Transformations of gamma-matrices through Pauli matrices transformations

Question

I have the transformation law of the Lorentz group for Pauli matrices:

$$\tag 0 (\sigma^{\mu})_{a \dot {a}}{'} = \Lambda^{\mu}_{\quad \nu} N_{a}^{\quad c}(\sigma^{\nu})_{c \dot {c}}(N^{-1})^{\dot {c}}_{\quad \dot {a}},$$ where $$N_{a}^{\quad c} = 1 + \frac{1}{2}\omega_{\mu \nu}\sigma^{\mu \nu}, \quad N^{\quad \dot {c}}_{\dot {a}} = 1 + \frac{1}{2}\omega_{\mu \nu}\tilde {\sigma}^{\mu \nu}.$$ Also I have the relation $$\tag 1 \gamma_{\mu} = \begin{pmatrix} 0 & \sigma_{\mu} \\ \tilde {\sigma}_{\mu} & 0 \end{pmatrix}, \quad (\tilde {\sigma}_{\mu})^{\dot {a} a} = \varepsilon^{ab}\varepsilon^{\dot {a} \dot {b}}(\sigma_{\mu})_{d \dot {b}}.$$ How exactly can I get the transformation law $$(\gamma_{\mu}){'} = \Lambda_{\mu}^{\quad \nu}\hat {S}\gamma_{\nu}\hat {S}^{-1}, \quad \hat {S} = 1 + \frac{1}{2}\omega_{\mu \nu}H^{\mu \nu},$$ $$H_{\mu \nu} = \frac{1}{4}\gamma_{[\mu} \gamma_{\nu ]}$$ from $(0)$?

I failed when I tried to write explicitly transformations of Pauli matrices in $(1)$: I can't pick out $\hat {S}^{-1}$, because I entangled in the indices.

This post imported from StackExchange Physics at 2014-03-06 21:15 (UCT), posted by SE-user Andrew McAddams