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QFT index question

  1. Sep 11, 2012 #1
    1. The problem statement, all variables and given/known data
    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]

    3. 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.
     
  2. jcsd
  3. Sep 13, 2012 #2

    dextercioby

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    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

    [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...
     
  4. Sep 13, 2012 #3
    [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:
     
  5. Sep 15, 2012 #4

    dextercioby

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    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...
     
  6. Sep 15, 2012 #5
    [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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