Solving Confusing Indices in Operator Calculus

[countable]

Sorry if this is a stupid question but I really want to understand this.

If we have an operator that's defined in the following way:

$$D_\mu \phi^a = \partial_\mu \phi^a - iA^{r}_{\mu}{(T_r)^a}_b\phi^b$$

How would we go about working out:

$$D_\mu D_\nu \phi^a$$

What's confusing me is that the operator D acts on one component of the scalar (the gauge field) on the LHS, but acts on two components of the scalar on the RHS.

FYI: the "answer" is that $$D_\mu D_\nu\phi^a=(\partial_\mu{\delta^a}_b - iA^{r}_{\mu}{(T_r)^a}_b)(\partial_\nu{\delta^b}_c - iA^{s}_{\nu}{(T_s)^b}_c)\phi^c$$

How does the above expression come about?

 

[TriTertButoxy]

The first thing to realize is that the your differential operators has two indices:

$$D_\mu \equiv (D_\mu)^a_{\phantom{a}{b}}$$.​

Therefore, the combination $D_\mu\phi^a$ is really

$$D_\mu\phi^a\equiv (D_\mu)^a_{\phantom{a}{b}}\phi^b$$.​

With this in mind, it should be pretty clear that

$$D_\mu D_\nu \phi^a\equiv (D_\mu)^a_{\phantom{a}{b}} (D_\nu)^b_{\phantom{b}{c}}\phi^c$$.​

Does that help?

 

[countable]

thanks that was a good explanation:)