Relations between Clifford algebra and Dirac matrices in the presence of families

The internal degrees of freedom of fermions are in the spin-charge-family theory described by the Clifford algebra objects, which are superposition of an odd number of $\gamma^a$'s. Arranged into irreducible representations of"eigenvectors"of the Cartan subalgebra of the Lorentz algebra $S^{ab}$ $(= \frac{i}{2} \gamma^a \gamma^b|_{a \ne b})$ these objects form $2^{\frac{d}{2}-1}$ families with $2^{\frac{d}{2}-1}$ family members each. Family members of each family offer the description of all the observed quarks and leptons and antiquarks and antileptons, appearing in families. Families are reachable by $\tilde{S}^{ab}$ $=\frac{1}{2} \tilde{\gamma}^a \tilde{\gamma}^b|_{a \ne b}$. Creation operators, carrying the family member and family quantum numbers form the basic vectors. The action of the operators $\gamma^a$'s, $S^{ab}$, $\tilde{\gamma}^a$'s and $\tilde{S}^{ab}$, applying on the basic vectors, manifests as matrices. In this paper the basic vectors in $d=(3+1)$ Clifford space are discussed, chosen in a way that the matrix representations of $\gamma^a$ and of $S^{ab}$ coincide for each family quantum number, determined by $\tilde{S}^{ab} $, with the Dirac matrices. The appearance of charges in Clifford space is discussed by embedding $d=(3+1)$ space into $d=(5+1)$-dimensional space.


I. INTRODUCTION
In the Grassmann graded algebra of anticommuting coordinates θ a there are in ddimensional space 2 d vectors, which define, together with the corresponding derivatives ∂ ∂θa , two kinds of the Clifford algebra objects: γ a andγ a [2,[6][7][8], both with the anticommutation properties of the Dirac γ a matrices, while the anticommutators among γ a andγ b are equal to zero.
The two Clifford algebras, γ a 's andγ a 's, are obviously completely independent and form two independent spaces, each with 2 d vectors [9].
Sacrificing the space ofγ a 's by defining with (−) B = −1, if B is an odd product of γ a 's, otherwise (−) B = 1 [7], we end up with vector space of 2 d degrees of freedom, defined by γ a 's only.
A general vector can correspondingly be written as where |ψ o > is the vacuum state.
We arrange these vectors as products of nilpotents and projectors ab (k): where k 2 = η aa η bb . Their Hermitian conjugated values follow from Eq. (1).
Vectors in Clifford space are chosen to be eigenstates of the Cartan subalgebra, Eq. (6), of the generators of the Lorentz transformations S ab in the internal space of γ a 's S 03 , S 12 , S 56 , · · · , S d−1 d , S 03 ,S 12 ,S 56 , · · · ,S d−1 d , Eq. (7) demonstrates that the eigenvalues of S ab on nilpotents and projectors generated by γ a 's differ from the eigenvalues ofS ab . Nilpotents are the superposition of odd number of γ a 's, projectors have an even Clifford character.
It is interesting to notice ( [8,9] and references therein): Vectors, which are superposition of an odd products of γ a 's, anticommute. Half of them can be taken as creation operators and the other half as annihilation operators. These creation and annihilation operators then fulfill the anticommutation relations postulated by Dirac [1] for second quantized fermions, and consequently explain [8,9] the Dirac's postulate.
In Sect. II the properties of products of nilpotents and projectors are discussed, arranged into eigenvectors of the Cartan subalgebra, when d = (3 + 1)-dimensional space is embedded into d = (5+1)-dimensional space. Nilpotents and projectors define the internal vector space of fermions so that the spin in d = (5, 6) manifests as a charge of fermions in d = (3 + 1).
In Sect. II C the matrix representation of vectors are presented.

II. PROPERTIES OF VECTORS IN CLIFFORD SPACE
In Refs. [8,9] the fact that the Clifford vectors, spanned by products of an odd number of γ a 's, are fulfilling the anticommutation relations postulated by Dirac for the second quantized fermions, are discussed. Let us illustrate how this happens in the case that d = (5 + 1).
Let us denote vectors in d = (5 + 1) of an odd Clifford character (they are superposition of an odd products of γ a 's), presented in Table I as products of nilpotents and projectors, bŷ b f † m (the third column on Table I). The member quantum number m = (ch, s) includes the charge ch and the spin s, the charge concerns the eigenvalue of S 56 , the spin the eigenvalue of S 12 . The corresponding Hermitian conjugated partner (the fourth column on Table I The first member m = ( 1 2 , 1 2 ) of the first family a, which is the product of three nilpotents, is correspondingly denoted byb a †    Table I The relations among creation and annihilation operators in Eq. (11) fulfill all the Dirac's requirements for the second quantized fermions.

A. Action
The Lorentz invariant action for a free massless fermion, describing internal degrees of freedom in Clifford space, is well known It leads to the Weyl equations of motion which fulfill also the Klein-Gordon equation γ 0 appears in the action to take care of the Lorentz invariance of the action.
Solutions of equations of motion, Eq. (13), for a free massless fermion with momentum p a = (|p 0 |, p 1 , p 2 , p 3 , 0, 0) and a particular charge 1 2 , are for any family f superposition of basic vectors |ψ m f >=b f † m |ψ o > with spin 1 2 and spin − 1 2 , both multiplied by e −i(p 0 x 0 − p x) , (see Eq. (97) in Ref. [8]). Coefficients in the superposition depend on the momentum p a .

B. Creation and annihilation operators in
The creation and annihilation operators of Table I are all of an odd Clifford character (they are superposition of odd products of γ a 's). The rest of the 2 4 creation operators of an even Clifford character can be found in Refs. [8,9].
Taking into account Eq. (1) one recognizes that γ a 's transform   or projector so that they are of an odd Clifford character), they "gain" charges as presented in Table I.
With the knowledge presented in Eq. (15) it is not difficult to reproduce Table II Table I.
The properties of vectors of Table II are analyzed in details in order that the correspondence with the Dirac γ matrices in d = (3 + 1) space is easy to recognize. Superposition of vectors with the spin ± 1 2 (either Clifford even or odd) solve the equations of motion, Eq. (13), for free massless fermions.
As seen in Table II    There are 2 4 = 16 basic vectors in d = (3 + 1), presented in Table II. They all can be found as well as a part of states in Table I with either nilpotent or projector, expressing the charge, added so that each state has an odd Clifford character belonging to one of 16 vectors of oddI in Table I. We make a choice of products of nilpotents and projectors, which are eigenstates of the Cartan subalgebra operators, Eq. (6), as presented in Eqs. (7).
The family members of a family are reachable by either S ab or by γ a , and represent twice two vectors of definite handedness Γ (d) in d = (3 + 1). Different families are reachable by eitherS ab or byγ a . Each state carries correspondingly quantum numbers of the two kinds of the Cartan subalgebra. In Table II also Γ (3+1) (= −4iS 03 S 12 ) andΓ (3+1) (= −4iS 03 S 12 ) are presented. Let us again point out that if we treat all the basic vectors in d = (3 + 1) as a part of vectors in d = (5 + 1), all of an odd Clifford character, so that they carry also a charge which is the spin S 56 , then the family members of a family are reachable by S ab only and families byS ab only.
When the basic vectors are chosen and Table II is made it is not difficult to find the matrix representations for the operators (γ a , S ab ,γ a ,S ab , Γ (3+1) ,Γ (3+1) ). They are obviously 16×16 matrices with a 4 × 4 diagonal or off diagonal or partly diagonal and partly off diagonal substructure.
Let us define, to simplify the notation, the unit 4 × 4 submatrix and the submatrix with all the matrix elements equal to zero as follows We also use (2 × 2) Pauli matrices It is easy to find the matrix representations for γ 0 , γ 1 , γ 2 and γ 3 from Table II γ manifesting the 4 × 4 substructure along the diagonal of 16 × 16 matrices.
The representations of theγ a do not appear in the Dirac case. They manifest the off diagonal structure as follows The operatorsS ab have again off diagonal 4 × 4 substructure, exceptS 03 andS 12 , which are diagonal.

III. CONCLUSIONS
We present in this contribution the matrix representations of operators, γ a 's, S ab 's,γ a 's, One can also observe the appearance of families, used in the spin-charge-family theory to explain families of quarks and leptons [3][4][5], when the Clifford space in d = (3 + 1) is embedded into d = (13 + 1).
There are 2 4 = 16 basic vectors in d = (3 + 1) and correspondingly all the matrices have dimension 16 × 16, which are for the operators, determined by γ a 's, by diagonal and for the operators, determined byγ a 's, off diagonal, exceptS 03 ,S 12 , which are the members of the Cartan subalgebra and correspondingly alsoΓ (3+1) = −4iS 03S12 . We keep the Clifford odd and the Clifford even vectors as the basic vectors. We treat in the Clifford odd part the creation and annihilation operators as they would all define the vector space, to point out, that if space of d = (3 + 1) is embedded in d ≥ 6, all the parts, even and odd, contribute to the enlarged vector space as factors.