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I think that the problematic part here is the notion of what demands what. For example, you state This is the standard argument by which Lorentz invariance is found to demand gauge invariance for massless particles. but I am not completely sure if I agree with this, or at least to the interpretation you are carrying with it. Taking a look at ...


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Commutators are Lie brackets, in this case on the algebra of differential operators. Asking whether $[\dot{},\dot{}]$ is a Lie bracket doesn't really make sense (since, for matrix groups, the Lie bracket is the matrix commutator, anyway). The field strength is not "just a definition", it is the natural curvature associated to the principal connection that ...


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I found this question while struggling with the same issue. The solution turns out to be quite simple. This works for any general gauge group $G$, with elements $g(x),\ g^{-1}(x),$ and $e$. $$ 0 = \partial_\mu(e) = \partial_\mu (gg^{-1}) = (\partial_\mu g) g^{-1} + g \partial_\mu g^{-1} $$ so $$ \partial_\mu g = - g (\partial_\mu g^{-1}) g $$ In your case ...



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