Spinorial representations ofSU(3)from a factored harmonic-oscillator equation

Abstract

We construct a seven-dimensional Dirac equation for quarks by extending the Klein - Gordon equation to seven dimensions, introducing a harmonic-oscillator term in the extra three dimensions, and taking the `square root' of the resulting equation. To facilitate solving the factored equation, we identify an SU(3) algebra in the higher dimensions and use the eigenstates of its Cartan subalgebra as basis functions. We find that the eigenstates of the extended Dirac equation are indeed exact representations of SU(3), one of which we identify with quarks. Indeed, a whole tower of SU(3) Dirac states appears. For principal quantum number , we find a singlet of mass zero; for , we find two 3s of mass , where is an adjustable constant; for , we find two 6s and two 3-bars, all of mass ; etc. Except for the ground state, all SU(3) representations appear in pairs. It is not clear that the pairs can be identified with weak-isospin doublets.