Why doesn't an electron influence itself in Maxwells theory?

[DerHutmacher]

Hello everyone, for quite some time I am struggling with the following question: If we consider the action for a single particle in Classical Electrodynamics
$$S[x(\tau),A(x)]=\int - m\ ds - \int d^4x\ A_{\mu}(x)j^{\mu}(x) -\frac{1}{4}\int d^4x F^{\mu\nu}(x)F_{\mu\nu}(x) $$ with $$ds= \sqrt{\eta_{\mu\nu}\dot{x}^{\mu}(\tau) \ \dot{x}^{\nu}(\tau) }d\tau$$ and $$j^{\mu}(x)=e \int d\tau \ \dot{x}^{\mu}(\tau)\delta^{(4)}(x-x(\tau))$$ ,where $$x(\tau)$$ is the parametrization of the world line, $$\eta_{\mu\nu}$$ the lorentz metric and $$F_{\mu\nu} = \partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$$ the field strength tensor. Now by variation of this, we get the usual equations of motion $$\partial_{\mu}F^{\mu\nu}=j^{\mu}$$ and $$-m\frac{d}{d\tau}\frac{\dot{x}_{\mu}}{\sqrt{\dot{x}^2}}=eF_{\mu\nu}\dot{x}^{\nu}$$. My confusion now comes from the appearence of the same Field strength tensor $$F_{\mu\nu}$$ in the equations of motion for the electron as well as being the object that is determined only by the electrons world line. Therefore I am asking, if in principle the electrons field is also having an influence on itself in classical electrodynamics, since only according to the given action it should. As given by Feynman (http://www.feynmanlectures.caltech.edu/II_26.html) the field of an uniformly moving electron diverges at its position, if we consider it to be a point charge. I would like to know if there is some kind of classical treatment for this issue and maybe someone can lift my confusion. If there is some explanation from the modern point of view I would also be glad to hear it.

Thank you
Lev

 

[Cryo]

Jackson (Classical Electrodynamics, Chapter 16) has a fairly long section about this. I think he ends up saying that, bar few special cases, when electron motion becomes rapid enough for it own electromagnetic radiation to affect its motion, the equations of motion start predicting unstable behaviour. Wasn't it Feynman who was trying to fix this problem (theoretically).

 

[Cryo]

Also, there is not necessarily a problem with the electromagnetic field of an electron diverging. One way to fix it is obviously allow for finite size of the electron, and then take that size to zero at the very end of the calculation -maybe it won't affect he result. Another one is to allow generalized functions into the expressions, e.g. for the point charge ($q$) at the origin the charge density is

$q\delta\left(\mathbf{r}\right)$

so electric field is

$\boldsymbol{\nabla}.\mathbf{E}=\frac{q}{\epsilon_0}\delta\left(\mathbf{r}\right)$

$\mathbf{E}=\frac{q\mathbf{r}}{4\pi\epsilon_0 r^3}$

Which works, in the generalized sense, even at the origin. And if you would ask for the derivative of electric field (along some vector $\mathbf{a}$), you would get:

$\left(\mathbf{a}.\boldsymbol{\nabla}\right)\mathbf{E}=\frac{q}{4\pi\epsilon_0 r^3}\left(\mathbf{a} - 3\frac{\left(\mathbf{a}.\mathbf{r}\right)\mathbf{r}}{r^2}\right) + \frac{q}{3\epsilon_0}\mathbf{a}\delta\left(\mathbf{r}\right)$

I might be getting it wrong, but first check seems to indicate this remains valid even at the origin

 

[Heikki Tuuri]

I wrote about this problem in my physics blog on May 17, 2019. The "self-force" on an accelerated electron, by its own changing electromagnetic field, is an open problem in physics. The self-force in many cases is so weak that we cannot measure it experimentally.

Larmor, Abraham, Lorentz, Feynman, Rohrlich, and others have worked on this problem. A specific question is, if there is a radiative "reaction" on a charge which is accelerated linearly with a constant acceleration. Wikipedia lists that as an open problem.

I am using a model where the electromagnetic field of an electron behaves like an elastic object attached to the charge. The mass of the elastic object might contain all the rest mass of the electron. When you accelerate the electron, the elastic object will deform and oscillate in various ways. Calculating the exact force it causes to the electron would be complicated.

You cannot calculate the behavior of the system from the current position, velocity, and higher derivatives of the electron alone. To calculate the reaction, we must know the full past path of the electron, so that we can calculate the current state and oscillations of the elastic object attached to the electron.

The Larmor formula and various other formulas try to derive the radiative reaction from badly incomplete information. That is the reason why they produce paradoxes like "signals from the future" (see the Abraham-Lorentz force in Wikipedia).

An analogue is water waves. If you throw a fishing float into water, it will produce complicated wave patterns. To calculate the force on the float, you need to solve the time development of the water waves - and that is not easy. An electron is like a fishing float in the electromagnetic field which fills spacetime.