A New Route to Symmetries through the Extended Dirac Equation

Abstract

A new route to the Dirac equation and its symmetries is outlined on the basis of the four-vector representation of the Lorentz group (LG). This way permits one to linearize the first Casimir operator of the LG in terms of the energy–momentum four-vector and enables one to derive an extended Dirac equation that naturally reveals the SU(2) symmetry in connection with an isospin associated with the LG. The procedure gives a spin-one-half fermion doublet, which we interpret as the electron and neutrino or the up-and-down quark doublet. Similarly, the second Casimir operator can be linearized by invoking an abstract isospin that is not connected with the LG, but with the two basic empirical fermion types. Application of the spinor helicity formalism yields two independent singlet and triplet fermion states—which we interpret as being related to U(1) and the lepton, respectively—to the SU(3) symmetry group of the three colors of the quarks. The way in which we obtain these results indicates the genuine yet very different physical natures of these basic symmetries. This new notion does not need the idea of grand unification. However, by still combining them in the product group SU(4)=SU(3)⊗U(1) and then further combining all groups into SU(2)⊗SU(4), one may get a symmetry scheme that perhaps supports the notion of unification by the group SU(8). We also argue that the simpler SO(4) group—instead of SU(4)—seems more appropriate for achieving unification.