Covariant deriv of matrix valued field(srednicki)

[LAHLH]

Hi

In ch84, Srednicki is considering the gauge group SU(N) with a real scalar field $\Phi^a$ in the adjoint rep. He then says it will prove more convienient to work with the matrix valued field $\Phi=\Phi^a T^a$ and says the covariant derivative of this is $D_{\mu}\Phi=\partial_{\mu}\Phi-igA^a_{\mu}\left[T^a,\Phi\right]$

Why is this covariant derivative not just $D_{\mu}\Phi=\partial_{\mu}\Phi-igA^a_{\mu}T^a\Phi$ ?

I understand $\Phi$ is a matrix and it does not commute with the generators, but I don't understand how this commutator is arising here in the second term of the covariant derivative? is it something to do with the adjoint rep?

 

[azntoon]

The covariant derivative as written by Srednicki indeed contains the commutator because it is the adjoint representation of the SU(N) group. The adjoint representation has a matrix structure, and for a given generator T^a, the operation [T^a, \Phi] is the matrix multiplication of the generator with the scalar field, which gives a matrix result. Thus, the second term of the covariant derivative contains this matrix multiplication.