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Partial and Covariant derivatives in invarint actions

  1. Nov 27, 2006 #1
    It's physics based but actually a maths question so I'm asking it here rather than the physics forums.

    [tex]I = \int \mathcal{L}\; d^{4}x[/tex]

    I is invariant under some transformation [tex]\delta_{\epsilon}[/tex] if [tex]\delta_{\epsilon}\mathcal{L} = \partial_{\mu}X^{\mu}[/tex] for some function/tensor/field thingy [tex]X^{\mu}[/tex]. This I've no problem with.

    However, is the same true for a covariant derivative? If [tex]\delta_{\epsilon}\mathcal{L} = D_{\mu}X^{\mu}[/tex] where [tex]D_{\mu}\varphi = \partial_{\mu}\varphi + g[A_{\mu},\varphi][/tex], as you get in nonabelian field theory. Is the action still invariant? Obviously the [tex]\partial_{\mu}[/tex] part of [tex]D_{\mu}[/tex] represents no problem but I don't know if the [tex]g[A_{\mu},\varphi][/tex] term vanishes or not within the integral.

    I've been doing some supersymmetry and a number of times I've got the answer the question has asked to find plus a covariant derivative of something. :cry: If I just got a mess of terms I'd know I'm way off, but the fact everything collects nicely into a covariant derivative makes me feel I'm at least on the right track.

    Thanks for any help :)
  2. jcsd
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