# The problem of self-force on point charges

Allow me to preface this by stating that I am a high school student interested in physics and self-studying using a variety of resources, both on- and off-line, primarily GSU's HyperPhysics website, Halliday & Resnick's Fundamentals of Physics, Taylor's Classical Mechanics, and ultimately the Feynman lectures (mirrored by Caltech). Hopefully this gives somewhat of a feel of my level of physics understanding so as to avoid any answers that fly far above my head.

As I've understood from previous reading of electromagnetism (for example, in Halliday), a point charge is not affected by its own electromagnetic field. Unfortunately, as I recently read in the Feynman lecture on electromagnetism, this appears to not be so:

For those purists who know more (the professors who happen to be reading this), we should add that when we say that (28.3) is a complete expression of the knowledge of electrodynamics, we are not being entirely accurate. There was a problem that was not quite solved at the end of the 19th century. When we try to calculate the field from all the charges including the charge itself that we want the field to act on, we get into trouble trying to find the distance, for example, of a charge from itself, and dividing something by that distance, which is zero. The problem of how to handle the part of this field which is generated by the very charge on which we want the field to act is not yet solved today. So we leave it there; we do not have a complete solution to that puzzle yet, and so we shall avoid the puzzle for as long as we can.

At first I figured I must've misunderstood, but upon rereading, it's clear Feynman states the that electromagnetic field due to a point charge does, in fact, influence said charge; I inferred this "self-force" must be somewhat negligible for Halliday to assert otherwise. What stood out to me was that Feynman states this problem had not yet been solved.

I suppose my first major question is simply, has this problem been solved yet? After a bit of research I came across the Abraham-Lorentz force which appears to refer exactly to this "problem of self-force". As the article states the formula is entirely in the domain of classical physics and a quick Google search indicates it was derived by Abraham and Lorentz in 1903-4, why is it that Feynman state the problem was still unsolved in 1963? Has it been solved in the classical case but not in QED?

Lastly, despite the Wikipedia article somewhat addressing the topic, is this problem of self-force present with other forces (e.g. gravity)? I believe it does state that standard renormalization methods fail in the case of GR and thus the problem is still present classically, though it does mention that non-classical theories of gravity purportedly solve the issue. Why is there not a similar Abraham-Lorentz-esque force possible in GR -- is there an underlying fundamental reason? Due to the relative weakness of gravity, can these self-force effects be ignored safely in practice?

I apologize for the long post size and appreciate any help I can receive. I only hope my post isn't too broad or vague!

-
If your question is specifically "What is the current state of knowledge about the self-force of the electron?" then I think that's a great question, though I'd be surprised if it hasn't been asked yet. –  user35033 Feb 15 at 7:25
thank you -- that captures the intent of my first question more succinctly. I don't know if it quite encompasses the second, however. –  oldrinb Feb 15 at 7:31
If you want good answers it's generally good to ask questions one at a time. But hopefully someone wiser than I can strike these two birds down. –  user35033 Feb 15 at 7:34
that was indeed one of my fears. If need be I can break this post into two, more specific ones. –  oldrinb Feb 15 at 7:36
If this question does actually address elementary particles and not point charges you should change the subject. As for point charges the answer is clear: Point charges are just an abstract mathematical model where the question does not make sense. –  Tobias Feb 15 at 8:42

I suppose my first major question is simply, has this problem been solved yet? After a bit of research I came across the Abraham-Lorentz force which appears to refer exactly to this "problem of self-force". As the article states the formula is entirely in the domain of classical physics and a quick Google search indicates it was derived by Abraham and Lorentz in 1903-4, why is it that Feynman state the problem was still unsolved in 1963? Has it been solved in the classical case but not in QED?

This is still only a theoretical problem, as a measurement of the expected self-force needs to be very sensitive and was never accomplished. Theoretically, self-force can be said to be described satisfactorily (and even there, only approximately) only for rigid charged spheres. For point particles, the common notion of self-force (Lorentz-Abraham-Dirac) is basically inconsistent (with basic laws of mechanics) and can be regarded as unnecessary - for point particles there exist consistent theories like Frenkel's theory or Feynman-Wheeler theory (with or without the absorber condition) and their variations without self-force (there are other works free of self-force too).

J. Frenkel, Zur Elektrodynamik punktfoermiger Elektronen, Zeits. f. Phys., 32, (1925), p. 518-534. http://dx.doi.org/10.1007/BF01331692

J. A. Wheeler, R. P. Feynman, Classical Electrodynamics in Terms of Direct Interparticle Interaction, Rev. Mod. Phys., 21, 3, (1949), p. 425-433. http://dx.doi.org/10.1103/RevModPhys.21.425

-

I'm not sure if this problem was ever solved in classical electrodynamics.

However, it is (somewhat) solved in quantum field theory electrodynamics (QED). In QED, self-interaction has noticeable effects on quantities such as the observed mass of a particle. Furthermore, the self-interaction effects create infinities in the theoretical predictions for such quantities (which is why I said "somewhat" above). But, these infinities can be cancelled out for any observable (such as energy or mass, etc...) This process of cancelling out the infinities is known as re-normalization.

To get a sense of how it works, imagine that your theory predicted the energy of a particle to be something like $$E_{theoretical} = \lim_{\lambda\to \infty} (\log\lambda + E_{finite})$$ where $\lambda$ represents the part of our calculation that becomes infinite. For example, if an integral diverges we can take set the upper bound of the integral to be a variable (such as $\lambda$) and then at the end take the limit as $\lambda$ goes to infinity. Methods such as these are called "regularization" (i.e. a way of rewriting the equation such that the divergent part of the calculation is contained within a single term).

Now in this limit, the total energy will be infinite. However, in the lab we can only measure changes in energy (that is, we need a reference point). So, let us then choose a reference point such that $E_{0,finite}=0$. In that case, we subtract the reference point from the theoretical energy to get $$\Delta E_{observed} = \lim_{\lambda\to\infty} (\log{\lambda} + E_{finite} - \log{\lambda} - 0) = E_{finite}$$ and all is well. This last step is called re-normalization.

-
I understand your heart is in the right place but that example does not convince me. –  user35033 Feb 15 at 17:40
@user35033 My heart has nothing to do with it. That is quite literally, what we do in QFT to make predictions. The predictions we make are extraordinarily accurate and well-tested. So, even though it certainly appears hand-wavy (and it somewhat is), it seems to work well. –  mcFreid Feb 15 at 17:51
Perhaps you could elaborate on what $\lambda$ stands for here. Your answer, as it stands, currently says no more than "we regularize the energy." –  user35033 Feb 15 at 18:01
I've made an edit that should elaborate on what $\lambda$ stands for. –  mcFreid Feb 15 at 18:29