# Is this EM field *due* to charged particle or external?

Suppose we're given the action

$S=-mc\int ds + \frac{q}{c}\int A_{\mu}(x) dx^{\mu}-\frac{1}{4c}\int d^{D}xF_{\mu \nu}F^{\mu \nu}$

The first two integrals are over the particle's worldline while the last is over spacetime. So I'm able to successfully vary the action with respect to the gauge potential to achieve Maxwell's equations but I'm confused as to what the gauge potential is in this case.

If the last term weren't there, this action would describe the physics of a charged particle in some external field $$A_{\mu}(x)$$. However, when we vary the action to yield Maxwell equation, we conclude that the current $$j^{\mu}(x)$$ *of the particle itself* is the source of the EM field. This is very confusing to me.

In the above action is the field the field caused by the particles motion or is it an external field? Or am I asking a silly question?

ShayanJ
Gold Member
The second term in the action describes the interaction between the external potential and the particle, and interaction is a two way road.

Thanks a lot, but I'm still confused. Does that mean that when I write $$F_{\mu \nu}$$ in terms of the gauge potential in the third term, I need a totally different potential than the one that shows up in the second term?

ShayanJ
Gold Member
Thanks a lot, but I'm still confused. Does that mean that when I write $$F_{\mu \nu}$$ in terms of the gauge potential in the third term, I need a totally different potential than the one that shows up in the second term?
No. Its just that there is a particle in an external field which we don't care how it is created. Then the evolution of the field is not only the free evolution, but there is an effect coming from the charged particle.
The equations of motion for both the particle and field can be achieved by varying the above action. That's it.

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vanhees71
Gold Member
The trouble, however, is that you run into serious problems. In principle, what you've written down is a closed equation for a single charged particle and the electromagnetic field. This includes the mutual interaction between the particle and the field, i.e., the motion of the particle in its own field. One solution, and that's the physical one, is the motion of the particle with a cosntant velocity and the corresponding Lorentz boosted Coulomb field. This is unproblematic.

The problems start when you include both, an external field, created from other charge-current densities than the particle and the particle's own field. The particle gets accelerated and this means its own field includes electromagnetic waves that carry energy away from the particle's kinetic energy. So the particle's motion is damped and this reacts of course back to the field. A fully self-consistent solution seems not to exist, because when naively writing down the equations you get pretty strange effects like the self-acceleration of a particle (which occurs as an obviously unphysical solution even if no external field is present) and singularities due to the point-like singularity in the field of the particle. Part of this is the appearance of an infinite contribution to the particle's mass due to it's own Coulomb field, which has infinite energy for a point-like particle. This infinity can partly removed by mass renormalization in this classical context, but there are still divergences left, which cannot be renormalized.

The only way out is to assume a finite extension of the particle and the assumption of forces keeping it together (the socalled Poincare stresses). There's no fully consistent theory of a classical point particle interacting with its own (radiation) field. This problem is partially solved by quantum electrodynamics, where all the singularities and divergences can be solved by renormalization of a few parameters (wave-function renormalization for the electron-positron and the photon fields, electron mass, and coupling constant), but that's only a solution in perturbation theory, i.e., there's no full solution of QED either. It's one of the big unsolved puzzles of modern physics!

ShayanJ
So the particle's motion is damped and this reacts of course back to the field. A fully self-consistent solution seems not to exist, because when naively writing down the equations you get pretty strange effects like the self-acceleration of a particle (which occurs as an obviously unphysical solution even if no external field is present) !

I believe a recent book came out which successfully addressed this problem, can't remember the name.
Also, there is the other issue of the equation themselves not making sense as you have a delta function in a set of non-linear DEs.

WannabeNewton