I want to check invariance of the action under Lorentz boosts for classical electrodynamics. The action is $$S = \int \mbox{d}^4x F_{\alpha \beta} F^{\alpha \beta} $$ I assumed that the fields transform infinitesimally as $$ A'^{\mu} = A^{\mu} + \omega^{\mu \nu} A_{\nu}$$ where $\omega$ is antisymmetric. So I plugged it in into the primed action as $ \partial_{\alpha} A'_{\beta} - \partial_{\beta} A'_{\alpha}$ and tried to calculate it but it doesn't cancel out. I think that I should change sign in lower case indices for the transformation to be $$ A'_{\mu} = A_{\mu} - \omega_{\mu \nu} A^{\nu}.$$ Am I correct?
My question is: Is it sufficient to just plug the transformed terms into the action and see that the action doesn't change under these transformations and then we have Noether invariance? Is there any quicker way?
