Internal quark symmetries and colour SU(3) entangled with Z_3-graded Lorentz algebra

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 Poincar\'e 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.


Introduction
In the current version of Quantum Chromodynamics the massive quarks are treated as Dirac fermions endowed with additional internal degrees of freedom. In the minimal version, Standard Model displays the exact SU (3) colour and the SU (2) flavour symmetries, as well as strongly broken SU (3) describing three quark families ( [1], [2], [3]).
If we introduce "master quark Dirac field" supposed to incorporate all internal quark symmetries, we should deal with 4 × 3 × 2 × 3 = 72-component fermionic master field (the first factor 4 corresponds to the four degrees of freedom of classical Dirac spinor, the next factor 3 stays for three colours, next factor 2 gives flavours, and the last factor 3 corresponds to the three families). Our aim here is to look for a framework introducing algebraic and group-theoretical structure which permits to incorporate all the internal quark symmetries enumerated above in some irreducible representations of Z 3 -graded generalization of Lorentz algebra.
Because in the quark sector of Standard Model e.g. u and d quarks behave as fermions, the two-quark states uu or dd should be excluded, unless there are extra parameters distinguishing the states in a pair. This was the origin of colour degrees of freedom, and of an exact SU (3) colour symmetry treating quarks as colour triplets which incorporate three distinct eigenstates, labeled as red, green, and blue. With such enlargement of the Hilbert space describing single quark states we arrive in Sect. 3 at new 12-component fermionic colour Dirac field, introduced in [4], [5], which is covariant under Z 3 symmetry group and contains colour Dirac Γ µ matrices whose structure is described symbolically by the following tensor product: M 3 (C) represents colour 3×3 matrices, H 2 are the 2×2 Hermitiean ones, and in our approach the generalized 12 × 12-dimensional generalized Dirac Γ µ -matrices employ in M 3 sector the generators of the particular ternary Clifford algebra discussed in Sect. 2; similar constructions were recently considered in [6], [7]. The colour symmetry is somehow hidden in Nature, because the states with non-zero colour charges are not observed in experiments due to the quark confinement mechanism ( [8]), so that in a quark model we deal only with composite hadronic states described asymptotically by the Dirac and Klein-Gordon (KG) equations. We argue that on the level of single quark states one should not postulate the description of colour quark triplets by standard Dirac fields, and we propose instead its 12-component colour generalization, incorporating the Z 3 -grading and generating colour entanglement.
In this new approach (see Sect. 3) free quark field components satisfy a sixthorder generalization of Klein-Gordon's equation, which factorizes into the triple product of the standard Klein-Gordon operator with real mass, and a pair of Klein-Gordon operators with complex-conjugated Lee-Wick masses (see e.g. [9], [10]). In such a way we obtain a set of three Z 3 -graded mass parameters m, jm and j 2 m, j = e 2πi 3 , covariant under the Z 3 -symmetry acting on complex energy plane. By a suitable choice of 12 × 12 colour Dirac equations one can introduce the colour-entangled quark triplet with one real mass and a pair of complex masses jm and its complex conjugate j 2 m forming together a Z 3 -graded triplet. 1 Such construction permits to introduce in Sect. 4 the vectorial realization of Z 3 -graded Lorentz group acting on a triplet of replicas of one and the same four-momentum vector (see also [11], [12]). 2 In Sect. 5, we present the full description of spinorial realizations of the Z 3 -graded Lorentz algebra which we introduced in [13] (see also [12], [14]).
In order to describe all quark symmetries, both the exact one (colour) and broken ones (flavour, generations), we consider in Sect. 6 in explicit way the action of Z 3 -graded Lorentz algebra L = L (0) ⊕ L (1) ⊕ L (2) on the sextet of generalized 12 × 12-component colour Dirac Γ µ matrices which provides the irreducible representation of algebra L. It appears that such a sextet requiresas a module of which L can act faithfully -six 12-component colour Dirac fields which span 72 quark states and takes into account all known internal symmetries of the standard quark model, describing colour, flavour and generations. Further, in Sect. 7 we introduce chiral flavour doublets and we show how to define chiral colour spinors in the framework using the colour Dirac equations.
In a short outlook in final Sect. 8 we are pointing out the differences between our approach and other results dealing with possible modifications of internal symmetries sector in the quark models. Some problems which may occur in the procedure of supplementing our model with gauge interactions are also briefly addressed.
2 Ternary Clifford algebra, Z 3 -symmetry and the SU (3) algebra In a recent series of papers ( [4], [16], [17], [11], [15]) ternary algebraic structures have been introduced and discussed. Among others, a ternary generalization of Clifford algebra with two generators (see e.g. in [6], [7]) is of particular interest for high energy physics due to its close relation with the Lie algebra of the SU (3) group appearing as an exact colour symmetry and as a broken symmetry mixing the three quark families. The standard 3 × 3 matrix basis of ternary Clifford algebra (which was first considered in XIX-th century by Cayley [18] and Sylvester [19], who called its elements "nonions" ) looks as follows: where j is the third primitive root of unity, and M † denotes the hermitian conjugate of matrix M. We see that all the matrices (2, 3) are non-Hermitian. To complete the basis of 3 × 3 traceless matrices, we must add to (2) and (3) the following two linearly independent diagonal matrices: In what follows, we shall often use alternative notation I A , A = 1, 2, ...8, with and can also add I 0 = l 1 3 . The Hermitian conjugation I † A (A = 1, 2, ..., 8) : provides the following permutation of indices A → A † : We can introduce as well the standard complex conjugation M →M, which leads to the relations which corresponds to another permutation of indices A, The 3 × 3 matrices Q 3 and Q † 3 are real, while Q 2 =Q 1 are mutually complex conjugated, as well as their Hermitean counterparts Q † 2 =Q † 1 . The matrices (2) and (3) are endowed with natural Z 3 -grading Out of three independent Z 3 -grade 0 ternary (i.e. three-linear) combinations, only one leads to a non-vanishing result. One can simply check that both j and j 2 ternary skew commutators do vanish as well as the odd permutation, e.g.
In contrast, the totally symmetric combination does not vanish but it is proportional to the 3 × 3 identity matrix I 0 = l 1 3 : with η abc given by the following non-zero components and all other components vanishing. The above relation can be used as definition of ternary Clifford algebra (see e.g. [21], [4]). Analogous set of relations is formed by Hermitian conjugates Q † a :=Q T a of matrices Q a , which we shall endow with dotted indicesȧ,ḃ, ... = 1, 2, 3. They satisfy the relation as well as the identities conjugate to the ones in (14) It is obvious that any similarity transformation of the generators Q a keeps the ternary anti-commutator (14) invariant. As a matter of fact, if we definẽ Q b = S −1 Q b S, with S a non-singular 3 × 3 matrix, the new set of generators will satisfy the same ternary relations, because it follows that and on the right-hand side we have the unit matrix which commutes with all other matrices, so that S −1 l Here is the full multiplication table of the associative algebra of eight basis matrices I A (A = 1, 2, ...8). It is also worthwhile to note that the six matrices Q a and Q † b together with two traceless diagonal matrices B and B † from (2, 3) form the basis for certain Z 3 -graded representation of the SU(3)-algebra, as it was shown by V. Kac in 1994 (see [22]).
All these matrices are cubic roots of the 3 × 3 unit matrix, i.e. their cubes are all equal to l 1 3 . One can observe that two traceless matrices I 2 = Q 2 and I 7 = B generate, by consecutive multiplications, full 8-dimensional basis of the SU (3) algebra. The full basis of 3 × 3 traceless SU (3) matrices is generated by all possible powers and products of B and Q 2 , and is displayed in Table 1 below.
We endow the two diagonal matrices B and B † = B 2 with Z 3 grade 0, the matrices Q a with Z 3 grade 1, and their three hermitian conjugatesQ˙b with Z 3 grade 2. Under matrix multiplication the grades are additive modulo 3.
The eight matrices B, B † , Q a , Q † b can be mapped faithfully onto the canonical Gell-Mann basis of the SU (3) algebra. The Lie algebra of the commutators between the generators I A is given in Appendix I. The linear combinations of matrices I A producing the Gell-Mann matrices are given in Appendix II.
Further we shall use the basis (6) for the description of the generators of colour algebra, which satisfies the Lie-algebraic relations with particular properties of complex structure constants (see Appendix I, relation 141).

The Z -graded Dirac's equation
We shall construct a generalized equation for quarks, incorporating not only their half-integer spin and particle-antiparticle content (due to charge conjugation, producing anti-quark states), but also the new discrete degree of freedom, the colour, taking three possible values.
Let us describe three different two-component fields (Pauli spinors), which will be distinguished by three colours, the "red" for ϕ + , the "blue" for χ + , and the "green" for ψ + ; more explicitly We follow the minimal scheme which takes into account the existence of spin by using Pauli spinors on which the 3-dimensional momentum operator acts through 2 × 2 matrix describing the scalar product σ · p.
To acknowledge the existence of anti-particles, we should also introduce three "anti-colours", denoted by a "minus" underscript, corresponding to "cyan" for ϕ − , "yellow" for χ − and "magenta" for ψ − ; here, too, we employ the twocomponent columns: As a result, the six Pauli spinors (19) and (20) will form a twelve-component entity which we shall call "coloured Dirac spinor". This construction reflects the overall Z 3 × Z 2 × Z 2 symmetry: one Z 2 group corresponds to the spin 1 2 dichotomic degree of freedom, described by eigenstates; the second Z 2 is required in order to represent the particle-anti-particle symmetry, and the Z 3 group corresponding to colour symmetry. The "coloured" Pauli spinors should satisfy first order equations conceived in such a way that they propagate all together as one geometric object, just like E and B components of Maxwell's tensor in electrodynamics, or the pair of twocomponent Pauli spinors which are not propagating separately, but constitute one single entity, the four-component Dirac spinor.
This leaves not much space for the choice of the system of intertwined equations. Here we present the ternary generalization of Dirac's equation, intertwining not only particles with antiparticles, but also the three "colours" in such a way that the entire system becomes invariant under the action of the Z 3 × Z 2 group.
The set of linear equations for three Pauli spinors endowed with colours, and another three Pauli spinors corresponding to their anti-particles characterized by "anti-colours" involves together twelve complex functions. The twelve components could describe three independent Dirac particles, but here they are intertwined in a particular Z 3 × Z 2 graded manner, mixing together not only particle-antiparticle states, but the three colours as well.
Let us follow the logic that led from Pauli's to (Z 2 -graded) Dirac's equation and extend it to the colours acted upon by the Z 3 -group. In the expression for the energy operator (Hamiltonian) the mass term is positive when it describes particles, and acquires negative sign when we pass to anti-particles, i.e. one gets the change of sign each time when particle-antiparticle components are interchanged.
We shall now assume that mass terms should acquire the factor j when we switch from the red component ϕ to the blue component χ, and another jfactor when we switch from blue component χ to the green component ψ. We remind that we use the notation introduced in (4), j = e 2πi 3 , j 2 = e 4πi 3 , j 3 = 1, and 1 + j + j 2 = 0.
The momentum operator will be non-diagonal, as in the Dirac equation, systematically intertwining not only particles with antiparticles, but also colours with anti-colours. The system that satisfies all these assumptions can be introduced in the following manner ( [4], [11]): Let us first choose the basis in which particles with a given colour and the particles with corresponding anti-colour are grouped in pairs: where ϕ ± , χ ± and ψ ± are two-component Pauli spinors defined by eqs. (19) and (20). In such a basis our "coloured Dirac equation" takes the following form in terms of six Pauli spinors: Let us remark that while in the Schroedinger picture the energy E and the momentum p are represented by differential operators in (22) we use their Fourier-transformed image, in which E and p are interpreted as multiplication by the corresponding numerical eigenvalues. The particle-antiparticle Z 2 -symmetry is obtained if m → −m and simultaneously (ϕ + , χ + , ψ + ) → (ϕ − , χ − , ψ − ) and vice versa; the Z 3 -colour symmetry is realized by multiplication of mass m by j each time the colour changes, i.e. more explicitly, Z 3 symmetry is realized by the following mappings: The system of equations (22) can be written using 12 × 12 matrices acting on the 12-component colour spinor Ψ build up from six "coloured" Pauli spinors.
In shortened form we can write where E = E l 1 12 , with l 1 12 denoting the 12 × 12 unit matrix, and the matrices M and P given explicitly below: The two matrices M and P in (27) and (28) are 12 × 12-dimensional: all the entries in M are proportional to the 2 × 2 unit matrix, and the entries in the second matrix P contain 2 × 2 Pauli's sigma-matrices, so P is as well a 12 × 12 matrix. The energy matrix operator E is proportional to the 12 × 12 unit matrix.
One can easily see that the diagonalization of the system is achieved only at the sixth iteration. The final result is extremely simple: all the components satisfy the same sixth-order equation, and similarly all other components. It is convenient to use the tensor product notation for the description of the matrices E, M and P. Using two 3 × 3 matrices B and Q 3 defined in (2), the 12 × 12 matrices M and P can be represented as the following tensor products: Let us rewrite the system (22) involving six coupled two-component spinors as one linear equation for the "colour Dirac spinor" Ψ, conceived as column vector containing twelve components of three "colour" fields, in the basis (21) given byΨ = [ϕ + , ϕ − , χ + , χ − , ψ + , ψ − ] T , with energy and momentum operators E and P on the left hand side and the mass operator M on the right hand side: Like in the case of the standard Dirac equation, let us transform this equation in a way that the mass operator becomes proportional the the unit matrix. For such a purpose, we multiply the equation (32) on the left by the matrix B † ⊗ σ 3 ⊗ l 1 2 . Now we get the following equation which enables us to interpret the energy and the momentum as components of the Minkowskian four-vector c p µ = [E, cp]: where we used the fact that ( The sixth power of this operator gives the same result as before, It is also worth to note that taking the determinant on both sides of the eq. (33) yields the twelfth-order equation: There is still certain arbitrariness in the choice of 3 × 3 matrix factors B † and Q 2 in the colour Dirac operator (33). This is due to the choice of j = e 2πi 3 as the generator of the representation of the finite Z 3 -symmetry group. If j 2 is chosen instead, in (33) the matrix B † will be replaced by B, Q 2 by Q 1 , which is its complex conjugate; the remaining terms keep the same form.
we get obviously Following the formulae (37) for the colour Dirac Γ µ -matrices we see that they are neither real (Γ µ = Γ µ ) nor Hermitian ((Γ µ ) † = Γ µ ). From the colour Dirac equation (33) one gets the following equations for complex-conjugatedΨ and Hermitean-conjugated Ψ † : whereΨ is a column, Ψ † is a row, Further, the second equation of (40) can be written in terms of the matrices (37) if we introduce the Hermitian-adjoint colour Dirac spinor Ψ H = Ψ † C, where the 12 × 12-matrix C satisfies the relation It can be also shown that neitherΓ µ nor (Γ µ ) † can be obtained via similarity transformation (38).
To obtain a general solution of the colour Dirac equation one should use its Fourier transformed version (see (36)). In the momentum space it becomes: The sixth power of the matrix Γ µ p µ is diagonal and proportional to m 6 , so that we have Now we should find the inverse of the matrix (Γ µ p µ − m l 1 12 ). Let us note that the sixth-order expression on the left-hand side in (44) can be factorized as follows: (45) The first factor can be expressed as the product of two linear operators, one of which defines the colour Dirac equation (36) (see also (43): Therefore the inverse of the Fourier transform of the linear operator defining the colour Dirac equation (43) is given by the following matrix: . (47) The inverse of the six-order polynomial can be decomposed into a sum of three expressions with second-order denominators, multiplied by the common factor of the fourth order. Let us denote by Ω the sixth root of (| p | 6 +m 6 ), along with five other root values obtained via multiplication by consecutive powers of the sixth root of unity, q = e 2πi 6 . Recalling relation (4) and that q 2 = j, we have the identity which leads to the decomposition formula or equivalently, As long as there is a non-zero mass term, we do not encounter the infrared divergence problem at | p |→ 0. Each of the three inverses of a second-order polynomial can be in turn expressed as a sum of simple first-order poles, e.g.
and similarly for other terms in (50). After such a substitution in (47), six Z 6graded simple poles do appear, Figure (1) illustrating the location of these six poles in the complex energy plane. In order to introduce the propagators in the coordinate space, one has to perform the contour integrals in complex energy plane. The first term in the decomposition (50) of the coulour Dirac propagator (50) presents two simple poles on the real line, while the second and the third terms display two simple poles each, located on complex straight lines Imp 0 = jRep 0 and Imp 0 = j 2 Rep 0 .
One can add that in the propagators given by formula (50) the non-standard residua ±j and ±j 2 should be justified by suitable form of the Z 3 -graded commutators describing quantum oscillator algebra of colour quark field excitations. ; see also [10].
It should be stressed that the colour Dirac equation (40) breaks the Lorentz symmetry O(1, 3) reducing it to O 3 , because the 3 × 3-matrices describing "colour" are different for the Γ 0 and Γ k components. However we shall show in the following Section 4 that one can introduce a Z 3 -graded generalization of the Lorentz transformations, acting in covariant way on three "replicas" of the energy-momentum four-vector introduced above. Analogous extensions of space-time were discussed in [29], [28].

Z 3 -graded set of three complex four-momenta and Z 3 -graded Lorentz transformations
The mass shell condition (35) for coloured Dirac equation can be decomposed into the usual relativistic Klein-Gordon invariant multiplied by a strictly positive factor which can be interpreted as generating the form-factor for quark propagator with given mass m.
The sixth-order polynomian C 6 can be further decomposed into the product of the following three second-order polynomials, Let us denote by superscripts (0), (1) and (2) the four-momenta with quadratic invariants given by From any real four-vector (0) p 0 µ one can define its two "replicas" 0 (1) p µ and (2) p µ with p 0 in the complex plane, obtained by the generalized Wick rotations by j and by j 2 in the following way. Let us introduce three 4 × 4 matrices acting on Minkowskian four-vectors: A = diag (j, 1, 1, 1), (57) providing a (reducible) matrix representation of the cyclic Z 3 group, where the superscripts (r+s) are added modulo 3, e.g. 1+2 → 0, 2+2 → 1, etc.
Acting on a given four-vector p µ = (p 0 , p) by one of the matrices (r) A we produce its three Z 3 -graded "replicas" belonging correspondingly to sectors (r) In what follows, we shall use for both the Lorentz boosts and the Wick rotations a short-hand notation: It should be stressed here that the spacetime remains Minkowskian, with one real time and three real spatial coordinates; however, the components of (1) p µ and (2) p µ can take on particular Z 3 -graded complex values. Three "replicas" (59) are the images of the same four-vector which can be obtained by Z 3 -valued Wick rotations in complex energy plane.
In particular, let us denote by where lower indices (00) mean that we transform p into a vector from another sector r, (r) p . Using the shorthand notations (60) and (61), we have: describes the Lorentz transformation from sector s onto sector r, and where the superscript (r − s) accordingly to the Z 3 -grading is taken modulo 3.
In order to provide the formulae for Z 3 -graded boosts in explicit form we will choose the four-vectore p µ = (p 0 , p) as restricted to the plane (0, 1), with the three-vector p aligned along the first spatial axis. In such a frame the Lorentz rotations reduce only to the boost in (0, 1) plane, which is given by the following transformation: Subsequently, we get the following triplet of homogeneous transformations: preserving respectively the bilinear forms (r) C 2 (see 55). The matrices (65) are self-adjoint: The generalized Lorentz boosts (65) conserve the group property: the product of two Lorentz boosts acting in the r-th sector is a boost of the same type. Indeed, we see from (65) that the product of two boosts acting in the r-th sector (r = 0, 1, 2) looks as follows (no summation over r!): If we look at three fourdimensional Lorentz boost transformations on planes (0, i), i = 1, 2, 3, the respective set of three independent "classical" Lorentz boosts belonging to Next, let us consider the general set of matrices (see (63)) transforming the s-th sector into the r-th one, There are two types of such matrices: raising and lowering the Z 3 by 1. For the sake of simplicity, let us firstly consider the two-dimensional case (i.e. µ, ν = 0, 1 in (69). The matrices 2 × 2 raising the Z 3 index (r) of the generalized fourmomenta ( The determinants of the matrices (70) are equal to j 2 . The matrices lowering the Z 3 index by one (or increasing it by 2, what is equivalent from the point of view of the Z 3 -grading) are: L12 = jchu j 2 shu j 2 shu chu , The determinants of the matrices (71) are equal to j. The above two sets of three matrices each are mutually Hermitian-adjoint: We recall that the superscript over each matrix (t) L rs is equal to the difference of its lower indices, i.e. (t) = (r − s).
The matrices (1) L rs and (2) L rs (r, s = 0, 1, 2) raising or lowering respectively the L rs they can be used as building blocks in bigger 12 × 12 matrices forming a Z 3 -graded generalization of the Lorentz group. This construction is possible due to the chain rule obeyed by these matrices, which due to the definition (63) display the group property. We have: In order to pass to arbitrary four-momentum vectors If we write a Z 3 -extended four-momentum vector ( p µ ) T as a column with 12 entries, we can introduce three boost sectors (r) Λ , (r = 0, 1, 2) of the generalized Z 3 -graded Lorentz group as 12 × 12 matrices as follows: It should be stressed that in each of the 12 × 12 matrices Λ -matrix depends only on three parameters defining three independent Lorentz boosts.
One can show that the matrices (74) display the following Z 3 -graded multiplication rules: where space rotations supplementing the Z 3 -graded boosts (74) are constructed as the following 12 × 12 matrices: where the choice of the colour generators Q † 3 and Q 3 is consistent with the colour Dirac equations (32-33).
The Z 3 -graded infinitesimal generators of the Lorentz boosts can be obtained by considering the matrices (r) Λ with infinitesimal boost parameters (i.e. taking the differential) what amounts to the replacements of the entries shu by 1, and of all other entries, chu and 1 alike, by 0. The resulting 12 × 12 matrices are the Lie algebra generators of the generalized Lorentz boosts, which we shall denote as (r) K i , r = 0, 1, 2. By taking their commutators we obtain the Z 3 -graded generators of the space rotations (r + s) modulo 3): In such a way we obtain the full set of generators of the Z 3 -graded Lorentz algebra which satisfy the following commutation relations: which were firstly introduced and studied in ( [13]).  Λ is Hermitean, we have

Let us consider
the matrix U  Λ and (2) Λ are also preserved after similarity transformation if the similarity matrices obey the same unitarity condition U † = U −1 .
In this way we introduced the symmetry SU (3) acting on the vector representation of the Z 3 -graded Lorentz group. The 3 × 3 matrices U appearing in the 12 × 12 matrices U during the unitary similarity transformations leave the 4 × 4 Lorentzian blocks unaffected, in agreement with the well known "no-go theorems" by Coleman and Mandula and O'Raifeartaigh ( [23], [24]).
We point out that in order to obtain the entire Z 3 -graded Lorentz group we should add as well the Z 3 -graded extension of space rotations, also represented as 12 × 12 matrices, with building blocks . made of 4 × 4 matrices, just like the Z 3 -graded boosts. As in the case of Lorentz boosts, besides the rotations that leave the transformed 3-momentum in the same sector, one gets also12 × 12 matrices with non diagonal 4×4 entries, which map one of the Z 3 -graded sectors onto another one.
We conclude that the full set of Z 3 -graded O(3) subgroup elements can be represented by 12 × 12 matrices and incorporated in the Z 3 -graded Lorentz group.
In this Section we were considering the vectorial realizations of the Z 3 -graded Lorentz group which can be also extended to the realizations of Z 3 -graded Poincaré algebra (see also ([12])) In the next two sections we will present our main result: how the sextet of the colour Dirac matrices Γ µ appears in the construction of faithful spinorial 72 × 72 matrix representation of the Z 3 -graded Lorentz algebra. In such a way we will be able to incorporate all internal symmetries of quark sector appearing in the Standard Model into a group-theoretical framework.
The two standard commutators of Γ µ matrices, namely provide only the first step towards the construction of the generators of a Z 3 -graded Lorentz algebra. Surprisingly, one can check that the generators satisfying the standard D = 4 L orentz algebra relations can be defined by double commutators of 12 × 12 matrices J i , K l as follows: Using the definition of standard colour Γ µ -matrices (37) and substituting it in (82 and 85), we get In order to introduce the Z 3 -graded Lorentz algebra where j ), one should supplement the relations (85) by the pairs of other possible double commutators: In particular, besides the representation (86) we get the following realizations: The three-linear double commutators in (85) and (88) are related with Z 3grading; when taken into account, the full set of Z 3 -graded relations defining the Z 3 -graded Lorentz algebra introduced in [13] (where r, s, = 0, 1, 2, r + s are taken modulo 3), results in the following set of commutation relations ( [13]): From the commutators [K (1) , K (1) ] J (2) and [J (1) , J (1) ] J (2) one gets the realization of remaining generators of L, The formulae (86), (89) and (91) describe the spinorial realization of the Lie algebra L which is implied by the choice (37) of matrices Γ µ . Let us introduce a unified notation englobing all possible choices of Γ µ -matrices (A = B) where I 0 = l 1 3 , I A with A = 1, 2, ..., 8 are given in (6), and α, β = 2, 3 but {σ α , σ β } + = 0 i.e. we always have either α = 2, β = 3 or α = 3, β = 2.
The characteristic feature of "colour" Γ-matrices is that the 3 × 3 matrices I A appearing as the first tensorial factors in (92) are different for temporal and spatial components of the matrix-valued 4-vector Γ µ . We see that the choice of the colour factor in (92) depends on two sets of values of the four-vector index: µ = 0 or µ = i where i = 1, 2, 3. This property can be interpreted as the entanglement of colour and Lorentz symmetry degrees of freedom. In the notation (93) basic Γ-matrices (37) derived in Section 3 can be denoted as In order to get a closed formula for the adjoint action S we should introduce the following pairs of Γ µ -matrices where we have chosen in (93) α = 3 and β = 2. Although for any choice of the first factor I A in Γ µ (A;α) 's (see 92) we have the boosts K It follows from (96), (97) that the standard Lorentz covariance requires the pair of coloured Dirac equations described by the doublet (Γ µ ,Γ µ ) of coloured Dirac matrices (see 95), which we shall call "Lorentz doublets". In particular, the Γ µ matrices (37) from Sect. 3 should be supplemented by the following Lorentz doublet partner: Further we will show that the Lorentz doublets of Γ µ -matrices required by the standard Lorentz covariance can be useful for the description of weak isospin (flavour) doublets of the SU (2)×U (1) electroweak symmetry. In such a way one can show that the internal symmetries SU describes the Γ µ -matrix (37) and respectively,Γ µ , its doublet partner (98). By calculating the multicommutators of J (1) i , K (1) l ∈ L (1) with the set Γ µ (a) , (a = 1, 2...6), we will show that the following sextet of Γ-matrices which break the Lorentz covariance is closed under the action of L (1) : It is easy to see that from the six components of the sextet (99) one can construct as well the set of six Γ µ -matrices Γ µ (A;α) , A = 2, 4, 8 and α = 2, 3, which can be described as well as three Lorentz doublets (95), with (A, B) = (2, 8), (2,4) and (4,8).. More explicitly, where (I A , I B = Q 2 , Q † 1 , B † ) and α = 2, 3. The construction of the Z 3 -graded Lorentz generators J (r) k , K (s) m employed as first tensorial factor the 3 × 3 matrices l 1 3 for r, s = 0, Q 3 for r, s = 1 and Q † 3 for r, s = 2. From the remaining six generators of the SU (3) Lie algebra in the Kac basis, only three do appear in the sextet (100). In order to implement the full SU (3) colour symmetry, the remaining matrices Q 1 , Q † 2 and B should be included in the module on which acts via commutation the spinorial representation of the Z 3 -graded Lorentz algebra. This means that the following sextet should be also taken into consideration, obtained by replacing the matrices I A by their complex conjugates, and keeping the remaining tensorial factors unchanged: The realization of L (2) sector acting on colour Γ µ matrices is obtained by introducing the Hermitean-conjugate sextet Γ µ linearly related with the tilded Γ-matricesΓ µ (ȧ) = (Γ µ (a) ) † which are required by standard Lorentz covariance described by the grade 0 sector L (0) (see (83), (84) and (87), (88)).

Lorentz doublets and classical Lorentz symmetrysector L (0)
The action of zero-grade rotation generators J (0) i on coloured matrices Γ µ is described by the eq. (96). In particular, the space rotations leave the temporal component Γ 0 (A;α) invariant, and transform the space components as the coordinates of a D = 3 three-vector, while the commutators of boosts K (0) i with (Γ 0 (A,α) , Γ i (B;β) generate new Γ µ -matrices which permit to introduce the "Lorentz partners".
Let us start with the first "standard" choice of colour Γ µ -matrices (see 93) When iterated, the commutators of boosts K (0) i with Γ µ (1) -matrices yields the following result: Apparently, we obtain a classical Lorentz doublet Γ µ , (see 99). It appears that in an analogous manner one can introduce classical Lorentz doublets for each colour Γ µ -matrix listed in (99) by adding to , where b = a + 4 (mod 6) and c = a + 2 (mod 6).

Sextet of colour Γ µ -matrices and representations of
Calculating the commutators of matrices Γ µ (a) with the generators (J m ∈ L (0) was rather easy, because the only non-commuting tensorial factors were the 3 × 3 "colour" matrices, while in the remaining two Z 2 × Z 2 factors matrices σ i commuted with the 2 × 2 unit matrices. However, when we consider the commutators of the operators ( K m ), r, s = 1, 2 with two colour Dirac matrices Γ µ (1) , Γ µ (2) defined above, we generate subsequently new commutators we need to calculate.
Let us observe how the new set (99) of Γ µ -matrices is produced. Calculating the commutators with the grade 1 generators we use the multiplication rule for tensor products of matrices: with a·c and b·d denoting ordinary matrix multiplication. The following formula will be helpful in our calculations: We recall also the well known identities involving Pauli's σ-matrices: Let us start with grade 1 rotations acting on Γ µ (1) = (Γ 0 (8;3) , Γ i (2;2) ), forming the 12 × 12 matrix valued four-vector (37) appearing in the colour Dirac equation (36): With the use of the rules of matrix multiplication of tensor products, we arrive at the following sequences of commutators: where we use the following shortened notation for the coefficients appearing on the right-hand side: We see that with the relations (110), the six commutators with J (1) i close on the following 36-component multiplet of obtained from six colour Γ µ (a) matrices: which describes the following triplet of Γ µ -matrices given by formula (100) with α = 3: In a short-hand notation the relations (110) look as follows: Now let us generate a new sequence of commutators starting from Γ k (2;2) : The relations (117) can be also expressed with a short-hand notation as follows We see that all 72 components of the sextet (106) are needed in order to obtain the irreducible representation closed under the action of the boost generators K (1) i . The pattern of the coefficients appearing on the right-hand side of these 12 commutators bears the imprint of the underlying Z 3 × Z 2 symmetry. The six commutators of K (1) i with time-like components of Γ-matrices produce only the space-like components, multiplied by halves of all sixth-order roots of unity, i.e. ± 1 2 , ± j 2 , ± j 2 2 , while the commutators with spatial components Γ k contain again the spatial components, multiplied by the coefficients ± α 2 , ± β 2 , ± γ 2 and time-like components Γ 0 (A;α) with the coefficients ± 1 2 , ± j 2 , ± j 2 2 , The full multiplication table of this Lie algebra over complex roots together with the diagram showing the structure constants on the complex plane are given in the Appendix I.
It is worth to observe that in the definitions (99) of the basic sextet Γ µ in the colour sector enters only the following triplet of colour generators (I 2 , I 4 , I 8 ) = (Q 2 , Q † 1 , B † ) which satisfies the following relations (see also the Table of commutators in Appendix I): The closure of the action of Q 3 on the multiplet (Q 2 , Q † 1 , B † ) leads to the covariance of the 72-dimensional multiplet (102) under the action of the gener- (1) , which do contain the matrix Q 3 as their first colour factor (see (91, 89)).
It can be recalled (see 102) that in order to construct the Lorentz doublets (Γ µ (a) ,Γ µ (a) ) (a = 1, 2, ..., 6) it is sufficient to use the components of the sextet of matrices Γ µ (a) suffice (see (106), and again the relations (119) imply the closure ofΓ µ (a) under the actions of generators belonging to L (1) .

Representations of Z 3 -graded Lorentz algebra -sector
Because from (119) follows the closure of the triplet under the action of Q † 3 , (compare with (6)) one can as well reproduce the covariant action of L (2) on the Hermitean-conjugate doublets of Γ µ -matrices.
The general pattern of commutators in (126)-(128) better explains why the irreducible representation is described by the sextet of colour Γ µ matrices. The generators J m contain as their 3×3 matrix factors the elements Q 3 and Q † 3 , which therefore cannot appear in the coloured Γ µ -matrices; the boosts contain as their second factor the matrix σ 1 , which as well can not appear in the sextet (99). Starting from the first "standard" colour Dirac operator whose Γ-matrices contain B † and Q 2 , commutators with Q 3 and Q † 3 can generate in the colour sector only the third colour matrix Q † 1 , besides B † and Q 2 . This reduces the number of Γ µ matrices spanning the spinorial realization of the Z 3 -graded Lorentz algebra to six, characterized by three colour matrices I A (A = 2, 4, 8) and two Pauli matrices σ α (α = 2, 3).
If we start with complex conjugate Dirac operator (36), (37) with Γ 0 = B ⊗ σ 3 ⊗ l 1 2 and Γ i = Q 1 ⊗ (iσ 2 ) ⊗ σ i (note that B is the complex conjugate of B † and Q 1 is the complex conjugate of Q 2 ), we get the alternative sextet describing coloured Dirac equations for the complex-conjugated fieldsΨ (see (40)), which contains as its colour factors the matrices Q † 2 , Q 1 and B.
7 Irreducible realizations of the Z 3 -graded Lorentz algebra and the full set of quark symmetries

Chiral colour doublets and flavour states
The flavour quark eigenstates in the Standard Model are represented by chirally projected Dirac spinors (see e.g. [25], [26]). If we introduce the D = 3+1 Clifford algebra defined by the relations and define then the standard chiral Dirac spinors are defined as The ψ ± denote the four-component Dirac spinors satisfying the chirality conditions P ± ψ ± = ψ ± , P ∓ ψ ± = 0, where P ± = 1 2 (l 1 2 ±iγ 5 ) are the chiral projection operators.

The flavour and generations in quark models and chiral colour Dirac multiplets
Three generations of quarks (called also "three families") are known, each formed by a flavour ("weak isospin") doublet: • (u, d), or the "up -down" doublet ("First generation"); • (s, c), or the "strange -charm" doublet ("Second generation"); • (t, b), or the "top -bottom" doublet ("Third generation"); The flavour SU (2) symmetry is visible only if we consider chiral (left-handed) quarks, described by the doublet (u, d). In the case of 2 nd and 3 rd generation, possible flavour symmetries exchanging c with s or t with b are strongly violated and the internal symmetry is used mostly for the classification purposes. 3 It is well established that the first generation splits into an SU (2) chiral flavour doublet u + , d + and a pair of anti-chiral flavour singlets u − and d − . In order to introduce three quark generations as described by second (non-colour) SU (3) internal symmetry, one should also assume analogous chiral structures for the doublets (s, c) and (t, b). However, in phenomenological Lagrangeans, if we take into consideration quark interactions without imposing a priori chiral structure of Feynman graph vortices, it appears quite reasonable to keep the doublets (s, c) and (t, b) as non-chiral ones (see e.g. [26]) Having introduced chiral and anti-chiral colour Dirac matrices, we can define respective 12-component chiral colour Dirac spinors and corresponding chiral colour Dirac equations (see (22)). The chiral and anti-chiral states can be formed by pairs of quarks (s, c) and (t, b) of other generations, i.e. one can use the same scheme for the doublets in each of three generations.
The chiral structure of the flavour sector of the Standard Model becomes important when we consider together leptons and quarks, with leptons interacting weakly as a kind of fourth colour (see e.g. ( [27])). In our model one can introduce leptons as colourless quarks just by replacing the 3 × 2 matricees B and Q a appearing in the tensor products defining the generalized 12 × 12 Dirac matices by the 3 × 3 unit matrix.
In the Standard Model the fact that leptons and coloured quarks are coupled weakly in analogous way leads to an imporant feature of the chiral anomaly cancellation. We hope that in the next stage of development of our model with interaction vertices introduced, such a cancellation mechanism can be also naturally achieved.

Outlook
The Standard Model (SM) of elementary particles is without doubt very successful experimentally tested part of theoretical physics; however, its grouptheoretical structure still requires further investigations. The internal symmetries are the product of three unitary groups SU (3) × SU (2) × U (1), with chiral SU (2) sector describing weak interactions and basic role of colour SU (3) group describing strongly interacting gluons and quarks which are not observable as free asymptotic states. The full spectrum of quarks requires still another SU (3) symmetry due to appearance of quark generations, which should be interpreted with the help of an additional geometric structure, describing from group-theoretical point of view the full set of all quarks as given by irreducible 72-dimensional representations of a new group which intertwines Lorentz and colour symmetries.
Unifying efforts in the literature ((see e.g. [25], [28], [30], [31], [32]) went along various paths, with two basic ways of unification: the first preserves the tensor product structure of space-time and internal symmetries, while the other one is more radical, intertwining the relativistic and internal colour symmetries. A well-known example of secon type of unification scheme is provided by the known passage from bosonic symmetries to supersymmetries, which describe the supermultiplet containing commuting bosons and anti-commuting fermions both incorporated in one common Z 2 -graded algebraic structure.
In our paper we deal exclusively with fundamental quark degrees of freedom, described by the collection of anti-commuting fields, with triplets of quark states with three different colours, which permit the introduction of Z 3 -graded algebraic structure. The Z 3 -grading does not change the fermionic statistics of quarks, but leads to particular link between relativistic (Lorentz) symmetry and internal (colour) symmetries, which in colour Dirac equations cease to be described by a tensor product group structure. The colour generators are naturally expressed in a Z 3 -graded ternary basis (see Sect. 2) and the SU (3) colour symmetries provide all possible 3 × 3 matrix choices of ternary basis (see the end of Sect. 4). It is the Z 3 -covariance which in a field-theoretic description of quarks provides the passage from the three copies of 4-component standard Dirac fields to the 12-dimensional colour Dirac field (see Sect. 3).
The physical observation that the quarks are described by six colour triplets did lead us to the idea that one should look for an algebraic scheme which would provide a unifying 72-dimensional module incorporating all six colour triplets of fermions inside a single irreducible representation. We demonstrate in this paper how this goal can be achieved by introduction of the Z 3 -graded Lorentz symmetries, which can be extended to Z 3 -graded Poincaré group. We consider the vectorial representation of the Z 3 -graded Poincaré algebra in Sect. 4, and the spinorial representation of the Z 3 -graded Lorentz algebra in Sect. 5 and 6.
Usually the efforts to incorporate all existing quarks in a unique irreducible multiplet are restricted only to the discussion of internal degrees of freedom. Our approach, which leads to a Z 3 -graded extension of standard relativistic symmetries, implies as well the modification of quark dynamics, what can be seen already from the wave equations for free quarks which imply dispersion relations of the sixth order, satisfied by all the components of the sextet of free Z 3 -graded quark fields. The dispersion relations can be described as a triple product of mass shells, one with real mass and a pair of mutually conjugate complex ones, which lead to the appearance of complex wave vectors (see Sect. 3 and 4) and provide damped exponential solutions along with freely propagating waves.
Our model is formulated only on the preliminary kinematic level, without defining neither the Lagrangean, nor the interaction vertices. Still on the kinematic level it is important to describe besides quarks, also the leptons and gauge fields. Before we proceed further we observe that: i) One should complete the quantum field-theoretic description of free quantum Z 3 -graded fermionic quark fields, with algebra of field oscillators and Green function.
In this paper we provided only the formulae for the quark propagators (see (47), (50), (51)). These formulae follow from the decomposition of basic quantum quark field satisfying sixth order equation into three Klein-Gordon like fields with Z 3 -graded set of complex masses m s = j s m, (s = 0, 1, 2). Further one should consider the Z 3 -covariant set of quantized free Klein-Gordon fields with respective oscillator algebras describing the field quanta that lead to three residua (1, j, j 2 ) of three propagators in eq. (50). These residua describe three respective metrics in the Hilbert-Fock spaces associated with our three Klein-Gordon type free quantum fields.
ii) One should provide the Z 3 -covariant interaction vertices, in particular the prescriptions for gauge field couplings. In order to obtain such results we should study the role of Z 3 -graded Lorentz transformations in space-time ( Sect. 3 and 4 were restricted only to the four-momentum space).
In order to be able to construct the action density and covariantize the space-time derivatives by introducing gauge fields we should find out how the Z 3 -graded Lorentz and Poincaré generators act on the space-time coordinates. This is going to be the subject of our future research.
The mapping between the Cartan subalgebras, B and B † on one side and λ 3 and λ 8 on the other side, is given by the following linear combinations: or more explicitly,