Internal quark symmetries and colour SU(3) entangled with Z3-graded Lorentz algebra

Abstract

In the current version of QCD the quarks are described by ordinary Dirac fields, organized in the following internal symmetry multiplets: the SU(3) colour, the SU(2) flavour, and broken SU(3) providing the family triplets. In this paper we argue that internal and external (i.e. space-time) symmetries are entangled at least in the colour sector in order to introduce the spinorial quark fields in a way providing all the internal quark's degrees of freedom which do appear in the Standard Model. Because the SU(3) colour algebra is endowed with natural Z3-graded discrete automorphisms, in order to introduce entanglement the Z3-graded version of Lorentz algebra with its vectorial and spinorial realizations are considered. The colour multiplets of quarks are described by 12-component colour Dirac equations, with a Z3-graded triplet of masses (one real and a Lee-Wick complex conjugate pair). We show that all quarks in the Standard Model can be described by the 72-component master quark sextet of 12-component coloured Dirac fields, which is required in order to implement the faithful spinorial representation of the Z3-graded Lorentz transformations.