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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?

knzhou
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Piotr
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  • See: http://quantummechanics.ucsd.edu/ph130a/130_notes/node452.html, $$\mathcal{L} \propto \mathbf{E}^2 - \mathbf{B}^2$$ – Sean E. Lake Mar 14 '17 at 23:48
  • Thanks but that calculation gets to the exact same point I got to but then they take a derivative and get an expression for $\frac{\partial L}{\partial(\partial_\nu A_\mu)}$. That's not what I'm looking for - I'm trying to show the equivalence of the two terms I mentioned or to find where I go wrong. – Piotr Mar 15 '17 at 00:00
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    This is an exact duplicate of the question I linked above (and answered). It might be worth keeping this question around, though, because this seems like a really common confusion that's hard to google. – knzhou Mar 15 '17 at 05:40
  • Ah, brilliant, I knew it was something like that. Thanks a lot! – Piotr Mar 15 '17 at 09:37

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