About 150 years ago, people began to do the experiments that would lead to the discovery of the electron.
They would study the electrical conductivity of rarefied gases. Finally, in 1896, J.J. Thomson and his collaborators proved that the colorful fog coming from the cathodes is composed of individual discrete corpuscles, the electrons.
It just happened that within 10 years, electrons were seen by Henri Becquerel in a completely different context – as radioactive beta-radiation of some nuclei. In 1909, Robert Millikan and Harvey Fletcher performed their oil-drop experiment. The charge of an oil drop may be as small as one electron's elementary charge which translates to an elementary force acting on the drops in an external electric field. Another force acting on the oil drops is the friction force and the equilibrium between these two forces determines the asymptotic speed of the oil drops.
The 1910s and 1920s were all about the "quantum motion" of the electrons: people started to understand the structure of the atoms. First, Bohr offered his classical planetary model of the atom equipped with the extra ad hoc quantization rules for the orbits. Finally, in the mid 1920s, the correct laws of quantum mechanics were found and replaced Bohr's model that wasn't quite correct even though it had some desirable properties that had partially captured the "spirit" of the coming quantum revolution.
Motion of the electron vs structure of the electron
I want to spend the bulk of this article by discussions about the internal structure of the electron – and how it was evolving over the years. That's why I find it very important to clarify a widespread misconception. What is it? People tend to confuse the wave function of the electron in space with the internal structure of the electron. They're completely different things.
The wave function of the electron in a hydrogen atom is a "cloud" of radius 0.1 nanometers or so. This distance scale doesn't determine the order-of-magnitude size of the electron; instead, it determines the size of the atom. Imagine that an electron is a ball of radius \(r_e\) for a while. The center of this ball may be located at various points in space. In the hydrogen atom, the location of the center of this electron ball is undetermined and the uncertainty is approximately 0.1 nanometers (Bohr radius times two). But the size of the ball – the electron – is a completely different, independent length that is much shorter.
Classical radius of the electron
Hendrik Antoon Lorentz began to discuss the radius of the electron as early as in 1892, four years before Thomson's discovery of the electron. He would offer a clever idea that the electrostatic potential energy of the electron – a distribution of the electric charge – is equal to the latent energy \(E=mc^2\) stored in the electron's rest mass. However, it was 13 years before Einstein found his special theory of relativity so all these "early glimpses" of relativity always had some bugs in them. For example, Max Abraham would claim that the aether theory implied \(E_0=(3/4)m_e c^2\) as the relationship between the interaction energy and the mass. Many of the dimensionless numerical coefficients of order one were simply wrong.
By the classical electron radius, we usually mean\[
r_e = \frac{1}{4\pi\varepsilon_0} \frac{e^2}{m_ec^2}\approx 2.818 \times 10^{-15}\,{\rm m}.
\] If you place two charges \({\mathcal O}(e)\) at the distance \(r_e\) from one another, the electrostatic potential energy between them will be \(m_e c^2\). It's not shocking that the electrostatic potential energy of a "diluted electron" represented as a sphere of radius \(r_e\) or a ball of this radius is of the same order, up to coefficients such as \(3/5\) or \(1/2\).
You should note that the classical electron radius is some 10,000-100,000 times shorter than the Bohr radius. The electron is much much smaller than the atoms. For the sake of completeness, let me also mention that there exists another length, the Compton wavelength of the electron \(h/m_e c\), which is exactly the geometric average of the classical radius of the electron and the Bohr radius. Up to numbers of order one, it is exactly in between, about \(2.4\times 10^{-12}\,{\rm m}\). The longer Bohr radius and the shorter classical electron radius are the fine-structure-constant times longer and shorter, respectively.
But let's return to the main story. If the electron is visualized as a sphere of radius \(r_e\), the electrostatic potential energy is of order \(E_0=m_e c^2\) which allows us to say that all of electron's mass actually arises from the electrostatic energy. Of course, for this "explanation" to be a meaningful one rather than an example of circular reasoning, we should also find a theory explaining why the electron wants to keep its radius at \(r_e\), why it doesn't want to blow up. Recall that the like-sign charges repel so "halves" or other "pieces" of the electron want to repel from the rest, too.
If you want to succeed in this task (a task that is misguided, however, as we will mention in a moment), linear electrodynamics is no good. People proposed various non-linear modifications of classical electrodynamics, most famously the Born-Infeld model in the 1930s. The Lagrangian\[
\mathcal{L}=-b^2\sqrt{-\det\left(\eta+{F\over b}\right)}+b^2
\] reduces to \(-F_{\mu\nu}F^{\mu\nu}\) if \(F_{\mu\nu}\to 0\) but for larger values of the electromagnetic field strength, it develops some non-linear terms that prevent the electron from shrinking to \(r_e\to 0\) or exploding to \(r_e\to\infty\). There are infinitely many nonlinear deformations of classical electrodynamics one could think of. But remarkably enough, the Born-Infeld action is exactly what was derived in the 1990s as the only right answer for the electromagnetic fields on D-branes in string theory. This fact showed that Born and Infeld had a remarkably good intuition for the "right equations". Everyone who finds an equation that later turns out to be "unique" by insights of string theory may count as a visionary of a sort.
Renormalization: how the attempts to regularize the electron became obsolete
If the electron radius were exactly \(r_e=0\), its electrostatic interaction energy would be infinite. This is true even in classical (non-quantum) physics. However, these infinities arising from short distances – and \(r_e\to 0\) is an example of short distances – reappear in quantum physics all the time. While the Born-Infeld action was an attempt to get rid of the infinities in classical physics, many attempts to remove analogous divergences have been made in quantum physics in general – and quantum field theory in particular.
The first image, (a), is the leading correction to the electron's self-energy. It's a Feynman diagram most directly corresponding to the classical electron's self-interaction, electrostatic energy. Because of this diagram, the electron mass (and other quantities) is modified by an infinite amount. The infinity arises from the part of integrals in which the two interaction points (events in spacetime from the Feynman diagram) are very close to each other i.e. \(x\to 0\) or, equivalently, in which the loop momentum \(p\to\infty\). This region of the integration variables is known as the ultraviolet (UV, short-distance) region. The other two diagrams – and many others – contribute their own UV divergences to the dynamics, too.
However, starting from the 1940s, people learned how to subtract these UV divergences. Even though the individual terms may be infinite, when all of the terms are properly summed and a finite number of "genuinely physical" parameters is set to their measured values, all the infinities cancel and the predictions for all "genuinely physical" quantities will be finite. This is the magic of the renormalization.
Renormalization has been emotionally frustrating to various people – including giants such as Paul Dirac – but this dissatisfaction was always irrational. What's important in science is that one has a well-defined procedure to extract the physical predictions and these predictions agree with the observations. QED and other quantum field theories supplemented with the renormalization technique can't get a different grade than A, at least in the subject of science. They could get a D or worse from philosophy or emotions but a bad grade from philosophy may often be a reason to boast.
Ken Wilson's concept of the renormalization group from the 1970s gave us a new way to understand why renormalization worked. Before Wilson, people would think it was necessary to imagine that the electron had to be exactly point-like and the subtraction of the infinities – and they had to be strict infinities – was an essential part of the game. However, after Wilson, people would interpret the renormalization differently. They would say that they work with an "effective theory" that is able to predict all sufficiently long-distance, low-energy processes.
This effective theory doesn't force you to believe that the electron is exactly point-like. Instead, you may imagine that it has a nonzero size and its inner architecture may be "pretty much anything". The effective theory allows you to be agnostic about the inner architecture of the electron. It allows you to prove that whatever the internal structure of particles is, the predictions for all the long-distance phenomena will only depend on a finite number of constants such as \(m_e,m_\gamma,e\) – which may be calculated as functions of the internal architecture of the particles. But one may prove that all the other details about the internal structure will make no impact on the long-distance, low-energy observables! We say that the long-distance, low-energy predictions only depend on the internal architecture of particles through a finite number of constants such as \(m_e,m_\gamma,e\).
To say the least, it's a way of thinking about the divergences that makes the whole process of renormalization more acceptable, more philosophically pleasing. The divergent terms may be finite, after all. And while the calculations become most beautiful if the electron is strictly point-like, you're allowed to imagine it is not exactly point-like but you may prove that almost all the dependence on the messiness of a "finite-size electron" evaporates if you study long-distance, low-energy processes only. According to the renormalization group's philosophy, infinities in renormalization that cancel are shortcuts for unknown large yet finite numbers whose detailed value is mostly irrelevant.
Going to high energies
Well, while it's perfectly enough for all questions that affect atomic physics, you may still want to know what is hiding inside the electron; you may want to go to high energies and short distances, either theoretically, or experimentally. When you look at the electron, it seems obvious that its size has to be much smaller than the classical electron radius, probably at least 2-3 orders of magnitude shorter than that. The Standard Model allows you to "create" electrons by the Dirac field \(\Psi(x,y,z,t)\) which depends on the point in spacetime – so it's apparently created at a single point only. And the interactions are perfectly local, too. In this sense, the electron is exactly point-like in the Standard Model – although the electron is obviously acting on objects in its vicinity in various ways as well. And we know that the Standard Model is a good theory for all distances longer than \(10^{-19}\,{\rm m}\) or so which implies that the internal structure of the electron can't be longer than that.
So what is inside the electron according to the cutting-edge theories?
In the 1970s, people proposed preons. Quarks and leptons could be composite particles much like protons and neutrons. It wouldn't be the first time when the "indivisible" particles of our time were divided to smaller pieces. In other words, it wouldn't be the most original idea about the way how to do further progress in physics. However, when one looks at the preon models, they don't seem to work well, they predict lots of new particles that don't exist according to the experiments, and they don't seem to be helpful to solve any open puzzles in physics.
In other words, the evidence is now pretty strong that if you want to stay at the level of point-like quantum field theories, electrons are strictly point-like particles. It doesn't mean that electrons are strictly point-like in general, however. If you upgrade your physics toolkit to string theory, the only known (and quite possibly, the only mathematically possible) framework that goes beyond that of point-like-particle-based quantum field theory, the effective field theories derived from all the convincing and viable vacua of string theory will look at the electron as an exactly point-like particle. However, if you look at the electron with the string accuracy, it's still a string.
In most vacua, the electron is a closed string although models where the electron is an open string exist, too. An electron as a compact brane is in principle possible as well but it is much more exotic and maybe impossible when all the known empirical constraints are imposed. The typical size of the string is of order \(10^{-34}\) meters although models where it's a few orders of magnitude longer also exist. At any rate, the size of the string hiding in the electron is incomparably shorter than the classical electron radius.
You may interpret a string as a chain of "string bits". In this sense, a string is a composite system that has many internal degrees of freedom, much like atoms and molecules. However, the stringy compositeness has some advantages that allow us to circumvent problems of the preon models. I discussed them in the article Preons probably can't exist three months ago.
Because string theory suggests that the internal size of the "things inside the electron" is much shorter than the classical electron radius, you may rightfully conclude that the most modern research has led to the verdict that the classical electron radius isn't such an important length scale. You may calculate it from the electron mass and the elementary charge; however, nothing too special is happening at the distance scale comparable to the classical electron radius. Instead, the size of "things inside the electron" may be much shorter than the classical electron radius and the electron emerges as a rather light particle because its interaction with the mass-giving Higgs field is rather weak – and because the Higgs condensate is rather small, too (the latter fact seems to be "unlikely" from a generic short-distance viewpoint: this mystery is known as the hierarchy problem).
Long distances: everything is clear
I think it's appropriate to emphasize once again that all these ambitious questions about the internal structure of the electron make pretty much no impact on the behavior of the electron in atoms and other long-distance situations. If we know that there is one electron in a state, this electron is fully described by its momentum (probabilistically, by a wave function) or position (by another wave function) and by its spin (one qubit of quantum information). And yes, the electron is spinning, after all.
The transformation of the spin degree of freedom to another basis – a basis connected with a different axis than the \(z\)-axis – is completely understood and dictated by the group theory applied to the group of rotations. Many people tend to be misguided about this point. They think that if they "deform" the electron by equipping it with some non-linear terms or by seeing its internal stringy or preon-based structure or by acknowledging the fields around the electron, the states of the electron will deviate from the simple Hilbert space whose basis is given by the states \(\ket{\vec p,\lambda}\) where \(\lambda\) is "up" or "down".
But this isn't possible. Even if you incorporate all the facts about the electron's structure, its interactions with all other fields, preons or (more likely) strings that may be hiding inside, as well as higher-order interaction terms that we neglect in the Standard Model, it's still exactly true that what I wrote is the basis of the electron's Hilbert space. This claim follows from the spacetime symmetries – and the basic, totally well-established facts about the electron such as \(J=1/2\). The symmetries are not only beautiful but, as the experiments show, they hold in Nature. Your full theory – including all the corrections and subtleties – must conform to them.
The behavior of the electron at long enough (atomic and longer) distances is described by the Dirac equation. When the speed of the electron is much smaller than the speed of light, you may simplify the Dirac equation to the Pauli equation which is nothing else than the non-relativistic Schrödinger's equation with an extra qubit, two-fold degeneracy for the spin (but the operator \(\vec S\) doesn't enter the Hamiltonian, at least not in the leading approximation in which the spin-orbit and other relativistic interactions are neglected).
Well, the electron also acts as a tiny magnet whose magnetic moment is a particular multiple of the spin, \(\vec\mu\sim \vec S\): we have to add \(-\vec\mu\cdot \vec B\) to the Hamiltonian. The coefficient \(\vec \mu/\vec S\) may be calculated from the Dirac equation, up to the 0.1% corrections of loop processes in Quantum Electrodynamics. The magnitude of the Dirac-equation-calculable magnetic moment is twice larger than what we would expect from a classical "spinning charge/current" of the same magnitude.
The electron may hide lots of wonderful new structure inside. However, the particle's behavior in the atoms is independent of these not-yet-settled mysteries. It's both good news and bad news. It's good news because our understanding of atoms and similar, relatively long-distance physical situations may be rather complete despite the incompleteness of our understanding of the internal structure. It's bad news for the same reason: the observations of the atomic and other phenomena can't tell us anything about the very short-distance physics even though we would love to learn about it.