Gauge invariance of stress-energy tensor for EM field

[Trave11er]

For free EM field:
L=-$\frac{1}{4}$F_abF^ab
Then the stress-energy tensor is given by:
T^mn=-F^ml∂^vA_l+$\frac{1}{4}$g^mnF_abF^ab
The author then redefines T^mn - he adds ∂_lΩ^lmn to it,
where Ω^lmn=-Ω^mln.
The redefined tensor is:
T^mn=-F^m_lF^vl+g^mv$\frac{1}{4}$F_abF^ab
It is gauge invariant and still satisfies ∂_mT^mn=0.

The question: is why the addition is allowed? - to my uneducated mind the procedure seems like changing the energy-momentum tensor arbitrarily.

 

[dextercioby]

It comes from the fact that the whole theory is gauge-invariant, i.e. the 4-potential is unique up to a sp-time derivative of an arbitrary function epsilon. It follows that the energy-momentum is also arbitrary so we can add arbitray functions to it to have it the way we want it, conserved and gauge-invariant. The de n A l in the original formulation spoils gauge invariance.

 

[Trave11er]

But in case with four-potential gauge invariance leaves the fields intact as well as the EM field tensor, whereas the components of stress-energy tensor have physical meaning as energy density, flux etc.

 

[Avodyne]

The original stress-energy tensor is not symmetric; you can fix the arbitrariness in it by demanding that it be symmetric, and this also renders it gauge invariant.

 

[Trave11er]

So this is all due to initial arbitrariness in Lagrangian?

 

[Avodyne]

No, it's due to arbitrariness in definitions of local fluxes of energy and momentum.

 

[Trave11er]

Pardon, but I don't understand. Aren't the fluxes of energy and momentum physical things. I mean, if you have a value for 1 kg*m/s for a bullet in x-direction, then you have some value for a the stress energy tensor of a field in some small volume - how can you redefine that?

 

[Bill_K]

Yes. And No.

In naive field theory, no. It is always possible to add a four-divergence to the Lagrangian density without changing the physical content of the theory. Thus the Lagrangian is not unique, and neither is the energy-momentum tensor. But its integral, the total energy and momentum of the field, is a well-defined and unique quantity. (Note that this is unrelated to electromagnetic gauge invariance, as the same argument applies to other fields as well.)

When you go on to look at the angular momentum density, you find it will not even be conserved unless the energy-momentum tensor is symmetric, which the canonical form is not. So you must, by hand, symmetrize it.

In general relativity, yes. The energy-momentum tensor is defined as the source of the gravitational field: T^μν ≡ 2δL/δg_μν, and is unique, and automatically symmetric. The moral is: use this definition and get the correct answer at once, even if you're not doing general relativity.

 

[Trave11er]

Thank you a lot Bill_K. This is an greatly insightful answer. I appreciate your help very much.