Conservation of energy in EM with point charges

[JustinLevy]

This is probably a product of how I was taught, but I am unsure of the status of conservation of energy in classical EM with point particles. Here is the background:

When I was taught in undergrad, general arguments were used to show that there needs to be a back-reaction on a point particle moving in electromagnetic fields due to coupling with the very fields it is producing (self-action), and that this can lead to difficulty including run-away solutions. Heck even the Griffith's textbook makes the following comments on these difficulties: "Perhaps they are telling us that there can be no such thing as a point charge in classical electrodynamics, or maybe they presage the onset of quantum mechanics." (pg 467, 3rd edition) We did not dwell on this in class.

However in grad school, similar comments were made about the self-action, but the professor gave us two journal papers, one seeming to show quite generally that there was a problem with the self-action of point charges in classical electrodynamics, while the other claimed that careful analysis showed there was no problem, but went into enough mathematical detail for a case that it was unclear how general the 'solution' was. (Unfortunately, I didn't think I'd ever refer to my notes again, so I threw them out. I don't remember what papers they were.) Again we didn't dwell on it, and his attitude was 'be aware of the issue', and decide for yourself based on the math.

Then, I saw electrodynamics rewritten in terms of a lagrangian and hamiltonian. With the lagrangian, since it is clearly time translation invariant, Noether's theorem should give us that there is a conserved energy. So at least in the Lagrangian formulation, it appears that any trace of 'run-away' solutions are gone ... forbidden by the very symmetries of the theory. We never discussed this in class, and I didn't realize this connection to the self-action 'problem' until a recent discussion.

However, if the argument can be settled that simply, then why is there even still discussion of this? Am I glossing over important subtle details of Noether's theorem here that lend to problems? I've tried talking to several professors about this, and they all said they've never seen the self-action 'problem' in classical electrodynamics approached from the Lagrangian point of view so they were not sure ... and likewise were very hesitant to agree since if it was truly that simple, clearly that would be the preferred argument and everyone would use it.

Help please?
In particular, if anyone knows of textbooks or definitive journal review articles on this, that would be of tremendous help.

 

[Stingray]

There are problems with boundary terms. The energy of an electromagnetic field is found by integrating $E^2+B^2$ over all space. For a general compact radiating system, the fields decay like $1/r$ near infinity. The energy integral therefore doesn't converge (globally). Things aren't any better for the linear or angular momentum. Except in special cases, you only have conservation laws in finite regions. These can be hard to use in practice due to the fluxes through boundary surfaces.

If you insist on trying to treat point charges, there are additional problems. The energy integral diverges both at infinity and at the particle's location. There is an infinite amount of binding energy. People imagine compensating for this by an infinitely negative "bare mass." Once you start allowing (infinite!) negative energies, all kinds of crazy things become possible. I don't think it's even very interesting to discuss energy conservation in these cases. I could go on, but I think that trying to discuss point particle motion in classical Maxwell EM is silly. It doesn't work.

All of that being said, there are good derivations of radiation reaction for extended objects. These all use local energy and momentum conservation as their starting points. As such, they cannot be inconsistent with it unless later approximations cause problems. IMO, the most complete derivation of this phenomenon is in http://www.iop.org/EJ/abstract/0264-9381/26/15/155015". Much of the material there is more complicated than necessary to treat the case of flat spacetime. It also doesn't discuss energy issues directly, but does use them as a starting point.

 

[Bob_for_short]

There are problems with boundary terms. The energy of an electromagnetic field is found by integrating $E^2+B^2$ over all space. For a general compact radiating system, the fields decay like $1/r$ near infinity. The energy integral therefore doesn't converge (globally). Things aren't any better for the linear or angular momentum. Except in special cases, you only have conservation laws in finite regions. These can be hard to use in practice due to the fluxes through boundary surfaces.

I never heard of such difficulties. If the charge motion is considered to be known (given functions of space-time), the radiation energy-momentum leaving the system is finite and even small with respect to the charge kinetic energy in the system. It is not hard at all to use it in practice and it is used widely, you exaggerate.

Another thing is how to take into account the finite radiated energy in the mechanical (charge) equations. P. Dirac has done such a derivation in the most strict way (in the frame of self-action ansatz). The resulting mechanical equations contain the third derivative and give runaway solutions.

 

[Dale]

My favorite page about energy conservation in EM is here: http://farside.ph.utexas.edu/teaching/em/lectures/node89.html

The derivation is all done in terms of current densities and charge densities in a finite and bounded region. So, it shouldn't be surprising that when you have an infinite charge density (i.e. a point charge) you can occasionally get nonsensical infinities in the results. I believe some small aspects remain an unresolved problem, but I personally think a bigger deal is made of it than it really merits. Particularly in light of the uncertainty principle.

 

[Stingray]

I never heard of such difficulties. If the charge motion is considered to be known (given functions of space-time), the radiation energy-momentum leaving the system is finite and even small with respect to the charge kinetic energy in the system. It is not hard at all to use it in practice and it is used widely, you exaggerate.

Another thing is how to take into account the finite radiated energy in the mechanical (charge) equations. P. Dirac has done such a derivation in the most strict way. The resulting mechanical equations contain the third derivative and give runaway solutions.

You're right that the electromagnetic power leaving a system is finite in normal cases. But you have to be careful in interpreting that when the total energy is not defined. In particular, you might imagine any finite amount of electromagnetic binding energy being converted to kinetic energy without radiation.

Dirac's work is probably about the closest one can come to meaningfully discussing point particles in this context, but it still has mathematical problems. The "caps" of his flux tube are not considered properly (although he was well-aware that his answer was a kind of "fudge"). This is exactly the place where everything blows up.

Despite this, his essential point -- that objects move as though the only part of the self-field that they feel is 1/2 (retarded-advanced) -- was validated in the paper I cited above. That applies to arbitrary extended objects in flat spacetime without approximation. The problem is then shifted to deriving suitable relations between the fields and the body's motion. This is not trivial in general. As far as I know, nobody has shown runaway behavior for matter models that obey standard energy conditions. It is probably possible to prove directly that it cannot occur, although I have not seen this done (nor tried to do so).

 

[granpa]

I just realized that a moving point charge will have an infinite amount of energy in its magnetic field so it should require in infinite amount of energy to change its velocity by a finite amount.

 

[heldervelez]

point particle : idealized and lack of spatial extension.

I do not remember of any physical experiment ( local physical laws, here and now) that shows evidence of any infinity.
When we integrate some quantity to the infinite is, again, an aproximation because the observed universe is finite in extension and in time.

Point particle is an convenient abstraction in EM or Newton gravity to look into a system from far apart.
If we try to look 'closer' we must enclose it by a 'volume' and proceed.

In Particle Physics the 'point particle' is applied to non divisible entities (electron,..) and are found without spatial extension.

IMO, if we consider that all 'particles' are just some sort of 'field' then if we try to look closer we will never find 'spatial extension' in the tradicional way 'as an object', but we can mention some waveLength.
This way all 'objects' in the universe have spatial extension and we can evade from all kind of infinities.
(the measure of the upper limits of the electron radius are closing to 0 as our skills to measure it increase, afaik).