# The size of an electron

Considering that an electron is a quantized excitation of the Dirac field, why are there still discussions regarding the "size" of an electron? Isn't the "size" of an electron simply defined as the expanse of the Dirac wave function?

Of course this means that, depending on the situation, the "size" of the electron changes (for example, bound to a proton the "size" of an electron is ~$a_0$, and as a plane wave free particle of well defined momentum it would be theoretically infinite in expanse), but I don't see how that is a problem.

I understand that the question of the distribution of an electron has been addressed here, and elsewhere, before. My intent here is to ask, why do we still even ask the question? Is there something wrong with the viewpoint I expressed above?

Edit: Asked another way (in response to comments), in what physical pictures is the "size" of the electron, independent of the expanse of the wave function, useful or meaningful?

in what physical pictures is the "size" of the electron, independent of the expanse of the wave function, useful or meaningful?

Wave functions describing particles are in the framework of first quantization. This is useful for specific problems and boundary conditions , to get spectra of atoms for example. It is not useful for the study of elementary particles, and electrons are elementary particles. Here is the table:

This table of elementary particles has been deduced from innumerable elementary particle experiments. These are point particles, zero extent, and enter the Lagrangian formulation of the Standard Model of particle physics, a very successful mode that encapsulates practically all data up to now. The calculations of the scattering of particles on other particles are done in the framework of second quantization.

In second quantization the wave function solutions define the ground state on which field operators are defined, which create or annihilate particles at all relevant points in space. The perturbative expansion of the solutions is represented by Feynman diagrams where the creation and annihilation operators are used to define the integrals that will give the solutions for the specific interactions. In all this formalism the elementary particles enter as point particles.

Thus the wavefunction solutions are relevant only for specific problems, and the representation of the electron as a dirac wavefunction, with a probability of spatial extent, is only useful for simple problems, as also the wavepacket representations. These have been superseded by the second quantization formalism as far as calculations and fitting elementary particle data go.

I'll quote the Nobel Lecture of Hans Dehmelt:

With the rise of Dirac’s theory of the electron in the late twenties their size shrunk to mathematically zero. Everybody “knew” then that electron and proton were indivisible Dirac point particles with radius R = 0 and gyromagnetic ratio g = 2.00. The first hint of cuttability or at least compositeness of the proton came from Stern’s 1933 measurement of proton magnetism in a Stern-Gerlach molecular beam apparatus. However this was not realized at the time. He found for its normalized dimensionless gyromagnetic ratio not g = 2
...

Today everybody “knows” the electron is an indivisible atomon, a Dirac point particle with radius R = 0 and g = 2.00.... But is it? Like the proton, it could be a composite object. History may well repeat itself.

No, the size of the electron is not given by the size of the wave function. At least that is not what is meant by the size of the electron. The size of the wave function of an electron can be arbitrarily big. One can, for instance, send a single electron through a very tiny hole in a metal sheet. The wave function of the electron will expand after the hole. Imagine that I use such a device to launch an electron from earth toward Jupiter. By the time it reaches Jupiter, depending on the experimental parameters, the wave function of the electron may be larger than Jupiter!

What we mean by the size of an electron is the size of the particle that one would observe in some detection process. How would one be able to determine this size? Typically, the detection process is a scattering process. In other words, one measures the amount of scattering that is observed when one electron is scattered by another electron (or some other point particle). Then one plots this scattering as a function of energy. If the electron has a particular size, then converted to an energy scale, this size would show up in the scattering graph as some scale dependence.

However, what has been observed in such scattering experiments is that the graph is scale invariant. From that one concludes that the electron is a point particle, at least up to the highest energy used in the experiment. In other words, the electron size must be smaller than the size associated with the highest energy of the experiment.

For a more mathematical treatment of this topic, one can read up on Bjorken scaling. However, it is applied to more than just electrons.

Considering that an electron is a quantized excitation of the Dirac field, why are there still discussions regarding the "size" of an electron? Isn't the "size" of an electron simply defined as the expanse of the Dirac wave function?

You could say that. I quite like saying the electron's field is what it is. The electron isn't some billiard-ball thing that has a field. You cannot separate the electron from its field. Field is what it is. It's quantum field theory. How anybody ever came to think the electron was an R=0 point particle absolutely beats me. What about the Einstein-de Haas effect which "demonstrates that spin angular momentum is indeed of the same nature as the angular momentum of rotating bodies as conceived in classical mechanics". How does that work for a point particle? Or electron magnetic moment wherein the electron "indeed behaves like a tiny bar magnet". A bar magnet is like a solenoid, reduce it to one turn of wire and there's a current going round and round. How does that work for a point particle? Or how about the Poynting vector for a static field: "While the circulating energy flow may seem nonsensical or paradoxical, it is necessary to maintain conservation of momentum". Or atomic orbitals, where electrons "exist as standing waves". Seen any spherical harmonics for point particles recently? Me neither.

There's wealth of hard scientific evidence out there that says the electron is not a point particle. Indeed, the evidence says there is nothing solid in the middle. And yet people will point to scattering experiments and say they establish an upper size to the electron. That just doesn't fit with all the other scientific evidence. IMHO it's like hanging out of a helicopter probing a whirlpool with a bargepole saying I can't feel the billiard ball, so it must be really small. The wave nature of matter is a fact of life. We can diffract electrons. The electron has a wave nature. A standing wave nature. Standing wave, standing field. And that field is what the electron is. Only it doesn't have an outer edge. So the electron doesn't have a size.

Of course this means that, depending on the situation, the "size" of the electron changes (for example, bound to a proton the "size" of an electron is $~a_0$)

It isn't, not really. The fields largely cancel, but not totally. There's a bit of a field left over, extending out into space. But we don't call it an electromagnetic field or an electron field. We call it something else.

I understand that the question of the distribution of an electron has been addressed here, and elsewhere, before. My intent here is ask, why do we still even ask the question? Is there something wrong with the viewpoint I expressed above?

No. But there's plenty wrong with the claim that the electron is a point-particle. That's elevating mathematical convenience higher than a whole wealth of hard scientific evidence.

Edit: Asked another way (in response to comments), in what physical pictures is the "size" of the electron, independent of the expanse of the wave function, useful or meaningful?

You understand that there is no magical mysterious action-at-a-distance. You understand that electrons and protons aren't really throwing photons at one another, as if hydrogen atoms twinkle and magnets shine. You understand why electrons and positrons move the way that they do. You understand that each is a chiral dynamical "spinor" in frame-dragged space.