[jules]

E&M is simple in the sense that the equations that govern the relationship between the charge distribution and the electromagnetic field are quite simple. And the equations governing how charges move in an electromagnetic field are quite simple. I think this is what Dyson has in mind. Understanding the implications and limitations of those equations is not simple at all.

An even more blatant difficulty with E&M, related to your third point, is that it is schizophrenic. Given the trajectories of charges it will tell you what the electromagnetic field will be, and given the electromagnetic field it will tell you how charges will move. Unfortunately, these two parts of the theory seem to be incompatible, and the theory will not tell you how fields + charges will evolve in time.

[hansen]

> Given the trajectories of charges it will tell you what the electromagnetic field will be, and given the electromagnetic field it will tell you how charges will move. Unfortunately, these two parts of the theory seem to be incompatible, and the theory will not tell you how fields + charges will evolve in time.

The Problem is not coupling charged matter fields to the EM field. You get a well defined set of coupled and (now) non-linear PDEs. The problems arise when you try to model point charges. Then the theory is plagued by infinities that originate in the infinite charge and current densities.

The infinities in QED are actually far less problematic than those in the classic theory. They just seem to be more problematic because you cant (approximately) ignore the backreaction of EM and matter fields.

[weedwarrior]

what you expounding upon is [1] of the indicators that we dont completely understand physics at the "point charge" scale, and very possible there is no such thing as a point charge, rather there is a centroid of field intensty/probability I.E. a wave function. point charges are likely an overly simplified view, and artefactual convienience of extrapolation.

[hansen]

The problems in the classical theory are easily understood. The charge and current densities of point particles are not smooth functions but distributions (think of the Dirac δ-“function”). If they act as the sources of the EM field the EM field itself becomes singular. Now if you try to solve the full Maxwell equations including the backreaction of matter & radiation fields you would have to multiply distributions which is ill defined.

There are similar problems in the quantum theory but the divergences are less severe and can be dealt with in a systematic way. Most physicist believe they will totally disappear in some more fundamental underlying theory. From a mathematicians point of view there is the hope that at least some QFTs are finite and the divergences are just an artifact of the construction & pertubation theory.

[DoctorOetker]

with schizophrenic you mean a system of coupled differential equations? in that case I wouldn't agree with schizophrenic.

or are you referring to the fact that mathematically speaking the system of differential equations is indeterministic due to mathematically perfectly valid advanced potentials? in this case I agree that there may be much more to learn from the Maxwell equations, and I have no qualms with naming this feature (highly) schizophrenic.

[jules]

I mean that the theory splits into two parts: the part that tells you the fields given the charges, and the part that tells you the charges given the fields. For instance, it doesn't tell you what two charges with given initial velocities will do. (not even if you also give an initial electromagnetic field)

[Certhas]

Well, the Lorentz force does though. And that follows from energy conservation and the Maxwell equations, e.g.:

https://physics.stackexchange.com/a/77028

So that's not really terribly deep or even true.

[jules]

So what are the equations for those two charges?

[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.

[aj7]

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