Self-energy of electron from classical reasoning

If it takes energy to group charge together(self energy) how can it be possible for every single electrons, etc, to have exactly same amount of charge? (think of if we hold some sand in our hand, then surely the masses of each of the sand in our hands are slightly different.)

I never studied elementary particles, but I wonder if it is the fall of electrodynamics/classical field theory? (Self-energy predicting something hold electron together.)

• I don't understand your question - the electron is, as far as we know, indivisible as a fundamental particle. If there were electrons with different charges, we would call them different names - the charge of a particle is one of its defining characteristics. – ACuriousMind Oct 23 '14 at 15:30
• Sorry for the lousy expression. But does electrodynamics alone predict the existence of fundamental particle? My question is if there is such a thing called self energy, then it is quite unlikely for all electrons share exact same amount of charge. Or let me restate the question: What holds electron together and how? (sorry I really don't have any working knowledge of particle physics, btw I like your profile picture) – Shing Oct 23 '14 at 15:39
• Nothing holds an electron together so far as we know. It is a quantum field. A particle and a wave. The physical object has no volume to be held together. Pick one – Jim Oct 23 '14 at 15:54

There is no notion of quantization of charge in classical electrodynamics. Charge is a continuous, infinitely divisible quantity there, and there's nothing at all that would indicate what carries the charge. The electron (or any other particle, for that matter) is not predicted by classical electrodynamics, and thus none of the classical notions of self-energy apply to it.

The quantization of charge and the electron as an indivisible particle can only be understood by quantum electrodynamics, where the four-potential of electrodynamics and matter fields are properly treated as quantum fields. There is such a thing as self energy in this picture, and it is the cause for a difference between the "bare" and "dressed" charges of the electron, of which only the latter is measurable (as the ordinary charge we know), so we should be cautious to assign any meaning to the former.1

There is nothing "holding particles together", as far as we know. At least, our best (quantum field) theories do not say anything about internal structure of fundamental particles - these things just are, and no experiment has hitherto indicated any kind of substructure of the electron.

If you think the charges "inside" an electron should repel each other and thus rip it apart, then you are commiting a category error - you apply the classical notion of force on a scale where quantum effects are dominant, and the classical thinking therefore invalid.

1"Bare" and "dressed" quantities are terms appearing when renormalizing a quantum field theory. If we define our theory through a Lagrangian, then the charge $e_0$ appearing in the matter-gauge field coupling part $e_0\bar\psi\gamma^\mu A_\mu\psi$ is called the bare charge, which has, in the course of renormalisation, to be taken infinite in some sense to yield a finite number for the renormalized dressed charge $e(\Lambda)$ which is dependent on the energy scale $\Lambda$ of the process we are looking at. That is, the measured "charge" of an electron is actually not constant when we go to higher process energies, but its coupling to the electromagnetic field becomes stronger with increasing energy, giving rise to a problem known as the Landau pole, where the coupling blows up and becomes first non-perturbative, and then infinite. Note that one often looks at the fine-structure constant instead of the charge itself, but it is essentially the charge squared.

• um...I don't understand quite a few things:1.) what meant by four-potential? – Shing Oct 24 '14 at 14:55
• 2.)if I haven't misunderstood you, you were saying there is no such thing as self-energy in classical physics. why? (as it is a mathematical consequence of total energy of whole system. ) anyhow, thanks. – Shing Oct 24 '14 at 15:01
• 3.) would you elaborate a bit more about "bare" and "dressed" charges of the electron? – Shing Oct 24 '14 at 15:03
• @ShingLau: I updated the answer for 1.) and 3.). As for 2.), I am not saying that classical physics has no notion of self-energy, I am saying that it doesn't apply on the quantum scale, i.e. you cannot understand the electron with classical physics, and, in particular, you cannot talk about its self-energy as if it were a classical collection of charges. – ACuriousMind Oct 24 '14 at 15:16
• Sorry for being late, I was taking my final, and then forgot everything lol – Shing Mar 23 '15 at 9:27

For grains of sand, each grain is made of atoms, and the number of atoms in each grain can be different.

There is no evidence that electrons are composed of plural particles.

let me restate the question: What holds electron together and how?

That depends on theory (view) of the electron.

In the beginning of 20th century, people thought electron was a small marble packed with charged particles. These would repel each other with tremendous force so some balancing forces are needed in this model. This view was always quite complicated (continuum mechanics with unknown balancing forces) and I do not think it was particularly fruitful.

On the other hand the view that electron is a point particle characterized by two numbers - mass and charge - is much simpler and is used every day, both in classical and quantum theory.

So far the prevalent view on the electron is that it is point particle. In this view, it has no space distributed parts which could repel each other, so no balancing forces are needed.

• Thanks, so under this content, we must postulate quantization of charge in classical electrodynamics, right? – Shing Oct 24 '14 at 14:44
• Quantization of electric charge is usually assumed in microscopic theory as additional independent assumption. This was motivated by experiments like Millikan's oil drop experiment and continues to be part of the picture of matter. However, theories with continuously changing amount of charge can be formulated too. It would be harder to use them to explain some experiments, though - check out the Millikan oil drop experiment. – Ján Lalinský Oct 24 '14 at 19:16