# Z_3 - graded colour Dirac equations for quarks, confinement and generalized Lorentz symmetries

## TL;DR

This paper introduces a novel Z_3-graded colour Dirac equation framework that unifies quark confinement, internal symmetries, and generalized Lorentz symmetries through a 12-component colour spinor and algebraic grading.

## Contribution

It presents a new mathematical model combining Z_3 grading with Dirac equations to describe quark confinement and internal symmetries in a unified algebraic structure.

## Key findings

- Introduces a 12-component colour Dirac spinor with Z_3 grading.
- Unifies SU(3), SU(2), U(1) symmetries with generalized Lorentz symmetry.
- Requires 24 colour Dirac multiplets for covariance.

## Abstract

We propose a modification of standard QCD description of the colour triplet of quarks describing quark fields endowed with colour degree of freedom by introducing a 12-component colour generalization of Dirac spinor, with built-in Z_3 grading playing an important algebraic role in quark confinement. In "colour Dirac equations" the SU(3) colour symmetry is entangled with the Z_3-graded generalization of Lorentz symmetry, containing three 6-parameter sectors related by Z_3 maps. The generalized Lorentz covariance requires simultaneous presence of 24 colour Dirac multiplets, which lead to the description of all internal symmetries of quarks: besides SU(3) \times SU(2) \times U(1), the flavour symmetries and three quark families.