Rewriting Maxwell Lagrangian

Asked Dec 19 '22 at 13:50

Active Dec 19 '22 at 13:58

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I'm having some problems with rewriting the Maxwell Lagrangian. The text states, \begin{align}\mathcal{L}&=-\dfrac{1}{4}F_{\mu\nu}F^{\mu\nu}-A_\mu J^\mu \\ &= -\dfrac{1}{2}(\partial_\mu A_\nu)^2 + \dfrac{1}{2}(\partial_\mu A^\mu)^2-A_\mu J^\mu.\end{align}

I see that we are just using $F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$ and I understand how we get the first term. But, in order to get the second term we would need $$(\partial_\mu A_\nu)(\partial^\nu A^\mu) = (\partial_\mu A^\mu)^2.$$

Am I missing something here and if not why is $(\partial_\mu A_\nu)(\partial^\nu A^\mu) = (\partial_\mu A^\mu)^2$?

edited Dec 19 '22 at 13:58

Qmechanic

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asked Dec 19 '22 at 13:50

SMYK

Which text? Which page? – Qmechanic Dec 19 '22 at 14:00
3

The two formulae given for the Lagrangian density differ, but only by a total derivative, so obtain the same physics. In particular, you should find the difference is a multiple of $\partial_\mu(A^\mu\partial_\nu A^\nu-A_\nu\partial^\nu A^\mu)$. – J.G. Dec 19 '22 at 14:18
Duplicate 1 : Squaring the E&M (Maxwell) field strength tensor. – Frobenius Dec 19 '22 at 20:30

Rewriting Maxwell Lagrangian

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