Lorentz-Abraham-Dirac equation

[Milsomonk]

Good morning all,
Hope everyone is staying safe and well. I am currently trying to understand the classical mechanism for radiation reaction as described by the LAD equation, and I am a little confused by the mass renormalisation that Dirac carried out.

\begin{equation}
m\frac{d u^\mu}{ds} = e F^{\mu\nu}_{Ext} u_\nu + \frac{2}{3}e^2\left( \frac{d^2 u^\mu}{ds^2} + \frac{du^\nu}{ds}\frac{du_\nu}{ds}u^\mu \right)
\end{equation}

I understand that one needs to know the value of the electromagnetic field at the position of the electron, but at this point the current diverges, so the electron is modeled as a sphere with radius a. Then the mass in the equation takes the following form:

\begin{equation}
m=m_0+\delta m
\end{equation}
\begin{equation}
\delta m \propto \frac{e^2}{a}
\end{equation}

Clearly then dm diverges as a goes to zero, but I do not understand how this makes m finite?

Aditionally, is it true to say that the small mass term dm is a contribution from the electrons own field (self interaction)?

Any illumination on the subject would be much appreciated.

 

[vanhees71]

That's the whole crux here: You take into account the back reaction of the electron's own field to the electron, and since you (unphysically) model it as a classical point charge, this field diverges and also the electromagnetic field energy and thus its contribution to the mass of the electron diverges either.

The idea is to regularize this divergent expression for the mass and lump it into the finite total mass of the electron, which means you have to omit this divergent contribution $\propto \mathrm{d} u^{\mu}/\mathrm{d} s$ and lump it to the left-hand side, making the finite term $m \mathrm{d} u^{\mu}/\mathrm{d} s$ with $m$ the physical mass of the electron.

Note that at this point the trouble still is not over since the LAD equation has very unphysical properties (admitting self-acceleration as well as pre-acceleration violating causality). The way out of this is to use another approximation, known as the Landau-Lifhitz equation. That's why I recommend the treatment of the problem in Landau&Lifshitz vol. 2.

Another great paper is about the treatment of the classical pointcharge as a Born-rigid body of finite extent by Medina:

https://arxiv.org/abs/physics/0508031v3

 

[Milsomonk]

Many thanks for your reply, this certainly addresses my confusion. If the mass $m$ is finite and then since $\delta m$ is divergent can we then infer that $m_0$ is also divergent? if so what is the source of this divergence?

Thanks for your help, I will certainly turn my attention to the LL equation, I just wanted to get the full historical picture. Thanks also for the paper recommendation, I shall have a read.

 

[vanhees71]

The source of these divergences (both in QED, where it's milder and in classical electrodynamics) is the abuse of mathematics, but it's also unphysical. A point particle has no place in a field theory, and the LAD equation clearly shows that it is not a consistent concept.

The reason, why the point-particle concept in classical electrodynamics works quite well is that in many cases you can neglect the radiation reaction, and the best treatment of the issue I know staying just within the point-particle picture is the book by Landau and Lisfhitz though I'd like to translate this to a purely relativistic treatment without referring to the non-relativistic interpretation first. On the other hand, indeed the technique used there is a kind of non-relativistic expansion (formally in powers of $1/c$), as shown in the book. If I'd have to give a lecture in an advanced E&M course, I'd choose this approach (hopefully being able to give a fully relativistically covariant version, though Landau and Lifshitz (Vol. 2) finally get it with some arguments in Paragraph 76).

Of course the other strategy is to choose some model of an extended electron with Poincare stresses and all that and then discuss the limit to a point particle. This is done in Medina's paper, which clearly shows that the classical point-particle limit is unphysical and it makes it at least plausible why nobody of all the great physicists couldn't find a satisfactory answer.

Finally there's another approach used by the very practitioners of this physics, i.e., the accelerator physicists who have to deal with the motion of "particles" in the accelerators at high precision to design these accelerators. There also relativistic hydrodynamical or transport (BUU) models are used, treating the charged particles as a fluid or a plasma. Some time ago I've heard a talk by somebody simulating the motion of the charged particles with such continuum models and then asking which of the approximations of the point-particle approach best describe the results from the continuum approach. The amazing answer is that it's the Landau-Lishitz approximation of the LAD equation! Unfortunately I cannot find the reference to this interesting result right now.

 

[Milsomonk]

Many thanks for your contributions, I understand the issue much better now.

In fact I am currently reading about computational models for radiation reaction and the comparisons between classical, semi-classical and QED models. I've found this article, which may even be the one you mentioned, just in case you're interested. It summarises some experimental and computational results testing the validity of different models in recent years.

https://iopscience.iop.org/article/10.1088/1361-6587/ab20f6

 

[vanhees71]

This is of course also a very interesting topic of ongoing research, the QED of strong fields. Here the holy grail is to demonstrate experimentally the Schwinger pair-production process, i.e., the production of electron-positron pairs due to strong electromagnetic fields, as produced by lasers.