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## 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:

[tex]F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}[/tex]

so then my issue comes in with this part of the Euler-Lagrange equation:

[tex]\frac{\partial\mathcal{L}}{\partial (\partial_{\mu}\phi)}[/tex]

## 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 [itex]\mu\rightarrow\lambda[/itex] and [itex]\nu\rightarrow\gamma[/itex] 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)

[tex]\frac{\partial (\partial_{\mu}A_{\nu})}{\partial (\partial_{\lambda}A_{\gamma})}-\frac{\partial (\partial_{\nu}A_{\mu})}{\partial (\partial_{\lambda}A_{\gamma})}[/tex]

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 [itex]A_{\mu}[/itex] and [itex]A_{\nu}[/itex] as separate fields, so that I would only do [itex]\mu\rightarrow\lambda[/itex] (again, only for the euler-lagrange equation)

and get two equations with terms similar to:

[tex]\frac{\partial (\partial_{\mu}A_{\nu})}{\partial (\partial_{\lambda}A_{\mu})}-\frac{\partial (\partial_{\nu}A_{\mu})}{\partial (\partial_{\lambda}A_{\mu})}[/tex]

[tex]\frac{\partial (\partial_{\mu}A_{\nu})}{\partial (\partial_{\lambda}A_{\nu})}-\frac{\partial (\partial_{\nu}A_{\mu})}{\partial (\partial_{\lambda}A_{\nu})}[/tex]

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.