[Certhas]

For any number of particles, the equation for each of them is this one here:

https://en.wikipedia.org/wiki/Covariant_formulation_of_class...

Edit: Coupled differential equations.

If I have an equation for x in terms of y, and one for y in terms of x, then in total I have a set of equations for x and y.

e.g.:

  dx/dt = y
  dy/dt = -x

then the solution is

  x = C e^(i t)
  y = i C e^(i t)

with C a constant determined by the initial conditions. Nothing mysterious.

[jules]

I understand differential equations ;-)

That page describes Maxwell's equations and the Lorentz force law. That a naive approach cannot work can be seen by considering the following example. Take one charge initially at rest with huge mass, and another light charge orbiting around it. According to Lorentz law it will orbit in a circle for the right initial conditions. However, then according to Maxwell's equations it will radiate electromagnetic waves, violating energy conservation. The field of the accelerated charge will affect the charge itself, but this is not easy to take into account, because that field is infinite at the location of the charge.

[Certhas]

Then stop playing with words and say what you mean.

Point charges make sense without qualification in EM, that much is true.

But that doesn't mean that there is something strange or fishy going on here. You can formulate Lorentz forces using mass distributions and you're fine (it's right there in the next subsection of the wiki). The pathologies don't appear. Of course the resulting equations are non-linear, and that is the origin of the failure of point sources to make sense.

If you then try to recover point sources by taking a limit, the details of the assumptions you make matter a great deal [1]. For a very easy example, your point source might have a dipole.

This problem is kind of well known and studied in the General Relativity literature where the non-linearity of the field equations forces you to confront similar type of problems already when trying to derive geodesic motion [2].

I suspect all this is familiar to you. I would phrase the conclusion as such: The Newtonian idea of focusing exclusively on the motion of the centre of mass, which reflects the influence of distributed forces on a rigid body, breaks down in any relativistic theory. This is simply because the notion of a rigid body breaks down. As such the point mass, which was a conceptual cornerstone of Newtonian mechanics, along with the notion of a force acting on it, are relegated to a technical tool useful only when non-linearities are neglected.

[1] https://arxiv.org/abs/0905.2391

When googling for this paper I found a nice set of slides by Wald about this whole business: http://www.math.utk.edu/~fernando/barrett/bwald1.pdf

[2] https://arxiv.org/abs/gr-qc/0309074

[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

[DoctorOetker]

define radiate? if you calculate the poynting vector you see energy is not really leaving the system in this case, although there certainly is electromagnetic oscillation/rotation/circulation, so there is no energy violation.

1) If we consider the ground state of a system this behaves as expected quantum mechanically: there is motion in the ground state but no energy leaves the system! Why hold Maxwell equations to a perhaps higher or perhaps falser standard than we hold quantum mechanics??

2) But if a system is not in its ground state we do expect (by the observation that excited molecules emit light) this oscillation/rotation of the electromagnetic field to radiate away energy that leaves the system. (Actually the simplest, most widely taught formulations of quantum mechanics, also don't predict decay of an excited energy state: The eigenstates of energy are valid solutions, and the solutions rotate in the complex plane indefinitely, so here it is both deterministic solutions to Maxwell Equations and Introductory QM predicting incorrectly no decay, with the important difference that Introductory QM can already calculate the energy spectrum, and hence absorption and emission spectrum, glossing over the fact that the transition is not in fact yet predicted)

3) As I said, I am willing to agree to calling Maxwell's equatins schizophrenic, but not because of their coupled equations nature, but because of its indeterministic nature (again a similarity with quantum mechanics!). If we subdivide the worldline of each charge's motion into individual infinitessimmal but continuous motion events, then we can spend a total of one Lienard Wiechert potentials (say 1.0 delayed + 0 advanced, the other way around, 0.7 delayed and 0.3 advanced, or -10.4 delayed and +11.4 advanced, etc!) on each such event individually, i.e. we can have this division that sums to 1 depend on both particle and time, as long as the total applied force in the past and future respect are compatible with the trajectory of the particle... this is quite schizophrenic indeed!

I since long suspect Maxwells Equations to contain quantum mechanical behaviour that has not been explored yet.

EDIT: There are roughly speaking 2 kinds of physicists/students: 1) those who either deny, ignore, gloss over this indeterminacy problem, or point at a pseudo-contradiction, and ignore they are holding Maxwell equations to a higher standard than Introductory QM and "just move on to QM, btw shut up and calculate!" 2) those who recognize explicitly or implicitly this schizophrenia (of the second indeterminacy kind, not the coupledness kind) and have attempted various approaches or formulations to rigorously enumerate/solve for all mathematically valid (self-consistent) histories/trajectories/solutions of Maxwell's equations. Einstein was famously trying to resolve this issue (among others) for the rest of his life, Feynmann worried about it (you can read this between the lines in his Absorber Theory), and so on. I believe many of those approaches and attempts will turn out to be different but equivalent perspectives, once the problem of solving for all self-consistent histories of ME's with given initial conditions has been solved...

I obviously belong to group 2) and given your dissatisfaction, I presume you too belong to group 2). However I believe that the issue is not the maxwell equation's themselves, but our lack of mathematics to solve for all self-consistent histories... Those who do not express dissatisfaction at the lack of complete enumeration of solutions to this insidious/schizophrenic/indeterminacy of ME, or just move on to QM belong to camp 1). There is no shame in that as long as they don't ridicule/demotivate those of us trying to bridge (semi-)classical physics with quantum mechanics... and make the analogies Dyson points out between Maxwell Equations and QM (the first and second layer, etc) more complete.

Perhaps this problem has already been resolved between Maxwell's publication of the ME's, and today. Perhaps the solution is present in the literature, but with very low uptake because most people are in camp 1). Perhaps the person who solved the issue was too modest in describing the significance of what he has concluded, and we will read about it in 70 years...

[Certhas]

jules is pointing out that, as EM + distributed matter is a non-linear set of equations, you can't easily give meaning to point masses. This is true, but the original implication, that the backreaction on matter is ill-defined or "shizophrenic" does not follow. This is a general feature of relativistic theories where you can not have rigid bodies.

[jules]

> if you calculate the poynting vector you see energy is not really leaving the system in this case

Isn't there? That wasn't what I learned. If you set the light charge in motion around the heavy charge, then space will fill with radiation that has energy. No?

> As I said, I am willing to agree to calling Maxwell's equatins schizophrenic, but not because of their coupled equations nature, but because of its indeterministic nature

Well, if you don't set a boundary condition then the solution isn't determined. So in some sense that's obviously demanding too much. Math can't tell you what a differential equation will do if you don't give it boundary conditions. The retarded/advanced potentials are Green's functions, and Green's functions depend on boundary conditions.

The problem I'm talking about is in addition to this problem. If you somehow decided that you're only using the purely retarded potential, then you could calculate the EM field given the trajectories of the charges, but that still doesn't tell you how charges move. If you decide that a charge is affected by the entire EM field including its own field then you run into infinities, and if you decide that a charge is affected only by the combined field from all other charges in the universe then you run into the issue that I mentioned above. Simple ways to address the problem fail, e.g. if you replace the charges by a sphere with smeared out charge, then the solution is no longer relativistically invariant because rigid spheres aren't.

> I since long suspect Maxwells Equations to contain quantum mechanical behaviour that has not been explored yet.

I don't know about unexplored, but if I recall correctly, Schrodinger was highly influenced by Maxwell's equations.

Basically, he noted the correspondence between ray optics (Fermat's least time principle) and classical mechanics (least action principle). Maxwell's equations are the wave version of ray optics, so he asked himself whether there is a wave version of mechanics. In other words, ray optics is to classical mechanics as Maxwell's equations are to what? So there's not just an analogy between Maxwell's equations and QM, but that's actually partially how QM came to be in the first place! Maybe you could even claim that Maxwell's equations should be classified as quantum physics not classical physics. After all, if the paradoxical nature of the double slit experiment is called quantum physics then surely Maxwell's equations are quantum physics because they correctly predict the result, as opposed to ray optics which does not. Maxwell's equations are a quantum theory of a single photon, ray optics are the classical theory of a single photon.

[aj7]

As a depressed MIT freshman I modeled my mood as a pair of coupled equations. I got growing oscillations!