For the Dirac equation, $i\gamma^\mu\partial_\mu\psi-m\psi=0$, $\overline{\psi}\gamma^\mu\psi$ is a conserved current. I feel like I've known this since I was three. $\overline{\psi}\gamma^\mu\psi$ is also a conserved constant, however, for the related parameterized set of equations,

$$i\gamma^\mu(1+i\alpha_1\gamma^5)\partial_\mu\psi-m(\alpha_3+i\alpha_2\gamma^5)\psi=0,$$

provided $\alpha_1$, $\alpha_2$, and $\alpha_3$ are real constants. A further constraint in QFT is that the matrix

$$M=k_\alpha\gamma^\alpha\left[k_\mu\gamma^\mu(1+i\alpha_1\gamma^5)-m(\alpha_3+i\alpha_2\gamma^5)\right],\quad \mbox{where } k_\mu k^\mu=m^2,$$

must be positive semi-definite for the two-point VEVs to be positive semi-definite, as they must be for us to construct a free field Fock-Hilbert space, which is satisfied only if $\alpha_1^2+\alpha_2^2+\alpha_3^2\le 1$. Given that, we have a class of free quantum fields. If $\alpha_1^2+\alpha_2^2+\alpha_3^2=1$, $M$ has a 2-dimensional zero eigenspace, as for the usual Dirac equation, $\alpha_3=1,\alpha_2=\alpha_1=0$, or in it's conjugate form, $\alpha_3=-1,\alpha_2=\alpha_1=0$, so achieving the bound is perhaps preferred so as not to introduce too many DoFs.

Is this generalized Dirac equation discussed in the literature? It seems *possible* that electrons, muons, and tauons *might* satisfy this equation with different values of $\alpha_1$, $\alpha_2$, and $\alpha_3$, but yet with the same conserved current, and that this difference might make a difference, or at least that someone must have shown that this either isn't useful or is equivalent to the usual Dirac equation. We also might investigate symmetries that transform between different values of $\alpha_1$, $\alpha_2$, and $\alpha_3$, etc., etc.

References preferred, or else an explanation of why it's obvious why this isn't useful. Thanks.