QFT Index Question Homework: Solving Euler-Lagrange Equations w/ F_{mu,nu}

[spacelike]

Homework Statement

I'm learning QFT and trying to do a basic problem finding the equations of motion from the Euler-Lagrange equation given a lagrangian.

The lagrangian is in terms of:
$$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$$

so then my issue comes in with this part of the Euler-Lagrange equation:
$$\frac{\partial\mathcal{L}}{\partial (\partial_{\mu}\phi)}$$

The Attempt at a Solution

Now, I'm not sure if I am supposed to treat this as two separate fields or not. My first attempt to solve this I made a change of index from $\mu\rightarrow\lambda$ and $\nu\rightarrow\gamma$ in the Euler-Lagrange equation so that I got terms that look something similar to:
(there's more terms and factors but I'm just showing the relevant part)
$$\frac{\partial (\partial_{\mu}A_{\nu})}{\partial (\partial_{\lambda}A_{\gamma})}-\frac{\partial (\partial_{\nu}A_{\mu})}{\partial (\partial_{\lambda}A_{\gamma})}$$
This then results in delta functions which multiply the other factors in the equations and I get the final answer.

OR

Am I supposed to only change ONE index and treat $A_{\mu}$ and $A_{\nu}$ as separate fields, so that I would only do $\mu\rightarrow\lambda$ (again, only for the euler-lagrange equation)
and get two equations with terms similar to:
$$\frac{\partial (\partial_{\mu}A_{\nu})}{\partial (\partial_{\lambda}A_{\mu})}-\frac{\partial (\partial_{\nu}A_{\mu})}{\partial (\partial_{\lambda}A_{\mu})}$$
$$\frac{\partial (\partial_{\mu}A_{\nu})}{\partial (\partial_{\lambda}A_{\nu})}-\frac{\partial (\partial_{\nu}A_{\mu})}{\partial (\partial_{\lambda}A_{\nu})}$$


NOTE: I did it the first way and the answer looks reasonable to me, but I just want to make sure my technique was correct.

 

[dextercioby]

You should never have the same index in the differentiations, you need to adapt all indices so that the equations respect the correct covariance requirement

$$\left[\partial_{\mu}\left(\frac{\partial_{\sigma}A_{\lambda}-\partial_{\lambda}A_{\sigma}}{\partial\left(\partial_{\mu}A_{\nu}\right)}\right)\right] F^{\sigma\lambda}$$

So you can see that:
* the free index is $\nu$
* "fractions" corresponding to differentiations do not mix indices
* no index appears more than twice, twice iff summed over.

There's something wrong with the LaTex code...Hmmmmm...

 

[Oxvillian]

$$\left[\partial_{\mu} \left(\frac{\partial_{\sigma}A_{\lambda}-\partial_{\lambda}A_{\sigma}}{\partial \left(\partial_{\mu}A_{\nu}\right)}\right)\right] F^{\sigma\lambda}$$

 

[dextercioby]

$$\left[\partial_{\mu} \left(\frac{\partial_{\sigma}A_{\lambda}-\partial_{\lambda}A_{\sigma}}{\partial \left(\partial_{\mu}A_{\nu}\right)}\right)\right] F^{\sigma\lambda}$$

I missed an operator.

$$\left[\partial_{\mu} \left(\frac{\partial\left(\partial_{\sigma}A_{\lambda}-\partial_{\lambda}A_{\sigma}\right)}{\partial \left(\partial_{\mu}A_{\nu}\right)}\right)\right] F^{\sigma\lambda}$$

Now the \lambda is not correctly parsed...

 

[Oxvillian]

$$\left[\partial_{\mu} \left(\frac{\partial \left( \partial_{\sigma}A_{\lambda} -\partial_{\lambda}A_{\sigma}\right)}{\partial \left(\partial_{\mu}A_{\nu}\right)}\right)\right] F^{\sigma\lambda}$$

Problem seems to be that the bb software is fond of inserting spaces in inappropriate places in order to break up long space-less lines.