[wmnwmn]

E&M is far from "simple". It contains special relativity, for starters. Also it is incompatible with thermodynamics: solving this problem is why Planck invented quantum mechanics. Also point charges have infinite energy: this problem leads to renormalization theory. Also it introduces gauge invariance, an essential but complex part of all modern theories. And lastly, the mathematics of E&M is a big step up from Newtonian theory.

[nonbel]

>"E&M is far from "simple". It contains special relativity, for starters."

A theory (which is just a set of assumed first principles and rules of logic) can be simple but allow you to deduce vast complexity from it. In fact, it is ideal for a theory to be as simple as possible.

The "game of life" is not really a theory, but it demonstrates that simple rules can lead to surprising complexity: https://en.wikipedia.org/wiki/Conway%27s_Game_of_Life

EDIT:

I can't begin to imagine why this would trigger a downvote. Are there people out there who prefer complex theories that make things more difficult to understand than necessary? Perhaps because it makes them feel smart or something, I don't know.

But that is basically saying you dislike science, because the point of science is to synthesize information into a small set of simple "laws" that allow us to deduce accurate and precise predictions.

[weedwarrior]

you are being downvoted because of a problem with the nature of HN echo chamber being a symptom. if the voting/karma system was improved we wouldnt have problems with this kind of downvoting

[aj7]

I reflexively downvote the game of life. I have never heard of a single useful application or analogy of “finite automata.” Yes, there are speculations about extremely small Planck lengths, by no new physics has panned out. Finally I am sad that one of the greatest computer scientists and polymaths, Ed Fredkin, got sucked into this. We all have weaknesses.

[jle]

GoL wasn't used here because of any value or analogy... it's used as an example of how a simple set of rules can generate complex behavior. GoL can display complex behavior, but it would be a stretch to to say that the rules governing it are complex. If you think GoL is too simplistic, you are agreeing with the original comment.

[ckocagil]

Surely you mean cellular automata, not finite automata.

[aj7]

Yes, you are right. My ignorance shows, but I am still hanging on to my opinion of cellular automata.

[aj7]

It’s remarkable how many silent downvotes you get when you tweak something that people think is cool. C’mon guys. Refute what I said.

[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.

[tntn]

Dyson's statement was about Maxwell's theory, not all of E&M and its interactions with every other physical theory. There are no point charges in Maxwell's theory, for example.

[adrianratnapala]

I think the point is that electromagnetism really is a very complex and knotty phenomenon, and we moderns are spoiled by having

(1) Heavyside's deceptively elegant formulas for the theory (which we now call the Maxwell Equations) and,

(2) A well oiled pedagogy about how those equations the capture specific phenomena named after the likes of Ampere and Faraday, and

(3) Powerful formalisms for studying circuits, optics, transmission lines etc that can mostly be used without going back to the fundamentals.

Most of that was not available to Maxwell when he started out, so it's not surprising that he presented his theory in a way that reflected the real complexity of it all.

[weedwarrior]

special relativity is also simple, once again its the differential equations that provide a difficult explanation.