I am studying loop quantum gravity using the book by Pullin and Gambini. I am having some trouble understanding and getting past the chapter on Yang Mills theory, mainly because I am confused about some of the notation and concepts. I am hoping someone can shed some light on this for me.
First of all, the Yang Mills covariant derivative is defined and written as:
$$ D_{\mu} \equiv \partial_{\mu} - ig/2\sigma^iA^i_{\mu} $$
I understand that the superscripted indices, $i$, on $\sigma^i$ and $A^i_{\mu}$ are internal indices of the theory that run from 1 to 3. But is the $i$ in front of the coupling parameter, $g$, meant to indicate this internal index as well, or is this the imaginary unit? I am guessing the second, but the notation is a little ambiguous to me so I would like to have some clarification.
Secondly, I don't understand the relation between the commutator of the covariant derivative with itself and the field tensor. In the book the following is written:
$$ \left[D_{\mu},D_{\nu}\right] = -ig/2F^i_{\mu\nu}\sigma^i $$
Where I understand that $F^i_{\mu\nu}$ is the field tensor of the theory. I have a few questions about this. Firstly: if I write out the commutator explicitly, using the definition of the covariant derivative from above, I get:
$$\begin{align} \left[D_{\mu},D_{\nu}\right] = D_{\mu}D_{\nu} - D_{\nu}D_{\mu} &= (\partial_{\mu} - ig/2\sigma^iA^i_{\mu})(\partial_{\nu} - ig/2\sigma^iA^i_{\nu}) - (\partial_{\nu} - ig/2\sigma^iA^i_{\nu})(\partial_{\mu} - ig/2\sigma^iA^i_{\mu}) \\ &=\partial_{\mu}\partial_{\nu} - \partial_{\mu}ig/2\sigma^iA^i_{\nu} - ig/2\sigma^iA^i_{\mu}\partial_{\nu}+i^2g^2/4\sigma^iA^i_{\mu}\sigma^iA^i_{\nu} \\ &- \partial_{\nu}\partial_{\mu}+\partial_{\nu}ig/2\sigma^iA^i_{\mu}+ig/2\sigma^iA^i_{\nu}\partial_{\mu}-i^2g^2/4\sigma^iA^i_{\nu}\sigma^iA^i_{\mu} \end{align}$$
Now, I understand that the first and fifth terms on the right cancel, because the partial derivatives, $\partial_{\mu}$ and $\partial_{\nu}$, commute. But I don't understand how to manipulate the remaining terms to get to a something of the form $-ig/2F^i_{\mu\nu}\sigma^i$. Can someone please show me the full derivation for this?
Then the book goes on to say that if one indeed works out the commutator explicitly, one gets that the field tensor is given by
$$ F^i_{\mu\nu} = \partial_{\mu}A^i_{\nu}-\partial_{\nu}A^i_{\mu}+g\epsilon^{ijk}A^j_{\mu}A^k_{\nu}, $$
where, I assume, $\epsilon^{ijk}$ are the Levi-Civita symbols, right? Maybe the full derivation of the commutator will already explain this too, but how does one get from the commutator to this last equation.
Finally, and this is a purely conceptual question. Why is it that the commutator of the covariant derivatives yields the field tensor? Is this a just a definition in gauge theory?
Any help would be greatly appreciated!