Generalization of the Dirac equation admitting isospin and color symmetries

One possible generalization of the Dirac square root procedure =D _d ^k D _d is presented, based on the explicit introduction of chiral symmetry, which generates a set {d} of symmetry-constrained Dirac fieldsD _{d d} =0admitting isospin and color. A self-consistent discussion is given of the basic geometrical construction, the field equations, and their relationship to chiral symmetry, isospin, and color, and of the construction of the Lagrangian, including the interaction gauge fields. The correspondence of the theory with the standardSU _c(3) ×SU _w(2) ×U _y(1) formulation for quarks and leptons is shown.