Since we are regarding Spinors on the space time it is also necessary to consider the action of $D_\mu$ on the Dirac Spinor space.

Now we have to find out how this covariant derivative looks like. To do this, we assume that $D_\mu$ is of the following shape

$$D_m=e_m^\mu(\partial_\mu+i\omega_\mu).
\label{27}$$

Here we have introduced the vierbein $e_m^{\nu}$ describing the transformation between arbitrary global coordinates and the local coordinates in which the metric tensor looks like the metric tensor of flat Minkowski space time. This means that the following relation is valid

$$\eta_{mn}=e_{m}^{\mu}e_{n}^{\nu}g_{\mu\nu},
\label{28}$$

which can be seen as the definition of the vierbein. It expresses that if we know the vierbein at each space time point we can obtain the metric field meaning that it contains the same information about the structure of space time. So the vierbein represents another way to describe the gravitational field and can replace the metric tensor. Further, the vierbein fulfils the relations

$$e_{\mu}^{m}e_{n}^{\mu}=\delta^{m}_{n}\quad , \quad e_m^{\mu}e_{\nu}^{m}=\delta^{\mu}_{\nu}.
\label{29}$$

(An extensive introduction to the vierbein-formalism can be found in ref. [@[book3]]).

By inserting Eq. $(\ref{27})$ into the above transformation condition Eq. $(\ref{26})$ and rewriting it a little bit one gets

$$e_m^\mu(\partial_\mu+i\omega_\mu)\rightarrow\Lambda_m^n e_n^\mu\left[\partial_\mu+U(x)\partial_\mu U^\dagger (x)+iU(x)\omega_\mu U^\dagger (x)\right].
