the following Lie Algebra

$$[\Sigma_{\alpha\beta},\Sigma_{\gamma\delta}]_-=\eta_{\gamma\beta}\Sigma_{\alpha\delta}-\eta_{\gamma\alpha}\Sigma_{\beta\delta}
+\eta_{\delta\beta}\Sigma_{\delta\alpha}-\eta_{\delta\alpha}\Sigma_{\gamma\beta}.
\label{17}$$

If we want to analyse the action of Lorentz transformations on quantum mechanical states of fermions described by the Dirac equation we have to consider the Dirac Spinors representation. In this representation the generators are given by

$$\Sigma_{\mu\nu}=\frac{i}{4}[\gamma_\mu,\gamma_\nu]_-.
\label{18}$$

(For an extensive treatment of the theory of group representations in the context of physics including a derivation of Eq. $(\ref{17})$ and the Dirac Spinor representation of the Lorentz group we suggest ref. [@[book2]](sections 2.2-2.4 and section 5.4, resp.).)

It is a priori clear that the Dirac equation is invariant under arbitrary global Lorentz transformations, i.e. independent of the space time point because it was derived from this constraint. (An explicit proof of this invariance property can be found in ref. [@[book8]].) In contrast to this, the well known Dirac Lagrangian is not invariant under local Lorentz transformations, but these are exactly the ones needed in the case of general relativity. In general relativity flat Minkowski space time is replaced by a curved space time (this topic is originally presented in ref. [@Einstein:1916vd]). This means that at every point of space time the metric tensor $g^{\mu\nu}$ looks different. If one wants to compare two 