Ternary Z2 x Z3 graded algebras and ternary Dirac equation

TL;DR

This paper introduces a Z3-graded generalization of the Dirac equation, revealing connections to quark color symmetry and the Standard Model's gauge structure through a novel sixth-order wave equation.

Contribution

It develops a new Z3-graded Dirac operator, linking algebraic structures to fundamental particle symmetries and gauge groups in particle physics.

Findings

The generalized wave equation intertwines quark states and colors.

Solutions exhibit non-propagating behavior with damping factors.

The model naturally suggests the origin of Standard Model gauge groups.

Abstract

The wave equation generalizing the Dirac operator to the Z3-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 Z2 x Z2 x Z3. 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) x U(1) that arise automatically in this model, leading to the full bosonic gauge sector of the Standard Model.