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Covariant derivatives commutator - field strength tensor

  1. Jun 20, 2015 #1
    1. The problem statement, all variables and given/known data
    So I've been trying to derive field strength tensor. What to do with the last 2 parts ? They obviously don't cancel (or do they?)

    2. Relevant equations


    3. The attempt at a solution
    [tex] [D_{\mu},D_{\nu}] = (\partial_{\mu} + A_{\mu})(\partial_{\nu} + A_{\nu}) - (\mu <-> \nu) = [\partial_{\mu},\partial_{\nu}] + [\partial_{\mu},A_{\nu}]+[A_{\mu},\partial_{\nu}] +[A_{\mu}, A_{\nu}] = [/tex]
    [tex] \partial_{\mu}A_{\nu} - \partial_{\nu} A_{\mu} + [A_{\mu},A_{\nu}] -A_{\nu} \partial_{\mu} + A_{\mu} \partial_{\nu} [/tex]
     
  2. jcsd
  3. Jun 20, 2015 #2

    Orodruin

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    They are cancelled by the first two terms when the derivative acts on whatever function the entire operator is acting on. What remains is just the derivative acting on the A fields.
     
  4. Jun 20, 2015 #3
    Thank you
     
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