The interaction of a charged particle with its own field

Question

I have seen other questions regarding the mentioned topic. I want an answer to be different in some ways from the answers that were posted in response to the related questions.

What exactly are self-force and self-energy? I have read that a charged particle cannot interact with its own field. This statement of course indicates that a field is not simply a number (the force function defined at every point, with which a particle would be acted on if placed there) but it is something else. Now, I would like to know how did the expectation of a particle's interaction with its own field came about and what led it to be discarded? I would request for a complete definition of an $E$-field that would encompass the whole physical picture of it.

From the answers to some of my last questions I have gotten the information (or let's say fact) that in the advanced theory, namely QFT, fields are more fundamental but at the time these ideas were being dealt there was no QFT, so would like to hear classical perspective in this matter of interaction of a particle with its own field. (I also wonder how it would be to compare this problem of fields to Russell's paradox in set theory)

Answer 1

The classical concept of "self-energy" is the electromagnetic mass $$ m_\text{em} = \frac{2}{3}\frac{e^2}{rc^2}$$ which describes the "additional inertial mass" that a charged sphere of radius $r$ and charge $e$ acquires compared to an uncharged sphere. The smallness of $e^2$ and $\frac{1}{c^2}$ means that you will not notice this effect at everyday mesoscopic scales. Nevertheless, classical electrodynamics indeed predicts this "mass increase" since the Abraham-Lorentz force acts on every accelerated particle, and hence "tries to slow" the acceleration of any charged object.

Intuitively, you need more force to accelerate a charged than an uncharged body because accelerated charged bodies radiate away some of the energy you try to put into them. The closely related concept of electromagnetic self-energy (see Wikipedia link) may be thought of as the energy needed to assemble the charged object out of charges coming from spatial infinity.

Like the concept of relativistic mass, the concept of electromagnetic mass is obsolete, and what we call mass today is the invariant mass of an object given by the square of its four-momentum, which never changes.

Before the advent of relativity and quantum field theory, the concept of electromagnetic mass posed several problems - the radius of the electron seemed to be zero, leading to an infinite electromagnetic mass, i.e. it seemed impossible to ever accelerate it, and several methods to compute it seemed to not agree (see 4/3 problem). This might be seen as a parallel to the quantum self-energy that plays a decisive role in the renormalization process going from bare to dressed particles, also commonly described as "removing infinities".

It may also be worth mentioning that the Feynman-Wheeler absorber theory tried to remove the self-energy in a time-symmetric formulation of electrodynamics, but had precisely problems with the Abraham-Lorentz force. Feynman would later imply that trying to remove the self-interaction was an error, and indeed, the later quantum theory of electrodynamics would again have the self-energy of charged fields arising in the course of renormalization.

Answer 2

I'll give you an answer which is based on axiomatic approaches to such matters.

First the short answer: It depends on whether you want a theory of electromagnetism which is invariant in inertial frames or not.

Now the simple explanation:

1- Assume we have found an inertial frame S0.

2-let's say we have a stationary spatial charge distribution (not a point charge which is not clear how is defined and it is ignorantly and easily assumed very natural in most textbooks). Based on the principles of classical electrodynamics, it has a well-defined electrostatic field all over the world. And it also does not self-interact meaning that it does not impose force on itself. For instance if it the charge is distributed on the surface of a sphere, the field is zero (not undefined) on the surface and inside the sphere.

3- Now let's say our charge distribution moves with a uniform velocity v1 with respect to S0. So if I fix a frame S1 on this creature, S1 would be an inertial frame too.

4- Now comes the decisive moment: do you want a theory invariant in inertial frames? meaning that there would be no difference in inertial frames or not. If yes:

4-a: Since your creature in S0 does not self-interact, It should not self-interact in S1 as well. However, you need to bring up new physics to make this happen. For instance new sets of transformations in place of Galilean Trans. to make electromagnetism does not rely on a special inertial frame such as aether. Because Maxwell's theory made one of the inertial frames distinctive by assuming the existence of aether. It is whilst you need a theory which does not differentiate between inertial frames.

If no:

4-b: The field of a moving charge distribution, as a function of 3 independent variables and one free parameter and if assumed to propagate with a limited speed,i.e. the speed of light, could affect your charge from the past and from the future. Again here you need to come up with new physics and principles or modify electromagnetism or invalidate electromagnetism. The last one is not a good choice because of the success of electromagnetism.

p (electromagnetism) -> q (its predictions) is equivalent to not-q (invalidating the predictions) -> not-p (invalidating electromagnetism). Electromagnetism is highly successful, ergo you cannot invalidate its predictions very easily.