# QFT index question

## 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.

Related Advanced Physics Homework Help News on Phys.org
dextercioby
Homework Helper
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...

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

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

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.