QFT index question

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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.
 

Answers and Replies

  • #2
dextercioby
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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

[tex] \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} [/tex]

So you can see that:
* the free index is [itex] \nu [/itex]
* "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...
 
  • #3
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[tex]\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}[/tex]

:smile:
 
  • #4
dextercioby
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[tex]\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}[/tex]

:smile:
I missed an operator.

[tex]\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}[/tex]

Now the \lambda is not correctly parsed...
 
  • #5
196
22
[tex]\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}[/tex]

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.
 

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