I've started working my way through David Tong's excellent lecture notes in an effort to teach myself QFT and in his first lecture on classical fields he observes that the Lagrangian density for Maxwell's equations in a vacuum:
$$\mathcal{L}=-\frac{1}{2}({\partial_\mu}{A_\nu})({\partial^\nu}{A^\mu})+\frac{1}{2}({\partial_\mu}{A^\mu})^2$$
can also "up to an integration by parts" be written as:
$$\mathcal{L}=-\frac{1}{4}({\partial_\mu}{A_\nu}-{\partial_\nu}{A_\mu})({\partial^\mu}{A^\nu}-{\partial^\nu}{A^\mu})$$
I'm failing to manage to show this. Working in reverse, it's clear that:
$$-\frac{1}{4}({\partial_\mu}{A_\nu}-{\partial_\nu}{A_\mu})({\partial^\mu}{A^\nu}-{\partial^\nu}{A^\mu})\\ =-\frac{1}{2}({\partial_\mu}{A_\nu})({\partial^\nu}{A^\mu})+\frac{1}{2}({\partial_\mu}{A_\nu})({\partial^\nu}{A^\mu})$$
but I can't resolve the second term. I clearly need to brush up my tensor calculus skills - can someone show how to derive this?
