Abstract
A natural generalization of the original Dirac spinor into a multi-component spinor is achieved, which corresponds to the single lepton and the three quarks of the first family of the standard model of elementary particle physics. Different fermions result from similarity transformations of the Dirac equation, but apparently there can be no more fermions according to the maximal multiplicity revealed in this study. Rotations in the fermion state space are achieved by the unitary generators of the U(1) and the SU(3) groups, corresponding to quantum electrodynamics (QED based on electric charge) and chromodynamics (QCD based on colour charge). In addition to hypercharge the dual degree of freedom of hyperspin emerges, which occurs due to the duplicity implied by the two related (Weyl and Dirac) representations of the Dirac equation. This yields the SU(2) symmetry of the weak interaction, which can be married to U(1) to generate the unified electroweak interaction as in the standard model. Therefore, the symmetry group encompassing all the three groups mentioned above is SU(8), which can accommodate and unify the observed eight basic stable fermions.
Highlights
In the prevailing standard model (SM) of elementary particle physics the various fermions involved are assumed to be massless
We show that the Dirac equation for a massive and charged fermion has a rich hidden intrinsic symmetry that has long been overlooked theoretically, but in practice already emerged through experimental evidence and was included in the SM in the guise of the various SU(N) external symmetry groups by which the different massless fermion are assembled in multiplets
We have shown that the different variants of how to decompose the mass shell condition (Equation 6) leads plausibly to eight different but coequal representations of the Dirac equation, which can in a physically meaningful way be interpreted as describing eight massive fermions
Summary
In the prevailing standard model (SM) of elementary particle physics (see e.g., the modern textbook by Schwartz [1] or the earlier one by Kaku [2]) the various fermions involved (one lepton doublet and three quark doublets, respectively coming in three generations) are assumed to be massless This notion is in contradiction to the observations but enforced on the SM by the assumption that chiral symmetry is broken. The dual degree of freedom of hyperspin emerges from the consideration of the duplicity implied by the two related (Weyl and Dirac) representations of the Dirac equation This yields naturally the SU(2) symmetry of the weak interaction, which can be married to U(1) to generate the unified electroweak interaction as in the standard model. The apt symmetry group encompassing the three above mentioned groups would be SU(8), which can accommodate and unify all observed massive fermions
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