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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:
[tex]D_\mu \phi^a = \partial_\mu \phi^a - iA^{r}_{\mu}{(T_r)^a}_b\phi^b[/tex]
How would we go about working out:
[tex]D_\mu D_\nu \phi^a[/tex]
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 [tex]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[/tex]
How does the above expression come about?
If we have an operator that's defined in the following way:
[tex]D_\mu \phi^a = \partial_\mu \phi^a - iA^{r}_{\mu}{(T_r)^a}_b\phi^b[/tex]
How would we go about working out:
[tex]D_\mu D_\nu \phi^a[/tex]
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 [tex]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[/tex]
How does the above expression come about?