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. \noindent 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 $Z_3$-graded discrete automorphisms, in order to introduce entanglement the $Z_3$-graded version of Lorentz and Poincaré algebras with their 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 argue that all quarks in the Standard Model can be described by the $72$-component master quark sextet of $12$-component coloured Dirac fields.
In a previous work [1], we have argued that the algebra of non-abelian superselection rules is spontaneously broken to its maximal abelian subalgebra, that is, the algebra generated by its completing commuting set (the two Casimirs and a basis of its Cartan subalgebra). In this paper, alternative arguments confirming these results are presented. In addition, Lorentz invariance is shown to be broken in QCD, just as it is in QED. The experimental consequences of these results include fuzzy mass and spin shells of coloured particles like quarks, and decay life times which depend on the frame of o…
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 des…
Colour $SU(3)$ group is an exact symmetry of Quantum Chromodynamics, which describes strong interactions between quarks and gluons. Supplemented by two internal symmetries, $SU(2)$ and $U(1)$, it serves as the internal symmetry of the Standard Model, describing as well the electroweak interactions of quarks and leptons. The colour$SU(3)$ symmetry is exact, while two other symmetries are broken by means of the Higgs-Kibble mechanism. The three colours and fractional quarks charges with values $1/3$ and $2/3$ suggest that the cyclic group $Z_3$ may play a crucial role in quark field dynamics. In…
We investigate certain $Z_3$-graded associative algebras with cubic $Z_3$-invariant constitutive relations. The invariant forms on finite algebras of this type are given in the low dimensional cases with two or three generators. We show how the Lorentz symmetry represented by the $SL(2, {\bf C})$ group emerges naturally without any notion of Minkowskian metric, just as the invariance group of the $Z_3$-graded cubic algebra and its constitutive relations. Its representation is found in terms of Pauli matrices. The relationship of this construction with the operators defining quark states is als…
By attaching three anticommuting Lorentz scalar (colour) property coordinates to space-time, with an appropriate extended metric, we unify gravity with chromodynamics: gauge transformations then just correspond to coordinate transformations in the enlarged spacetime-property space.
The representation of the superalgebra SO(2,1) which is given by Eq. (3.1) and which resulted in the relativistic wave equation (4.1) is fully reducible. In fact, its even part is the direct sum of two spin 1/2 representations of the Lorentz group and does not represent spin 3/2 particle as we claimed. Consequently, we retract our claim and withdraw the manuscript.
The correlation of the fractionally represented hypercharge group with the isospin and colour group in the standard model determines as faithfully represented internal group the quotient group ${\U(1)\x\SU(2)\x\SU(3)\over\Z_2\x\Z_3}$. The discrete cyclic central abelian-nonabelian internal correlation involved is considered with respect to its consequences for the representations by the standard model fields, the electroweak mixing angle and the symmetry breakdown. There exists a further discrete $\Z_2$-correlation between chirality and Lorentz properties and also a continuous $\U(1)$-external…
The Bose--Fermi recoupling of particles arising from the $\bZ_{2}$--grading of the irreducible representations of SU(2) is responsible for the Pauli exclusion principle. We demonstrate from fundamental physical assumptions how to extend this to gradings, other than the $\bZ_{2}$ grading, arising from other groups. This requires non--associative recouplings where phase factors arise due to {\it rebracketing} of states. In particular, we consider recouplings for the $\bZ_{3}$--grading of SU(3) colour and demonstrate that all the recouplings graded by triality leading to the Pauli exclusion princ…
We define a superalgebra S2(N/2) as a Z2 graded algebra of dimension 2N+3, where N is a positive, odd integer. The even component is a three-dimensional abelian subalgebra, while the odd component is made up of two N-dimensional, mutually conjugate algebras. For N = 1, two of the three even elements become identical, resulting in a four-dimensional superalgebra which is the graded extension of the SO(2,1) Lie algebra that has recently been introduced in the solution of the Dirac equation for spinn 1/2. Realization of the elements of S2(N/2) is given in terms of differential matrix operators ac…
We systematically study Lorentz symmetry extensions in quantum field theories (QFTs) and their 't Hooft anomalies via cobordism. The total symmetry $G'$ can be expressed in terms of the extension of Lorentz symmetry $G_L$ by an internal global symmetry $G$ as $1 \to G \to G' \to G_L \to 1$. By enumerating all possible $G_L$ and symmetry extensions, other than the familiar SO/Spin/O/Pin$^{\pm}$ groups, we introduce a new EPin group (in contrast to DPin), and provide natural physical interpretations to exotic groups E($d$), EPin($d$), (SU(2)$\times$E(d))/$\mathbb{Z}_2$, (SU(2)$\times$EPin(d))/$\…
We have recently proposed a Lagrangian in trace dynamics at the Planck scale, for unification of gravitation, Yang-Mills fields, and fermions. Dynamical variables are described by odd-grade (fermionic) and even-grade (bosonic) Grassmann matrices. Evolution takes place in Connes time. At energies much lower than Planck scale, trace dynamics reduces to quantum field theory. In the present paper we explain that the correct understanding of spin requires us to formulate the theory in 8-D octonionic space. The automorphisms of the octonion algebra, which belong to the smallest exceptional Lie group…
A standard model is formulated in a Weyl space, $W_4$, yielding a Weyl covariant dynamics of massless chiral Dirac fermion fields for leptons and quarks as well as the gauge fields involved for the groups D(1)\,(Weyl), $U(1)_Y{\times} SU(2)_W$\,(electroweak), $SU(3)_c$\,(colour), SO(3,1)\,(gravity) and SO(4,1)\,(strong interaction, symmetry breaking). The dynamics is based on a gauge and Weyl invariant Lagrangean density ${\cal L}$. Gravitation is included from the beginning as the gauge aspect of the Lorentz group which is here extended in the hadronic sector of the model to the ten parameter…
