I'm looking for comments and references about an idea : gauging the Dirac representation of the Dirac matrices. What kind of field interaction would it give ?
Specifically, the Dirac equation is defined as this (free field, to begin with) : \begin{equation}\tag{1} \gamma^a \; \partial_a \Psi + i \, m \, \Psi = 0. \end{equation} By definition, the gamma matrices obeys the following relation : \begin{equation}\tag{2} \gamma^a \, \gamma^b + \gamma^b \, \gamma^a = 2 \, \eta^{ab}. \end{equation} Any set of 4 matrices which obeys this relation can be used in equation (1) above (usual Dirac representation, Weyl representation, Majorana representation, etc). All representations are related by an unitary transformation : \begin{align}\tag{3} \tilde{\gamma}^a &= U \, \gamma^a U^{\dagger}, \\[12pt] \tilde{\Psi} &= U \, \Psi. \tag{4} \end{align}
Now, suppose that the representation becomes a local symetry of the Dirac equation ; $U \Rightarrow U(x)$. We then need to change the partial derivative : \begin{equation}\tag{5} \partial_a \Rightarrow D_a \equiv \partial_a + i C_a(x), \end{equation} where $C_a(x)$ is a new gauge field.
I did not pursued further that idea by lack of time. But I would like to know if this idea has been explored by someone else (surrely it was already studied before !).
So what it gives ? What kind of interaction gauge field ? Is there any mathematical problem with this ?
EDIT : Just a few more comments :
The Lorentz group acting on the Dirac field is represented by $SL(4, \mathbb{C})$, and its elements aren't all unitary matrices : the rotations are represented by unitary matrices, but not the pure Lorentz transformations.
Gauging the Lorentz group gives gravitation (this is well known and is a part of classical General relativity). Then gauging the $\gamma$ representation will certainly interfere with the gravitation gauge field (veirbein and its spin connection), since some unitary matrices may represent some rotations (but not all unitary matrices !).
I don't think that the group of transformations that are changing the $\gamma$ representation is the same as the Lorentz group (i.e. $SL(4, \mathbb{C})$), but I may be wrong.
What is the full group that is defining the $\gamma$ representations ? Does it really need to be unitary, i.e. $SU(4)$ ? I suspect they are just similarity transformations, so any invertible 4 X 4 matrices may be good, not just unitary matrices.
In other words, is there a transformation of $SL(4, \mathbb{C})$ (from the Lorentz group) that may change the usual Dirac matrices to the Weyl matrices and to the Majorana matrices ?
