It's often said that the standard Electromagnetic Field, with no charges or currents, has rotational symmetry, which is what, via Noether's Theorem, implies the existence of EM field angular momentum and it's conservation.
How can that be proved? That is, how can it mathematically be shown that the Lagrangian of a standard EM has rotational invariance?
The Lagrangian density is proportional to $F_{ab}F^{ab} = g_{ab}g_{cd}F^{ac}F^{bd}$.
where $g$ is the minkowski metric.
Under a rotation, you'd send $F^{ab} \rightarrow \Theta^{a}{}_{m}\Theta^{b}{}_{n}F^{mn}$, where $\Theta$ is the standard 4D rotation matrix
but, from the way this is contracted above, it should be clear that this is mathematically equivalent to sending $g_{ab} \rightarrow \Theta^{m}{}_{a}\Theta^{n}{}_{b}g_{mn}$, and it is well known that the minkowski metric is invariant under rotations, so the Action is therefore invariant under rotations, too.
