Formalism of Non-abelian Gauge theory?

Question

Depending on the source I have seen two different definitions/formalisms for Non-abelian Gauge theories and was wondering how the two were related.

The first one is the more common where the gauge field $A_{\mu}$ is promoted to a matrix, which transforms like: $$A_{a} \rightarrow UA_{a}U^{-1} +\frac{i}{g}U\partial_{a}(U^{-1}) $$

and the curvature tensor is written with an additional commutator $$F_{\mu\nu} = \partial_{\mu}A_{\nu} - \partial_{\nu}A_{\mu} - ig[A_{\mu},A_{\nu}] $$

However in Quantum Field Theory for the Gifted Amateur I have seen the following definition for the above (specific for $SU(2)$) The gauge field is a matrix that transforms like $$\sigma \cdot W_{a} \rightarrow \sigma \cdot W_{a} + \frac{1}{g}\sigma\cdot\partial_{a}\alpha -\sigma\cdot\alpha \times W_{a} $$

Where $\alpha$ is phase rotation $U = e^{i\sigma\cdot\alpha/2}$ and $\sigma$ are the Pauli matrices.

Here the curvature tensor is defined as:

$$G_{\mu\nu} = \partial_{\mu}W_{\nu} - \partial_{\nu}W_{\mu} - g(W_{\mu} \times W_{\nu}).$$

How are these two definitions related? I understand that the first is more general to all groups, but didn't know how to derive the second definition from this.

Answer 1

The first notation hides the Lie algebra generators by writing $A_\mu = A_m^a {\boldsymbol \lambda_a}$ where the implied generators obey $$ [{\boldsymbol \lambda}_a, {\boldsymbol \lambda}_b]= i {f_{ab}}^c {\boldsymbol \lambda}_c. $$ The second notation makes the generators explicit and writes $$ {\bf A}_\mu\cdot {\boldsymbol \lambda}. $$

For SU(2), the generators are the Pauli matrices and commutator algebra of the Pauli matrices tells us that $$ [{\bf a}\cdot {\boldsymbol \sigma}, {\bf b}\cdot {\boldsymbol \sigma}]= 2i ({\bf a}\times {\bf b})\cdot {\boldsymbol \sigma} $$ and the rest should be straightforward.