\label{31}$$

So the covariant derivative $e_m^\mu(\partial_\mu+i\omega_\mu)$ obeys condition $(\ref{27})$ if $e^\mu_n$ and $\omega_\mu$ transform according to

$$e^\mu_n \rightarrow \Lambda^n_m e^\mu_n\quad,\quad \omega_\mu\rightarrow U\omega_\mu U^{\dagger}-iU(\partial_\mu U^{\dagger}).
\label{32}$$

The matrices $\omega_\mu$ describe the Dirac Spinor connection. They can be performed by the above generators of the Lorentz group of Eq. $(\ref{18})$

$$\omega_\mu(x)=\frac{1}{2}\omega_\mu^{mn}(x)\Sigma_{mn}.    
\label{33}$$

So we have found the Lagrangian which is invariant under local Lorentz transformations

$$\mathcal{L}=\bar \psi\left[i\gamma^m e_m^\mu\left(\partial_\mu+\frac{i}{2}\omega_\mu^{mn}\Sigma_{mn}\right)-m\right]\psi,
\label{34}$$

leading to the Dirac equation in general relativity

$$\left[i\gamma^m e_m^\mu\left(\partial_\mu+\frac{i}{2}\omega_\mu^{mn}\Sigma_{mn}\right)-m\right]\psi=0.
\label{35}$$

(For an extensive discussion of gravity as a gauge theory the reader is referred to ref. [@[book3]].)

Combining the well-known anti-commutation relations for the Dirac-matrices $[\gamma^\mu,\gamma^\nu]_+=2\eta^{\mu\nu}$ and $(\ref{28})$ leads to the new anti-commutation relations

$$[e^\mu_m\gamma^m,e^\nu_n\gamma^n]_+=2g^{\mu\nu},
\label{36}$$

for the Dirac matrices in curved space time. We will need this relation in the next chapter.\

Calculation of the Spinor coefficients
--------------------------------------

Now we have to determine the specific form of the Spinor connection coefficients $\omega_\mu^{mn}$. It is important to notice that our coefficients $\omega_\mu^{mn}$ contain one index $(\mu)$ referring to the global coordinate system and two indices $(m,n)$ referring to the local coordinate system where $g_{mn}=\eta_{mn}$. First, we express the above coefficients $\omega_\mu^{mn}$ by the usual Christoffel symbols $\Gamma^\rho_{\mu\nu}$ defining the parallel transport between 