# What is the electrodynamic action and its energy-momentum tensor?

1. Aug 25, 2011

### juanrga

I have studied Jackson, Landau, and Barut textbooks on electrodynamics, together with Weinberg's Gravitation and Cosmology textbook, and I find that the usual action

$S = S_f + S_m + S_{mf}$

is inconsistent and not well-defined. For instance, what is the meaning of $S_f$? A free-field term? Or an interacting-field term that diverges?

Moreover, the derivations of the energy-momentum tensors from the above action seem ad-hoc. For instance, Landau & Lifgarbagez just claim that during the derivation one must assume that the particles are non-interacting! That is, one must ignore the $S_{mf}$ term.

All of this mess is confirmed by papers as that by Feynman & Wheeler [1] and by Chubykalo & Smirnov-Rueda [2] where alternative actions are proposed to correct the deficiencies. However, I find still difficulties with those actions and no systematic procedure to get the corresponding energy-momentum tensors.

Does exist some well-defined and consistent action for electrodynamics leading to a well-defined and physically correct energy-momentum tensor?

[1] Rev. Mod. Phys. 1949: 21(3), 425.
[2] Phys. Rev. E 1996: 53(5), 5373. [Erratum] Phys. Rev. E 1997: 55(3), 3793.

Last edited: Aug 25, 2011
2. Aug 25, 2011

### bobloblaw

I'm not sure if I completely understand your confusion (or your notation) but the EM (no matter) action is $S = \int d^4x\sqrt{|g|}\mathcal{L}=\int d^4x\sqrt{|g|}F_{\mu\nu}F^{\mu\nu}=\int d^4x\sqrt{|g|}g^{\alpha\mu}g^{\beta\nu}F_{\mu\nu}F_{\alpha\beta}$ This is the correct action because you can show that by varying it with respect to the EM vector potential you get the free EM field equations.
To get the stress-energy tensor you just vary the action with respect to the metric tensor. This is the general procedure for finding the stress-energy tensor.

3. Aug 26, 2011

### juanrga

Thanks. I am using the notation in Landau & Lifgarbagez textbook on electrodynamics (see also [2]).

It is easy to obtain the stress-energy tensor for a free field from the action $S_f$ for a free field. The problems start when considering the whole action $S = S_f + S_m + S_{mf}$ on my first message.

No strange that several authors [1,2] substitute the classical electrodynamics action $S = S_f + S_m + S_{mf}$ by other actions. For example, the field term that you wrote does not exist in the Feynman & Wheeler action [1].

However, as said above, I find still difficulties with those actions and no systematic procedure to get the corresponding energy-momentum tensors.

For instance, the variations of the actions in [1,2] with respect to the metric tensor do not give stress-energy tensors compatible with the equations of motion.

My question remains open!

Last edited: Aug 26, 2011