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just a quick question on $F_{\mu\nu}^a$. I'm correct to think $F_{\mu}^{\mu,a}$ vanishes, aren't I? (Just want to make sure...) My reasoning is as follows:

The derivative terms cancel anyways - that's obvious - so the only "critical" term of $F_{\mu}^{\mu,a}$ is $f^{abc}A_{\mu}^b A^{\mu,c}$ but this vanishes because the combination of A's is symmetric but the $f$ totally antisymmetric. Am I right?

A friendly helper
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1 Answers1

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Yes, $$\sum_{\mu} F_{\mu}{}^{\mu}~:=~\sum_{\mu,\nu}F_{\mu\nu} g^{\nu\mu}~=~0$$ vanishes because it is a trace of a product of a symmetric and an antisymmetric tensor. It is irrelevant for the argument that $F_{\mu\nu}$ is Lie algebra valued.

Qmechanic
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