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I have seen distinct definitions of gauge covariant derivative (in Yang-Mills theory)

$$ D_\mu \phi = (\partial_\mu + igA_\mu) \phi $$

vs

$$ D_\mu \phi = \partial_\mu \phi + ig[A_\mu,\phi] .$$

I guess the first is the common definition, which is the same as the covariant derivative in QED. What is about the second?

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    $\begingroup$ The notation $A_\mu \phi$ is a placeholder for "whatever you get when you let $A_\mu$ act on $\phi$, given that they both live in representations of the gauge group". In the usual case it's the commutator (though it doesn't have to be) so the second notation is more explicit. $\endgroup$ – knzhou May 24 '17 at 15:03
  • $\begingroup$ The first one is incorrect in case of Yang Mills (see answer below). It is strictly for QED. $\endgroup$ – DanielC May 24 '17 at 15:09
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As AccidentalFourierTransform points out, the second expression is the non-abelian generalization of the former. The first one is only valid for the abelian case (QED), while in general Yang Mills the fields are matrices transforming in some representation of the gauge group and the correct form is the latter.

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