Ternary Z$_2$ × Z$_3$ Graded Algebras and Ternary Dirac Equation Article - 2018

The wave equation generalizing the Dirac operator to the Z$_3$-graded case is introduced, whose diagonalization leads to a sixth-order equation. It intertwines not only quark and anti-quark state as well as the u and d quarks, but also the three colors, and is therefore invariant under the product group Z$_2$ × Z$_2$ × Z$_3$. The solutions of this equation cannot propagate because their exponents always contain non-oscillating real damping factor. We show how certain cubic products can propagate nevertheless. The model suggests the origin of the color SU(3) symmetry and of the SU(2) × U(1) that arise automatically in this model, leading to the full bosonic gauge sector of the Standard Model.