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 S U ( 3 ) colour, the S U ( 2 ) flavour, and broken S U ( 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 S U ( 3 ) colour algebra is endowed with natural Z 3 -graded discrete automorphisms, in order to introduce entanglement the Z 3 -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 Z 3 -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 Z 3 -graded Lorentz transformations.