I am trying to work out Non-Abelian gauge theories but I couldn't get my head around the fact that gauge fields transform with an extra inhomogeneous term under the adjoint action of a group $G$, that is

$A_{\mu} \rightarrow gA_{\mu}g^{-1} - \partial_{\mu}g g^{-1} $

where $g \in G$. As far as I understand, the adjoint action of a group on a Lie Algebra valued object is given as

$Ad(g) T = gTg^{-1}$

where $T \in Lie(G)$. So how do (can) gauge fields transform differently even though the gauge fields themselves are Lie Algebra valued ? I understand we want to keep covariant derivatives gauge covariant but that doesn't make anything clear about this behavior of gauge fields.

  • $\begingroup$ The notation you use is not familiar to me. So when you refer to a Lie-Algebra valued gauge field, I guess that means that you are contracting the non-Abelian gauge field with its associated generator, but still I then would expect the first expression to contain a term that represents some derivative of a scalar field also contracted with the generators. Guess that is the second term, but with some notation that I don't understand. Then because it is non-Abelian theory, I'm also expecting to see a third term that contains the fabc's. $\endgroup$ Commented Sep 15, 2016 at 4:24
  • 2
    $\begingroup$ physics.stackexchange.com/questions/191981/… $\endgroup$ Commented Sep 15, 2016 at 14:49
  • $\begingroup$ Possible duplicate of Do gauge bosons really transform according to the adjoint representation of the gauge group? $\endgroup$ Commented Nov 10, 2018 at 16:05


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