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I have the following homework problem in theoretical electrodynamics:

Show that the gauge invariant Lagrange density $\epsilon_{\mu\nu\alpha\beta} F^{\mu\nu} F^{\alpha\beta}$ can be written as a total divergence of a four-vector.

This total divergence should be something like $\partial_\mu G^\mu$, right?

With $$ \hat F^{\mu\nu} = \frac 12 \epsilon^{\mu\nu\alpha\beta} F_{\alpha\beta} $$ I figured that I write this Lagrange density (I'll refer to it as $L_2$) could be written as: $$ L_2 = 2 \hat F^{\mu\nu} F_{\mu\nu} $$

I simplified this by using the matrix representations of each $F$ and got it down to: $$ L_2 = -\frac 4c B_i E^i $$

$B$ and $E$ can be expressed in Terms of $A$ like so: $$ B_i = \epsilon_{ijk} \partial_j A^j ,\quad E_i = -\partial_i A^0 -\partial_0 A^i $$

How would I continue to find that four-vector $G$?

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If you write $$F_{\mu\nu} = \partial_{[\mu}A_{\nu ]} $$ Then your Lagrange density is $${ \mathcal{L}} = \epsilon^{\mu\nu\alpha\beta}\partial_{[\mu}A_{\nu ]}\partial_{[\alpha}A_{\beta ]}$$

Now we want a vector whose divergence is ${ \mathcal{L}}$. The $\partial_{\mu}$ looks promising, so we ask if we can bring that outside so it acts on the remaining vector, i.e. $${ \mathcal{L}} = \partial_{\mu}(\epsilon^{\mu\nu\alpha\beta}A_{\nu }\partial_{[\alpha}A_{\beta ]}) \ \ (1)$$ We don't need to worry that we've lost the antisymmetrization brackets on $\mu$ and $\nu$ because the epsilon symbol forces this.

As you pointed out, this isn't quite what we started with because we have an additional term of the form $$(\epsilon^{\mu\nu\alpha\beta}A_{\nu }\partial_{\mu}\partial_{[\alpha}A_{\beta ]}) $$ However the total antisymmetry of the epsilon symbol means we can treat the $\mu$ $\alpha$ $\beta$ contribution as $$\partial_{[\mu}F_{\alpha\beta]} $$ which vanishes due to the Maxwell equations. Hence the anzatz (1) holds. (The vector whose divergence we take looks like the abelian Chern Simons current).

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  • $\begingroup$ You mean $\partial_\mu G^\mu = \partial_{[\mu} \epsilon^{\mu\nu\alpha\beta} A_{\nu]} \partial_{[\alpha} A_{\beta]}$ and that is it? But then that $\partial_\mu$ acts onto the $A_\beta$ as well. Don't I need to write $\epsilon^{\mu\nu\alpha\beta} (\partial_{[\mu} A_{\nu]}) (\partial_{[\alpha}A_{\beta]})$ which then prevents this pulling out? $\endgroup$ – Martin Ueding Jan 20 '13 at 12:44
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    $\begingroup$ If you write $G^{\mu} = \epsilon^{\mu\nu\alpha\beta}A_{\nu}\partial_{[\alpha}A_{\beta]}$ then $\partial_{\mu}G^{\mu}$ is what you want. The epsilon automatically antisymmetrizes $\mu$ and $\nu$ for you. $\endgroup$ – twistor59 Jan 20 '13 at 12:52
  • $\begingroup$ In your definition of $F$, the $\partial$ only acts on $A$. But with that $G$, the $\partial_\mu$ will act on both $A_\nu$ and $A\beta$ with the product rule. Or does some (anti-)symmetry remove that part? $\endgroup$ – Martin Ueding Jan 20 '13 at 12:55
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    $\begingroup$ Right, so you will get some terms like $\partial_{[\mu}F_{\alpha]\beta}$, but the epsilon will fully antisymmetrize over $\mu$, $\alpha$ and $\beta$, and this vanishes by Maxwell equations. $\endgroup$ – twistor59 Jan 20 '13 at 13:03
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    $\begingroup$ @EmilioPisanty OK as it was a homework question I was trying to keep the answer as lean as possible, but now it's pretty much there I should maybe beef it up a bit... $\endgroup$ – twistor59 Jan 20 '13 at 17:11

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