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How does a gauge covariant derivative in a non-abelian field theory act on various quantities which are not valued in the algebra, and why? In particular, how does it act on a scalar valued function $f(x)$ and a matrix-valued function $\chi(x)$ which is not valued in the [representation of the] algebra of the gauge symmetry group, and why?

My understanding is that the gauge covariant derivative acts as follows on $\psi$, which is in the fundamental representation and would be written as a column matrix function, and $\phi$, which is in the adjoint representation and would be written as an nxn matrix function (let "$Id$" be the identity matrix, and absorb the coupling constant):

1) $D_{\mu} \psi = \partial_{\mu} \psi + i A_{\mu} \psi$, where $A_{\mu}$ is hermitian matrix-valued.

2) $D_{\mu} \chi = \partial \phi + i [A_{\mu}, \phi]$

So then for a matrix-valued function $\chi$ which is not valued in the matrix representation of the algebra, and a scalar-valued function $f(x)$, I expect:

3) $D_{\mu} \chi = \partial_{\mu} \chi + i A_{\mu} \chi$, using a matrix product in the second term, and

4) $D_{\mu} f = \text{Id} \hspace{.1cm} \partial_{\mu} f$. However, I am wondering if it is instead $D_{\mu} f = \partial_{\mu} (f \hspace{.1cm} \text{Id}) + i A_{\mu} (f \hspace{.1cm} \text{Id})$.

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Covariant derivatives only act on sections which take values in some representation of the Lie algebra.

Scalar functions are a special case; they take values in the trivial representation.

It doesn't make sense to ask for the covariant derivative to act on matrix valued functions if the functions aren't valued in some matrix representation of the Lie algebra.

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