Iam Studying "Quantization of the electromagnetic field using Quantum Field Theory" by Lahiri and Pal.
But I don't get how they computed action in equation $8.23$.
$$A=-{1\over 4} \int d^4xF_{\mu \nu}F^{\mu \nu}=-{1\over2}\int d^4 x [(\partial _\mu A_ {\nu})(\partial ^\mu A^ {\nu})-(\partial _\mu A_ {\nu})(\partial ^\nu A^ {\mu}) ]~.$$
I don't get how they evaluated $~F_{\mu \nu}F^{\mu \nu}~$ and arrived at this result , can any one please help me?
Expand $F_{\mu\nu}$ and $F^{\mu\nu}$ and multiply. Since $\mu,\nu$ are summed over, in the next step, they can be interchanged so that $$(\partial_\mu A_\nu)(\partial^\mu A^\nu)=(\partial_\nu A_\mu)(\partial^\nu A^\mu).$$ Hope this helps!
I briefly state main steps:
1) Using antisymetry of $F_{\mu\nu}$ $$ F_{\mu\nu}\partial^\mu A^\nu = -F_{\mu\nu}\partial^\nu A^\mu $$ $$A=-{1\over 4} \int d^4xF_{\mu \nu}F^{\mu \nu}=-{1\over2} \int d^4 xF_{\mu\nu}\partial ^\mu A^\nu$$
2) You need use $F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$
