The electron is usually described as a fundamental or elementary particle. That doesn’t tell you much, but when you look for more information, it’s rather scant. You soon learn that the electron has a mass of 9.109 x 10-31 kg or 511keV/c². You learn that it has a charge of −1.602 x 10−19 Coulombs or -1e, the e being elementary charge. You also learn that it has spin ½. However you don’t learn much else. Instead you get mixed messages. Take a look at the enigmatic electron by Frank Wilczek. He said “to understand the electron is to understand the world”. That’s good. But he also said “there are several inconsistent answers, each correct”. That’s not good. Wilczek then said “the proper quantum mechanical description of electrons involves wave functions, whose oscillation patterns are standing waves”. That’s good. But he also said “the electron is a simple point-particle”. That’s not good. Nor is electrons “are understood with precision, and at the same time utterly mysterious”. I don’t think electrons are utterly mysterious. That’s because I’ve looked at the hard scientific evidence, and the history. Including Schrödinger’s quantization as a problem of proper values, part II. That’s where he said “let us think of a wave group… which in some way gets into a small closed ‘path’, whose dimensions are of the order of the wave length”. Compare and contrast with Dirac, who was enamoured with mathematical beauty but never understood the electron. In 1938, he still thought it was a point particle. In 1962 he thought it was a charged shell.
The electron is not a point particle
Many sources will tell you the electron is point-like. See for example the particle data group. The particle data group is “an international collaboration charged with summarizing particle physics. They produce a review which has been called the bible of particle physics, and has been cited in more than 50,000 papers. And they’ll tell you that electrons “are definitely smaller than 10-18 meters”:
Image by the particle data group
This is authoritative stuff, but when you look for the supporting evidence for this point-like electron, it is elusive. Take a look at the 2002 paper limits on sizes of fundamental particles and on gravitational mass of a scalar by Irina Dymnikova, Juergen Ulbricht, and Jiawei Zhao. They talk about the QED reaction e+e− → γγ(γ) at energies between 91GeV and 202GeV. That’s high-energy electron-positron annihilation to gamma photons. They say the interaction proceeds via the exchange of a virtual or “excited” electron with a mass greater than 402 GeV, and the characteristic size of this is less than 1.17 x 10−17cm. They also say that they assume that a fundamental particle must have a de Sitter vacuum core related to its mass, with a finite geometrical size defined by gravity. This de Sitter vacuum core is hypothetical. It cannot be employed to support a claim that relies upon a virtual electron exchange. Not when virtual particles only exist in the mathematics of the model. There’s no evidence that a 402 GeV electron is exchanged between the electron and the positron, and there’s no evidence for the de-Sitter vacuum core. So there’s no evidence here that the electron is small.
There’s no actual evidence that the electron is small
On Wikipedia you can read that “observation of a single electron in a Penning trap shows the upper limit of the particle’s radius is 10−22 meters”. But when you follow up on the references and read Hans Dehmelt’s 1989 Nobel lecture you realise that the upper limit is merely an extrapolation. It’s an extrapolation from a measured g value, which relies upon “a plausible relation given by Brodsky and Drell (1980) for the simplest composite theoretical model of the electron”. The extrapolation yields an electron radius R ≈ 10-20 cm, but it isn’t a measurement. Especially when “the electron forms a 1 μm long wave packet, 30 nm in diameter”. When you track back to Brodsky and Dell you can read the anomalous magnetic moment and limits on fermion substructure. And what you read is this: “If the electron or muon is in fact a composite system, it is very different from the familiar picture of a bound state formed of elementary constituents since it must be simultaneously light in mass and small in spatial extension”. The conclusion is effectively this: if an electron is composite it must be small. But there’s no actual evidence that it’s composite. So it’s a non-sequitur to claim that the electron must be small. So again there’s no evidence here that the electron is small.
But there is evidence for electron spin
Note that Brodsky and Dell weren’t talking about measuring the electron’s size. They were talking about “the very (almost incredibly) precise measurements of the electron and muon gyromagnetic ratios”. Also note that Dehmelt wasn’t measuring the electron’s size either. He was using a Penning trap and spin-flips to measure the electron’s magnetic moment. Check out the Wikipedia electron magnetic moment article. It says the electron’s magnetic moment is −9284.764 × 10−27 J⋅T−1. And that if the electron was a classical charged particle literally rotating about an axis with a spin angular momentum of L, its magnetic dipole moment would be μ = -eL/2me. However the measured value is different by the electron spin g-factor, which is known with great precision to be 2.00231930436146. See the spin magnetic dipole moment section of the Wikipedia article and note this: “The magnetic moment of an electron is approximately twice what it should be in classical mechanics. The factor of two implies that the electron appears to be twice as effective in producing a magnetic moment as the corresponding classical charged body”. No, Dehmelt wasn’t measuring the electron’s size. He was measuring its spin. The evidence is evidence of the electron’s spin, not its size.
The electron doesn’t rotate like a planet
Talking of which, I particularly like the discovery of electron spin by Samuel Goudsmit: “But don’t you see what this implies? It means that there is a fourth degree of freedom for the electron. It means that the electron has a spin, that it rotates”. That was in the autumn of 1925, after Wolfgang Pauli shot down Ralph Kronig for the same idea earlier that year. See the Wikipedia spin article: “When Pauli heard about the idea, he criticized it severely, noting that the electron’s hypothetical surface would have to be moving faster than the speed of light”. Also note the final paragraph: “in retrospect, the first direct experimental evidence of the electron spin was the Stern–Gerlach experiment of 1922. See the story of the bad cigar, which gives some detail about Otto Stern and Walther Gerlach. Also see Rod Nave’s hyperphysics for a picture of the Stern-Gerlach experiment:
Image from Rod Nave’s hyperphysics
If you’ve ever played football you’ll know the gist of how the Stern-Gerlach experiment works. Especially if you can bend it like Beckham. You’re twenty yards from goal, and you kick the ball left-of-centre with your right foot, aiming for a point a metre left of the leftside post and a metre above the bar. The ball curves right, into the top left corner. Goal! Or you kick the ball right-of-centre with your left foot, aiming for a point a metre right of the rightside post and a metre above the bar. The ball curves left, into the top right corner. Goal! See the Wikipedia Curl (football) article where you can read about the Magnus effect which causes a rotating ball to form a whirlpool about itself. For the Stern-Gerlach experiment you have to flip things over such that left or right, you’re aiming for the centre of the goal, but the spin takes it to the right post or the left. Silver atoms are like footballs in that there’s an outer electron which is spin-up or spin-down with respect to the rest of the silver atom. It’s like they all have topspin or backspin, and nothing in between. They go to one place or the other, not anywhere in between. Electrons are like footballs too, in that their spin is a real rotation. They don’t rotate like footballs, because they’re spin ½ particles. But nevertheless that spin is real.
Spin is real
See Hans Ohanian’s 1984 paper what is spin? I think it’s a rather telling and chilling account. He says “since the naïve mechanical picture of spin proved untenable, physicists were left with the concept of spin minus its physical basis, like the grin of the Cheshire cat”. He goes on to say Pauli pontificated that spin is an essentially quantum-mechanical property, and that the lack of a concrete picture was a satisfactory state of affairs. He then quotes from Pauli’s 1955 essay Exclusion Principle, Lorentz Group and Reflection of Space-time and Charge: “After a brief period of spiritual and human confusion caused by a provisional restriction to ‘Anschaulichkeit’, a general agreement was reached following the substitution of abstract mathematical symbols, as for instance psi, for concrete pictures. Especially the concrete picture of rotation has been replaced by mathematical characteristics of the representations of rotations in three-dimensional space”. Have you ever heard such astonishing arrogant ignorant nonsense? It’s particularly astonishing because Ohanian says this: “the means for filling the gap have been at hand since 1939, when Belinfante established that the spin could be regarded as due to a circulating flow of energy”. Frederik Belinfante’s paper was 80 years ago. Sigh. Follow the lead and search on Gordon decomposition, and you soon find Gordon decomposition of Dirac current: a new interpretation by Suresh C Tiwari. He says this: “wherever matter-radiation interaction is involved, eg the Newton-Lorentz equation or Dirac equation, factoring out e from electromagnetic quantities one always ends up with the combination e²/c that curiously has the dimension of angular momentum”. He also says he put forward a conjecture that electric charge is a manifestation of mechanical rotation. That was in 1997. And here we are in the 21st century still falling for that total lack of visualisation or vividness. It fair takes the breath away.
An electron doesn’t rotate like a planet
An old version of the Wikipedia Stern-Gerlach article more or less says what Pauli said: “Electrons are spin-½ particles. These have only two possible spin angular momentum values measured along any axis, +ħ/2 or −ħ/2. If this value arises as a result of the particles rotating the way a planet rotates, then the individual particles would have to be spinning impossibly fast. Even if the electron radius were as large as 2.8 fm (the classical electron radius), its surface would have to be rotating at 2.3 × 1011 m/s. The speed of rotation at the surface would be in excess of the speed of light, 2.998 × 108 m/s, and is thus impossible. Instead, spin angular momentum is a purely quantum mechanical phenomenon”. There’s just one little problem with that. An electron doesn’t rotate the way a planet rotates. So saying the speed of rotation would exceed c and therefore spin isn’t a real rotation is another non-sequitur. Electron spin is a real rotation. The Stern-Gerlach experiment proves it. And that’s not the only experiment that proves it. The Einstein-de Haas effect proves it too.
Spin is demonstrably a real rotation
The Einstein-de Haas effect dates from 1915. It’s also referred to as the Richardson gyro-magnetic effect, which goes back to 1908. It’s an experiment that measures the torque generated by a reversal of the magnetization of an iron cylinder. The cylinder is surrounded by a solenoid. When you turn on the current, the cylinder rotates. It’s not unlike the impulse that makes your garden hose reel rotate a little when you turn the water on. When you turn the water off, the reel jerks back to its original position. See The Quirky Side of Scientists where David Topper talks about this on page 11. Also see the Wikipedia Einstein-de Haas article. It says this: “Considering Ampère’s hypothesis that magnetism is caused by the microscopic circular motions of electric charges, the authors proposed a design to test Lorentz’s theory that the rotating particles are electrons”. There’s more in the Einstein digital papers under Einstein’s four papers and two notes on Ampère’s molecular currents. Three of the papers were written in collaboration with Wander de Haas, who was Lorentz’s son-in-law. They used alternating current tuned to the natural frequency of the cylinder to produce a measurable deflection. It’s all to do with conservation of angular momentum. The electron’s magnetic moment is called a moment for a reason, and it’s there because the electron behaves like a tiny bar magnet. If you put a bar magnet inside a solenoid, it lines up with the magnetic field, like a compass needle points north. In similar vein the electron intrinsic spins line up with the magnetic field of the solenoid. Reverse the current and these “compass needles” swing round and point the other way. Their angular momentum changes, for every action there is a reaction, so the iron cylinder swings round too, the other way. The crucial point is that the Einstein-de Haas effect “demonstrates that spin angular momentum is indeed of the same nature as the angular momentum of rotating bodies as conceived in classical mechanics”.
Spin angular momentum is real angular momentum
The crucial point to note is that spin angular momentum is a genuine angular momentum, as is the angular momentum of a macroscopic rotating body. If it wasn’t, the action of reversing the electron spin wouldn’t cause the cylinder to rotate. Or vice versa, as per the Barnett effect. That’s the inverse of the Einstein-de Haas effect. You can rotate an unmagnetized soft iron cylinder to magnetize it. See Novel NMR and EPR Techniques by Janez Dolinsek, Marija Vilfan, and Slobodan Zumer. They tell how in 1914 the Barnett effect “provided the first scientific evidence that the electron had an anomalous magnetic moment with a g factor of 2”. They also say many people are not aware of the Barnett effect because it’s mentioned so little in the literature:
The Barnett effect, image from Frontiers in physics, mechanical generation of spin current
Also note that the electron behaves like a tiny bar magnet, that a bar magnet behaves like a solenoid, and that a solenoid features current going around and around. So what’s going around and around in an electron?
The Poynting vector is real too
Take a look at what Feynman said in the Feynman lectures: “Suppose we take the example of a point charge sitting near the center of a bar magnet, as shown in Fig. 27–6. Everything is at rest, so the energy is not changing with time. Also, E and B are quite static. But the Poynting vector says that there is a flow of energy, because there is an E × B that is not zero. If you look at the energy flow, you find that it just circulates around and around. There isn’t any change in the energy anywhere – everything which flows into one volume flows out again. It is like incompressible water flowing around. So there is a circulation of energy in this so-called static condition. How absurd it gets!”
Feynman lecture Fig 27-6 Copyright © 1964, 2006, 2013 by the California Institute of Technology, Michael A. Gottlieb, and Rudolf Pfeiffer
Only it isn’t absurd at all. The Wikipedia Poynting vector in a static field article talks about a circular flow of electromagnetic energy. It shows the Poynting vector marked with an S. It goes around and around. The article says this: “While the circulating energy flow may seem nonsensical or paradoxical, it is necessary to maintain conservation of momentum. Momentum density is proportional to energy flow density, so the circulating flow of energy contains an angular momentum”. Feynman also said this: “we know also that there is momentum circulating in the space. But a circulating momentum means that there is angular momentum. So there is angular momentum in the field”. You bet there’s angular momentum in the field.
It’s light Jim, but not as we know it
The Wikipedia article is talking about a cylindrical capacitor, but it’s true for an electron too. Because the electron is a charged particle, and it behaves like a tiny bar magnet. Because it doesn’t have an electric field or a magnetic field, it has an electromagnetic field. People think this field is static, but something’s going round and round, as per the Poynting vector. And where else have we seen a Poynting vector? Why, in an electromagnetic wave:
Electromagnetic wave image © Blaze labs
So what’s going around and around in an electron? Let’s see now. We made the electron out of light in pair production, we can diffract electrons, as per the Davisson-Germer experiment and the Thomson and Reid diffraction experiment. We can refract electrons, we can perform electron optics with electrons, and when we annihilate the electron with a positron, what we get is light. It’s elementary my dear Watson. The thing that’s going around and around is light. Only when it does, we don’t call it a photon any more. We call it an electron.
A 511keV photon with toroidal topology
John Williamson and Martin van der Mark wrote a paper in 1991 called Is the electron a photon with toroidal topology? The answer is yes, and they’re not the only people to ask such questions. Also see Qiu-Hong Hu’s 2005 paper The nature of the electron:
Images by John Williamson and Martin van der Mark and by Qiu-Hong Hu
As for the radius of this toroidal energy flow, I think it’s important to remember that the photon takes many paths, just like a seismic wave takes many paths. It isn’t limited to some AB line, even if you wrap that line into a closed path. Hence the Wikipedia electron article says the issue of the radius of the electron is a challenging problem. No wonder, because the electron’s field is what it is. Lorentz knew that back in 1902. And we all know that this field has no outer edge. The field gets weaker and weaker with distance from the centre, but it doesn’t stop. So why do people think the electron has a radius? Because Lorentz used the expression m=e2/r0c2? If you’ve ever read Maxwell’s theory of molecular vortices, I think you may know the answer. See the heady collisions Columbia article which likens particles to tornadoes and hurricanes. Then take a look at a hurricane. It has a radius of maximum wind, which lies just within the eyewall. The eye of the storm has a radius, but the radius of the eye is not the size of the storm. The eye is where there is no wind. Where there is no storm:
In similar vein the classical electron radius is not the size of the electron, and nor is any other radius. Saying the electron is point-like is like hanging out of a helicopter probing a whirlpool with a bargepole, and then saying I can’t feel the billiard-ball so it must be really small. Even the Compton radius isn’t the size of the electron. Nor is the Compton wavelength. Wilczek said ”the electron is effectively a spinning ball of charge, which electromagnetism tells us, generates a dipole magnetic field. The size of that ball can be estimated to be roughly 2.4 x 10-12 meters. The electron is not a spinning ball of charge. It’s an electromagnetic wave going around and around that looks like it’s a ball of charge. The wavelength is 2.426 x 10-12 meters, but the wave is wrapped round twice around a twisting turning spin ½ path. That’s why the electromagnetic field variation looks like an all-round standing field and therefore a charged particle.
The radius of “the eye of the storm” for this electromagnetic standing-wave is the Compton wavelength divided by 4π. It isn’t a fluid-flow vortex like a cyclone or a Falaco soliton or a smoke ring. It’s a stress-energy vortex, an optical vortex, an energy flow. But it has intrinsic spin like a tornado has intrinsic spin. That’s what makes it what it is. Remove the rotation from a spinning coin and it’s still a coin. Remove the rotation from a tornado and it isn’t a tornado any more. All you’ve got is wind. It’s the same for the electron. Remove the rotation from an electron and it isn’t an electron any more. All you’ve got is light. Because the electron is a “spinor”, and a spinor does what it says on the can: “Spinors were introduced in geometry by Élie Cartan in 1913. In the 1920s physicists discovered that spinors are essential to describe the intrinsic angular momentum, or ‘spin’, of the electron and other subatomic particles”.
The electron is a standing-wave spinor
The electron is a spinor, with an intrinsic spin that makes it what it is: a standing wave. Hence in atomic orbitals electrons “exist as standing waves”. And outside of atomic orbitals, electrons still exist as standing waves. Standing wave, standing field. Like Wilczek said, “the proper quantum mechanical description of electrons involves wave functions, whose oscillation patterns are standing waves”.
GNUFDL spinor image by Slawkb, see Wikipedia
This is why the de Broglie hypothesis concerns the wave nature of matter, not the point-particle nature of matter. This is why after noticing a point made by Hermann Weyl, Erwin Schrödinger gave us the time-independent Schrödinger equation which “predicts that wave functions can form standing waves”. This is why we can diffract electrons. We can even refract them, as per Ehrenberg and Siday’s 1949 paper The Refractive Index in Electron Optics and the Principles of Dynamics. That’s the paper that predicted what’s now known as the Aharonov-Bohm effect. That dates from 1959, and demonstrates that electromagnetic four-potential is most fundamental, not point particles. We’ve known since 1933 that we can make electrons and positrons out of electromagnetic waves in pair production. See Patrick Blackett’s paper Some photographs of the tracks of penetrating radiation co-authored with Giuseppe Occhialini. We also know that we cannot separate an electron from its electromagnetic field. There are no neutral electrons. The electron’s electromagnetic field is part of what it is. In fact the electron’s electromagnetic field is what it is. There is a wealth of evidence for the wave nature of matter, and no evidence for the point-particle nature of matter. That’s why it’s quantum field theory, not quantum point-particle theory. Or should be.
The electron has spherical symmetry
To grasp what we’re dealing with, I think it’s useful to think of the spinor depiction as something like an uninflated inner tube. When we pump it up, we have a ring torus, akin to the Poynting vector depictions, reminiscent of a dipole, and without a cowlick. But that doesn’t go far enough. We need to keep on inflating. Our inner tube gets fatter, resembling a horn torus. Even that doesn’t go far enough. We have experimental evidence that the electron at rest has a spherical symmetry as per the physicsworld article search for electron’s electric dipole moment narrows. So we have to keep on inflating until our spinor is a spindle-sphere torus. Only then does it look like the s-orbital:
Gifs courtesy of Adrian Rossiter’s torus animations, S-orbital © copyright 2010 Encyclopaedia Britannica
Only this isn’t what the electron really looks like, because an electron has no outer surface. I’m tempted to say it has an inner surface, like the eye of the storm, but there is no inner surface in any real sense. The depiction shows the rotation, that’s all. And it isn’t the full picture because in a hydrogen s-orbital the electron’s electromagnetic field is largely countered by the proton’s electromagnetic field. Take the proton away, leaving a free electron, with its field in all its glory. Now what does it look like?
What the electron looks like
Of course, the electron doesn’t really look like anything. It isn’t some purple billiard ball, even if some cosmic joker has also shown it as such in the physicsworld dipole article. It has rotation and internal motion, like Born and Infeld were saying in 1935. A spin ½ wave motion, such that the wave ends up looking like a standing field. And it’s just field, like Gustav Mie said in 1913. But you can “see” a solenoid’s magnetic field with iron filings, so you can “see” the electron electromagnetic field too. Then you see that this field has angular momentum. It isn’t a simple spin, it’s a compound spin, a bispinor spin-½ spin with two orthogonal rotations, one toroidal, one poloidal, one at twice the rate of the other. It’s like a steering wheel spin and a smoke-ring spin combined, but with a toroidal topology and a spherical geometry. The result isn’t so different to the gravitomagnetic field. Oliver Heaviside developed gravitomagnetism as an analogy of electromagnetism. What does gravitomagnetism feature? A space-time vortex along with twisted spacetime and frame-dragging. What’s the difference between gravitomagnetism and electro-magnetism? One deals with curved spacetime, the other with curved space. That’s a big difference, but they’re not entirely different. Take a look at the Wikipedia gravitoelectromagnetism article: “Consider a toroidal mass with two degrees of rotation (both major axis and minor-axis spin, both turning inside out and revolving). This represents a “special case” in which gravitomagnetic effects generate a chiral corkscrew-like gravitational field around the object”. So what sort of electromagnetic field would we have for a spinor with two degrees of rotation? It would look something like this:
Poynting flux image found on physics stack exchange
It’s like you reach into a lattice with your right hand and twist, then reach round the side with your left hand and twist again. Perhaps one day somebody will set up animations to show it to you in an intuitive fashion, demonstrating how pair production really works, and how the electron is a dynamical spinor in frame-dragged space. But I’m going to show you a flat version instead. It’s easier to draw, and easier to grasp:
For some strange reason human beings seem to have difficulty thinking about curvature in three dimensions. It’s somehow easier to understand the electron when it’s depicted as a flat thing on a piece of paper. It’s easier to see why it’s a dynamical spinor in frame-dragged space, and why it moves the way that it does, and how a magnet works. It’s easier to see why the scientific evidence says this is what the electron is. The scientific evidence of the screw nature of electromagnetism. But first, a brief word on the positron.