I have a question that may be very simple and potentially for that very reason I can't find a sensible answer to it - everyone just skips over it. I have a EM Lagrangian given by:
$L -\frac{1}{4} F^{\mu \nu} F_{\mu \nu}$
And we have that
$F^{\mu \nu} = \partial_\mu A_\nu - \partial_\nu A_\mu $
And so:
$L = \frac{1}{2}(-\partial_\mu A_\nu\partial^\mu A^\nu + \partial_\mu A_\nu\partial^\nu A^\mu) $
and in the second term we can manipulate the indices using the metric so that:
$\partial_\mu A^\nu\partial_\nu A^\mu $
My lecture notes insist that:
$L = \frac{1}{2}(-\partial_\mu A_\nu\partial^\mu A^\nu + (\partial_\mu A^\mu)^2) $
And I just can't see how the corresponding second terms are equal. Surely the term I wrote contains terms like:
$\partial_0 A^i \partial_i A^0$
Whereas the term from the notes clearly doesn't have a mixed term like this. Where do I go wrong?
