Hi when trying to find the Dirac equation (in the metric (1,a^2,a^2,a^2)) from the action; \mathcal{S}_{D}[\phi,\psi,e^{\alpha}_{\mu}] = \int d^4 x \det(e^{\alpha}_{\mu}) \left[ \mathcal{L}_{KG} + i\bar{\psi}\bar{\gamma}^{\mu}D_{\mu}\psi - (m_{\psi} + g\phi)\bar{\psi}\psi\right]<br />
I can show that, i \bar{\gamma}^{\mu}D_{\mu}\psi - (m_{\psi}+g\phi)\psi =0 by varying the action. I know that D_{\mu}=\partial_{\mu}+\frac{1}{4}\gamma_{\alpha\b  eta}\omega^{\alpha\beta}_{\mu} and I know how to relate \bar{\gamma}^{\mu}to the flat space-time gamma matrices \gamma, I am just stuck trying to prove that \frac{1}{4}\gamma_{\alpha\beta}\omega^{\alpha\beta  }_{\mu}=\frac{3}{2}\frac{\dot{a}}{a} I think this term is equal to \frac{1}{4}\left( \gamma^{\alpha}\gamma^{\beta} - \gamma^{\beta}\gamma^{\alpha}\right) \left( e_{\alpha}^{\nu}(\frac{\partial}{\partial x^{\mu}})e_{\beta\nu}+e_{\alpha\nu}e_{\beta}^{\sig  ma}\Gamma^{\nu}_{\sigma\mu}\right) but if it is I can't get the above result, can anyone help