For the Dirac equation, iγμ∂μψ−mψ=0, ¯ψγμψ is a conserved current. I feel like I've known this since I was three. ¯ψγμψ is also a conserved constant, however, for the related parameterized set of equations,
iγμ(1+iα1γ5)∂μψ−m(α3+iα2γ5)ψ=0,
provided α1, α2, and α3 are real constants. A further constraint in QFT is that the matrix
M=kαγα[kμγμ(1+iα1γ5)−m(α3+iα2γ5)],where kμkμ=m2,
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 α21+α22+α23≤1. Given that, we have a class of free quantum fields. If α21+α22+α23=1, M has a 2-dimensional zero eigenspace, as for the usual Dirac equation, α3=1,α2=α1=0, or in it's conjugate form, α3=−1,α2=α1=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 α1, α2, and α3, 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 α1, α2, and α3, etc., etc.
References preferred, or else an explanation of why it's obvious why this isn't useful. Thanks.