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Commutator between covariant derivative, field strength

  1. Jul 29, 2013 #1
    Hello,
    i try to prove that
    μFμ[itex]\nu[/itex] + ig[Aμ, Fμ[itex]\nu[/itex]] = [Dμ,Fμ[itex]\nu[/itex]]
    with the Dμ = ∂μ + igAμ

    but i have a problem with the term Fμ[itex]\nu[/itex]μ ...
    i try to demonstrate that is nil, but i don't know if it's right...

    Fμ[itex]\nu[/itex]μ [itex]\Psi[/itex] = [itex]\int[/itex] (∂[itex]\nu[/itex]Fμ[itex]\nu[/itex]) (∂μ[itex]\Psi[/itex]) + [itex]\int[/itex] Fμ[itex]\nu[/itex]μ[itex]\nu[/itex] [itex]\Psi[/itex] = (∂[itex]\nu[/itex]Fμ[itex]\nu[/itex]) [[itex]\Psi[/itex] ] - [itex]\int[/itex][itex]\Psi[/itex]∂μ[itex]\nu[/itex]Fμ[itex]\nu[/itex] = 0

    with [itex]\Psi[/itex] a smooth function, nil at infinity

    if it's wrong please do you post the right answers? and why it is wrong...
    thank you
     
  2. jcsd
  3. Jul 30, 2013 #2

    vanhees71

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    2016 Award

    This is right. Of course, if you need it mathematically accurate, you have to think about, how fast you test function has to go to 0 at infinity, but if it's a physics question, what you did should be sufficient.
     
  4. Jul 30, 2013 #3
    Thank you (yes it's physics question)
     
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