special representation).

-   $\partial_\mu$ $\hat =$ $\frac{\partial}{\partial x^\mu}$.

-   $\gamma^\mu$ $\hat =$ Dirac matrices.

-   $\Lambda^\nu_\mu$ $\hat=$ Lorentz transformation matrices and $(\Lambda^{-1})^\nu_\mu$ $\hat=$ the corresponding inverse transformation.

-   U $\hat =$ arbitrary unitary operator and $U^{\dagger}$ $\hat=$ its hermitean conjugated operator.

-   $x^{[ab]}=x^{ab}-x^{ba}$

Besides we will make the arrangement that Greek indices refer to arbitrary global coordinates and Latin indices refer to the local coordinate system in which the metric tensor $g_{\mu\nu}$ takes the form of the special metric tensor of flat Minkowski space time $\eta_{mn}$. If a certain index adopts a specific value the special local coordinates are denoted with a bar to distinguish them from arbitrary global coordinates. As usual natural units will be used which means that $\hbar=c=G=1$.

The general shape of the equation
---------------------------------

Like in case of usual gauge theories one starts with the Lagrangian of the free field equation, in our case the Dirac equation (first published in ref. [@Dirac:1928hu])

$$\mathcal{L}=\bar \psi(i\gamma^\mu\partial_\mu-m)\psi.
\label{16}$$

In the framework of relativistic quantum mechanics a Lorentz transformation is represented by an unitary operator which acts on a state in Hilbert space. Infinitesimal transformations are provided by the generators of the group. The generators of the Lorentz group obey 