How do gauge fields transform with an extra inhomogeneous term even though they are Lie Algebra valued in Non-Abelian gauge theories?
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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 GG, that is
Aμ→gAμg−1−∂μgg−1Aμ→gAμg−1−∂μgg−1
where g∈Gg∈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−1Ad(g)T=gTg−1
where T∈Lie(G)T∈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.
I hope you like it if yes then make me as brainlist plz plz plz
Aμ→gAμg−1−∂μgg−1Aμ→gAμg−1−∂μgg−1
where g∈Gg∈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−1Ad(g)T=gTg−1
where T∈Lie(G)T∈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.
I hope you like it if yes then make me as brainlist plz plz plz
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