I’ve finally just now figured out how to combine the quark / lepton weak hypercharge and weak isospin quantum numbers with the generation numbers. This should allow the Koide mass formula to be extended to the quarks!
The weak hypercharge and weak isospin quantum numbers for the elementary fermions are:
, with the quantum numbers for the antiparticles with opposite handedness given by the negatives of the above.
The Koide mass formulas for the charged and neutral leptons can be derived from making the assumption that these particles are color neutral composite particles built from three preons that I’ve usually called “snuarks” and they are taken from a set of three mutually unbiased bases for the Pauli algebra.
In the density matrix language, a particle is not represented by a state vector, but instead by a state matrix. A state matrix gives the transition amplitudes for the states that are bound together. The diagonal entries correspond to the amplitude for the propagation of one of the snuarks without change. The off diagonal entries give amplitudes for the various ways a snuark can switch states. Consequently, in the language of quarks and gluons, the off diagonal entries represent the gauge bosons of the theory while the diagonal entries give the valence quarks. Using “r,g,b” as the indices for the matrix, the (r,g) entry gives the amplitude for transitions from G to R; in the quark / gluon language, this would be the action of a R/G gluon.