Dirac equation based on the vector representation of the Lorentz group

In this paper, we derive an expanded Dirac equation for a massive fermion doublet, which has in addition to the particle/antiparticle and spin-up/spin-down degrees of freedom explicity an isospin-type degree of freedom. We begin with revisiting the four-vector Lorentz group generators, define the corresponding gamma matrices and then write a Dirac equation for the fermion doublet with eight spinor components. The appropriate Lagrangian density is established, and the related chiral and SU (2) symmetry is discussed in detail, as well as applications to an electroweak-style gauge theory. In “Appendix,” we present some of the relevant matrices.