with a photon field, and it's mostly empty. You can think of it as an empty EM field as well. At every point in space there is a quantum harmonic oscillator for each possible spatial frequency, and thing about quantum harmonic oscillators is that only allowed energy levels come in steps of hw. The minimum energy of the oscillator is 3/2hw in 3 dimensions, and then it goes up to 5/2hw, then 7/2 hw, etc. One step above the zero-point level is considered one photon at that spatial frequency. The photon could have a range of frequencies, and be localized in some way, or be more spread out and less localized.

Just think of it of a field as an infinite set of harmonic oscillators at every point in space, and think of the particles as quantum vibrations of this field.

In a similar way, there is an electron field that fills of space with a zero-point energy, and it has certain linearly quantized energy levels above the zero level that indicate the number of electrons. This explains why every electron has exactly the same mass, charge, spin, and g-factor. Saying an electron is the same thing as saying a quantum vibration of the electron field, but the latter is too wordy. The electron vibration can be localized, as in a vibration around an atom, or more spread out like a free particle, or an electron in a double slit experiment.

The big difference between the electron field and the photon field is that with electron vibrations, they can't stack directly on top each other. This is described as the Pauli Exclusion rule. The electron field is a fermion field, described by the Dirac equation. Two electron vibrations can be in almost the same state very close to each other, but they can never occupy the same exact state.

I like to visual all quantum particles, whether they are photons or electrons, as 3 dimensional fuzz balls, and those fuzz balls oscillate and move around and sometimes disappear according the probabilistic laws of QFT. It's the sudden collapse of the fuzz balls that's most shocking to me, (wavefunction collapse is mysterious).

Suppose we had a 50 or 100 GeV electron beam, like SLAC or the CERN LEP accelerator, and we shot it through this field of an infinite set of harmonic oscillators, or photons. What would happen?

In the normal Compton scattering, where the electron is at rest in the Lab, above a few MeV photon energy, an inelastic Compton scattering would begin to produce real electron-positron pairs, and we would see the extra positrons and electrons. The cross section for Compton (Klein-Nishina) scattering is ≈0.665 barns (6.65 x 10^-25 cm²).

Now gamma shift into the reference frame where the electrons are 50 or 100 GeV, and we should see the electrons colliding with the infinite set of harmonic oscillators or photons in the vacuum (and even with the CMB). Shouldn't we see the inverse Compton effect, with high energy gammas, and possibly even positrons comming out of the vacuum chamber? Can't we use this test to put an upper limit on the density of photons or harmonic oscillators in the vacuum?

 

[mfb]

Most of these harmonic oscillators with significant energy levels are in their ground state, which means there is no photon to interact with. High-energetic charged particles in space can interact with the CMB without problems. For protons, this leads to effects like the GZK cutoff.

 

[Bob S]

Most of these harmonic oscillators with significant energy levels are in their ground state, which means there is no photon to interact with. High-energetic charged particles in space can interact with the CMB without problems. For protons, this leads to effects like the GZK cutoff.

Energetic electrons interact with the CMB (cosmic microwave background) via the SZ (Sunyaev–Zel'dovich) effect. See http://en.wikipedia.org/wiki/Sunyaev-Zel'dovich_effect. Is your "field of photons" density less than the CMB?

 

[mfb]

If you consider a volume in some object cooler than 3K and without any sources except blackbody radiation, yes.

 

[DonJStevens]

A comment by P. A. M. Dirac from Proceedings of the Royal Society of London (1962) may be helpful here. Title: Particles of finite size in the gravitational field.

"So from the physical point of view, the possibility of having a point singularity in the Einstein field is ruled out. Each particle (electron) must have a finite size no smaller than the Schwarzschild radius.

I tried for some time to work with a particle with radius equal to the Schwarzschild radius, but I found great difficulties, because the field at the Schwarzschild radius is so strongly singular, and it seems that a more profitable line of investigation is to take a particle bigger than the Schwarzschild radius and to try to construct a theory for such a particle interacting with the gravitational field."

The next larger significant size (not so strongly singular) is the radius 3Gm/c squared. This size could provide gravitational confinement without gravitational collapse to infinite (or unknown) density. This size is too small to measure.

 

[DonJStevens]

If we want to know, what exactly is an electron, we need to know how electrons (and positrons) are produced (materialized). So much has been learned about this process that it is difficult to keep up. First, we know electrons are produced from photons (produced from electromagnetic energy). We know an electron can absorb a photon. This absorption is a direct conversion of photon energy to mass. Energy added to the electron increases its mass. A photon consists of equal amounts of positive and negative electric field energy. And so, we cannot materialize an electron without also producing a positive charge particle. The electron has extremely high energy density with a radius less than 10 exp -18 meters while the photon with sufficient energy to produce an electron, positron pair has a wavelength that is large, with far less energy density than the electron. A photon, when absorbed by an existing (high density) particle will become a high energy density entity. This is the first step required to produce mass particles.

 

[Naty1]

As noted, nobody knows 'exactly' what any of the fundamental particle are.

The best we can do so far is to describe characteristics according to quantum mechanics...spin, mass, charge, etc, whatever is incorporated in the Standard Model of particle physics...A complementary and different perspective might be afforded via string theory and that is a nice approach since it relates all the elementary particles to one another...as energy vibrations.

 

[Naty1]

as luck would have it, a related string discussion is underway:
https://www.physicsforums.com/showthread.php?p=3981113#post3981113

 

[DonJStevens]

Much has been learned, and we can now discuss a limitation on the smallness of things called a "cutoff". A quote from Leonard Susskind follows:
"A cutoff sounds like a cop-out, but there is an excuse. Physicists have long speculated that the Planck length is the ultimate atom of space. Feynman diagrams, even those involving gravitons, make perfect sense as long as you cease adding structures smaller than the Planck length - or so the argument goes. This was the almost universal expectation about space-time -- that it would have an indivisable, voxelated structure at the Planck scale."
This is from the book, The Black Hole War (page 335).
If the electron radius is equal to the Planck length or 1.616x10^-35 meter, this is much larger than the electron Schwarzschild radius and so, (at first evaluation) we may find that the electron cannot collapse to its Schwarzschild radius, 2Gm/c^2 and it cannot collapse to the larger radius, 3Gm/c^2.
Another quote from Leonard Susskind follows:
"But extraordinary things are happening. In recent years, we have been accumulating evidence that the machinery in the interior of particles (electrons) is not mush bigger, nor is it much smaller than the Planck length." (page 214)
When the electron radius value is reduced to the Planck length (or slightly larger) due to gravitational time dilation (blueshift) and an equal amount of gravitational length contraction then the size (close to) 2Gm/c^2 is attainable.

 

[PhilDSP]

The best we can do so far is to describe characteristics according to quantum mechanics...spin, mass, charge, etc, whatever is incorporated in the Standard Model of particle physics

With the knowledge we collectively have we can probably do a lot better than that. One of the best known fundamental prescriptions of QM, Dirac's "Principles of Quantum Mechanics" describes the motion of the charge of the electron that travels at the speed of light. Why it should do that and what it means in terms of the electron's structure and observable parameters is a not-so-well-known but arguably important Physics cottage industry and there is quite a bit of literature on the subject.

 

[DonJStevens]

You are so correct, there is much literature on the subject of electron structure. With careful selection from available literature, we can do a lot better. Charge motion at speed of light is necessary to explain electron angular momentum (and magnetic moment). John A. Wheeler has suggested that the electron is the result of gravitational collapse. See page 1215 in the book, Gravitation. In the book, The Enigmatic Electron, author, Malcolm H. MacGregor writes, "One electromagnetic configuration --- is a current loop formed by a rotating point-like charge."

In a (2008) paper titled, The Dirac-Kerr-Newman electron, theorist Alexander Burinskii writes, "Recall that the angular momentum J = h bar/2 for parameters of electron is so high that the black hole horizons disappear and the source of the Kerr-Newman spinning particle (electron) represents a naked singular ring." We can see (in this concept) the electron size cannot be as small as its Schwarzschild radius because charge velocity greater than c would be needed to obtain angular momentum (h bar/2). The minimum radius with the charge moving at the speed of light is (3Gm/ c^2).

Malcolm MacGregor has said, "It remains to this day one of the most arcane subjects in particle physics." And later, "---the spin of the electron--is a mysterious internal angular momentum for which no concrete picture is available, and for which there is no classical analog." Can we put these pieces of the electron puzzle together to create an improved electron description? I will suggest that this can be accomplished.

 

[PhilDSP]

To that I'll add that Martin Rivas' "Kinematical Theory of Spinning Particles: Classical and Quantum" gives quite a thorough and integrated review of very many approaches to modeling spin, especially in the electron. Unfortunately both MacGregor's and Rivas' books are out-of-print these days.

 

[DonJStevens]

I will try to find a copy of the Martin Rivas book. I am saddened to learn that the MacGregor book out-of-print. His book is very readable. This quote (page 72) points out a significant requirement. "Thus we are forced from stability considerations alone, to introduce a non-electromagnetic force that holds the electron together. If we were to consider an extremely small size for the electron, -- then gravitational forces could be invoked to solve the stability problem."
Though MacGregor does not pursue this solution, a number of theorists, including Brian Greene, John Wheeler and Alexander Burinskii expect that electrons have some properties very much like a micro black hole.

 

[PhilDSP]

I should probably warn that the math comprehension requirements for Rivas' book are fairly steep. He kind of starts where Ballentine's QM textbook leaves off.