[jules]

I'm not trying to play with words...by EM I mean Maxwell's equations and the Lorentz force law. I think that's the conventional meaning. The point is that these two are taught in an EM class as if it's a single coherent theory that tells you what point charges do.

Mass distributions don't solve the issue in a satisfactory way, in my opinion. If you replace a particle with a finite size sphere you've solved the infinity but lost relativistic invariance. You could perhaps come up with a way to hold the charge distribution together in a relativistically invariant way, but that can hardly be considered part of EM, and might involve arbitrary choices. That a dipole behaves differently than a monopole is clear. That's already the case even if you ignore the self interaction.

The question "What happens if I put an electron in a uniform magnetic field?" or "What happens if I have two electrons?" seems like it should be answered by EM. One can hardly ask a simpler question. I'm pretty sure that most physics students who've had an EM course are under the impression that they should be able to answer this question. When I was in such a class it was never explained that this was even an issue, and when I asked about it the answer I got was "just wait for QED".

If you don't like the word "schizophrenic" for this issue, that's cool. I think it's descriptive, but YMMV. Wald's slides say:

> Classical Electrodynamics as Taught in Courses

> At least 95% of what is taught in electrodynamics courses at all levels focuses on the following two separate problems: (i) Given a distribution of charges and/or currents, find the electric and magnetic fields (i.e., solve Maxwell’s equations with given source terms). (ii) Given the electric and magnetic fields, find the motion of a point charge (possibly with an electric and/or magnetic dipole moment) by solving the Lorentz force equation (possibly with additional dipole force terms).

That's all I meant by it.

Wald's slides are interesting, thanks :)

[DoctorOetker]

I would like to respond to your reply at https://news.ycombinator.com/item?id=18846249 and I was going to use among other things the example of an electron in a uniform magnetic field. So I was totally surprised when I read this comment already mentioning

> ... "What happens if I put an electron in a uniform magnetic field?" ...

Either 1) this is pure coincidence (and you are contrasting the difficulty of the 2 electrons compared to the "simpler" electron in a magnetic field), or 2) you are referencing a certain 'issue' or puzzle about the electron in a uniform magnetiic field?

Could you clarify if it is 1) or 2) or something else? and if 2) clarify the puzzling issue regarding the "electron in a unifoorm magnetic field"?

Then I will feel more comfortable answering the other comment you made, so I can clarify my earlier reply to you :)

[jules]

It's fundamentally the same issue, and the same difficulty.

The reason I mentioned 2 charges orbiting around each other is to avoid getting into a discussion about the uniform magnetic field, and that maybe the extra radiation energy is just coming from the uniform magnetic field, and that energy is still conserved because the total energy was infinite to begin with.

[Certhas]

If you take a non-relativistic model of your matter content, then the theory becomes non-relativistic. That's trivial. So take a relativistic model for your matter and you have no problem [1]. EM gives you a theory of EM Fields and their interaction with matter. It shouldn't be surprising that EM doesn't give you a theory of matter.

I maintain there is no conceptual problem with EM, the problem is with your electron model which is unphysical. It might seem reasonable to you, but that's because of your intuition to build up matter from point particles, which is only justified by QFT considerations that came almost a century after EM.

[1] A simple matter model often used in GR is dust. I'm sure this would work for EM even better.

[jules]

I don't think making a relativistic theory of charged matter that approximates anything in the real world is as easy as you think. Charged dust will behave in very complicated ways, so I'd have to see a differential equation that models it.

I'm not saying that the point particle model is reasonable. I'm saying that it seems reasonable given what is said in a standard EM course.

Let me phrase it in a different way. In classical mechanics you have lots of problems of the form "the state of the system at time 0 is X, what is the state at time t?".

The problem with EM is that it doesn't have a relativistically invariant answer to such questions when point charges are involved. And, as far as I am aware, there also isn't a standard relativistically invariant answer involving a charge distribution, or at the very least it's not commonly taught.

Maybe you think that I shouldn't find this surprising, but given how EM is taught, I'd say that my surprise is fully justified.

[Certhas]

The teaching will vary greatly depending on teacher. But I remember that I was told that rigid bodies, and hence centre of mass thinking did not work in relativistic mechanics. It's very possible that as a student I never put that together with the inadmissibility of point charges in EM.

I must insist though that EM has no problem with initial value formulations. It simply doesn't provide you with a theory of matter. It turns out that that theory of matter really requires QM, hence in EM we never bother with non-QM models of relativistic matter. That's why you have to look in the GR literature.

As a pedagogical point I can agree that the limits of the conceptual foundations of our theories are never really explored enough. EM turns out to be fine, the field tensors are completely measurable, but that's a really cool paper that isn't taught either:

https://link.springer.com/chapter/10.1007/978-94-009-9349-5_...

As for charged dust, it works even if you switch on GR as well:

http://www.numdam.org/article/AIHPA_1973__18_2_137_0.pdf