Can we make the Dirac representation a gauge theory? 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 ?
 A: Expanding on my comment, I think the Rarita Schwinger field (spin 3/2) has exactly the gauge symmetry you want:
https://books.google.be/books?id=KFUhAwAAQBAJ&lpg=PA96&ots=vh0WtWM5rg&dq=rarita%20schwinger%20fermionic%20gauge%20symmetry&pg=PA95#v=onepage&q&f=false
This gauge symmetry removes the spin 1/2 component of the field so only the spin 3/2 part is left.
Now if you did the same gauging to the spin 1/2 field, you would gauge the entire spin 1/2 field away, the object would be made entirely out of nonphysical arbitrary gauge-stuff; I think.
A: Imposing local gauge symmetry on the Dirac equation produces the electromagnetic field interacting with it. 
See 
http://www.physics.rutgers.edu/~steves/613/lectures/Lec06.pdf
Before any down voting please see my comments below. The question was not about we whether Dirac's equation can be used to represent a spin 3/2 or higher fermion, though you could interpret it that way, and his example simply added a vector field as the gauge field. There is then no choice and it has to be electromagnetism. Weyl and Majorana fields are also consistent. See Peskin.  Btw, the spin 3/2 Rarita Schwinger field I understand has problems, though I am not an expert on it.
If this is totally off the mark, just explain please. 
