What is the Lagrangian for a relativistic charge that includes the self-force?

Question

The usual Lagrangian for a relativistically moving charge, as found in most text books, doesn't take into account the self force from it radiating EM energy. So what is the Lagrangian for a relativistic charge that includes the self-force?

Answer 1

classical electrodynamics mainly deals with two kinds of proplems: a) The action of a field on a charged particle and b) the fields arising from the motion of such a field. Of course, this can only be approximative but it turns out that a lot of phenomena can be described in this way.

However, you are right, an entire treatment would include a) and b) simultaniously - including the whole dynamics of such a system with radiative reaction (or, the Abraham-Lorentz-Force and its relativistic counterpart).

But as Qmechanic pointed out in a comment, there may be no fully consistent way to do so within the framework of classical electrodynamics. Jackson (Chapter 16) states:

The difficulties presented by this problem touch one of the most fundamental aspects of physics, the nature of an elementary particle. Although partial solutions, workable within limiting areas, can be given, the basic problem remains unsolved.

Thus, you will have to search for a really satisfactory answer within the description of Quantum Electrodynamics. Otherwise, you may include the effect phenomenologically. This was done e.g. by Barone and Mendes in Lagrangian description of the radiation damping (PRA, 2007) and they give a Lagrangian of the form

$$\mathcal{L}=\frac{m}{2}\dot{\mathbf{r}}_1\cdot\dot{\mathbf{r}}_2 - \frac{\gamma}{2}\epsilon\left(\dot{\mathbf{r}}_1,\ddot{\mathbf{r}}_2\right) -V\left(\mathbf{r}_1,\mathbf{r}_2\right),$$

with $\gamma := 2e^2/3c^3$, with $\epsilon = \epsilon_{ij}dx^idx^j$ beeing the Levi-Cevita tensor, and the $\mathbf{r}_i$ arise from the special treatment of the problem used in the paper employing some kind of image phase-space representation of the system. Furthermore, $V\,$ is the potential related to the Abraham–Lorentz–Dirac-Force which is given explicitely in the paper.

Answer 2

It is the usual CED Lagrangian for a charge and electromagnetic field; it includes the free field Lagrangian, a free particle Lagrangian and the interaction Lagrangian $jA$. Self-action is encoded in it when you consider the system as a whole, as coupled unknown variables. Of course, it yields rubbish: an infinite "electromagnetic mass" that you need to discard.

I say "rubbish" because, on one hand, we define the charge as a stable, fixed entity; on the other hand we introduce a self-force to get some dynamics due to it. Our definition contradicts our intentions.

EDIT: The infinite "electromagnetic mass" in the charge equation takes simply into account the self-induction effect that prevents the charge from accelerating ;-)

Answer 3

Solving both, Lorentz force and Maxwell's equation, does include radiation reaction. Radiation reaction emerges from the interaction of the charge with the field it has emitted itself, the self-field. If you do not want to solve Maxwell's equations and treat the fields as external, you can use Lorentz-Abraham-Dirac equation with some known inconsistencies for unphysical initial conditions, or the Landau-Lifshitz equation as approximation. Some authors claim that Landau-Lifshitz equation is not an approximation but agrees with solutions Lorentz-Abraham-Dirac equation for physical initial conditions. Lorentz-Abraham-Dirac and Landau-Lifshitz equations cannot be derived from Hamilton's principle because radiation emission is a dissipative process. For a pedagogical review see http://arxiv.org/abs/gr-qc/9912045.

Answer 4

I would say that it is already in the Lagrangian, problem being: in the classical ED lagrangian you describe point-like particles, and the self interaction is classically described as the classical lorentz force a charge creates on itself if it is a sphere-like particle, considering retarded potentials. A classical solution would be to include the shape in the lagrangian, but the lagrangian is local, so it deals with points. If someone talks to you about QED, it is because classically you reach the conclusion that the reaction creates a mass like term opposing acceleration, and this mass can become infinite, in QED you also have some infinites, but you can remove them. Nonetheless I would say you remove it because what actually interests QED is the measure you make, and you never really reach the size of a point while measuring, but it might be an over interpretation.

BUT I have a good news for you there exists a classical formulation that includes the retroaction, to the expense of retrocausality, or waves flowing backward in time, or a kind of action at a distance: it is the lagrangian you can find in the Feynman-Wheeler absorber theory. very interesting but Feynman abandoned it.

Answer 5

If the charge is point-like then its classical self-energy is infinite. As the self-force comes from the gradient of potentials, the result is undetermined.

Consider a theory with a local interaction. Then we must take into account some field, e.g. the electromagnetic field. Now for a massive relativistic particle moving at constant velocity there is always an inertial frame of reference where the particle is not moving. Therefore the magnetic field vanishes and the electric field is that given by Coulomb's law, i.e. $$\mathbf E(\mathbf r)=\frac q{4\pi\epsilon}\frac{\mathbf r}{\Vert \mathbf r\Vert^3}.$$ Since the particle stays at $\mathbf r=0$, the force is undefined.

So far the only massless particle known is the photon, which is neutral.