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Clifford Algebras and Lorentz Group

Finite-dimensional representations of the proper orthochronous Lorentz group are studied in terms of spinor representations of the Clifford algebras. The Clifford algebras are understood as an `algebraic covering' of a full system of the finite-dimensional representations of the Lorentz group. Space-time discrete symmetries P, T and PT, represented by fundamental automorphisms of the Clifford algebras, are defined on all the representation spaces. Real, complex, quaternionic and octonionic representations of the Lorentz group are considered. Physical fields of the different types are formulated within such representations. The Atiyah-Bott-Shapiro periodicity is defined on the Lorentz group. It is shown that modulo 2 and modulo 8 periodicities of the Clifford algebras allow to take a new look at the de Broglie-Jordan neutrino theory of light and the Gell-Mann-Ne'emann eightfold way in particle physics. On the representation spaces the charge conjugation C is represented by a pseudoautomorphism of the complex Clifford algebra. Quotient representations of the Lorentz group are introduced. It is shown that quotient representations are the most suitable for description of the massless physical fields. By way of example, neutrino field is described via the simplest quotient representation. Weyl-Hestenes equations for neutrino field are given.


Introduction
Importance of discrete transformations is well-known, many textbooks on quantum theory began with description of the discrete symmetries, and famous Lüders-Pauli CP T -Theorem is a keystone of quantum field theory. However, usual practice of definition of the discrete symmetries from the analysis of relativistic wave equations does not give a full and consistent theory of the discrete transformations. In the standard approach, except a well studied case of the spin j = 1/2 (Dirac equation), a situation with the discrete symmetries remains vague for the fields of higher spin j > 1/2. It is obvious that a main reason of this is an absence of a fully adequate formalism for description of higher-spin fields (all widely accepted higher-spin formalisms such as Rarita-Schwinger approach [51], Bargmann-Wigner [6] and Gel'fand-Yaglom [27] multispinor theories, and also Joos-Weinberg 2(2j+1)-component formalism [36,66] have many intrinsic contradictions and difficulties). The first attempt of going out from this situation was initiated by Gel'fand, Minlos and Shapiro in 1958 [29]. In the Gel'fand-Minlos-Shapiro approach the discrete symmetries are represented by outer involutory automorphisms of the Lorentz group (there are also other realizations of the discrete symmetries via the outer automorphisms [45,40,58]). At present the Gel'fand-Minlos-Shapiro ideas have been found further development in the works of Buchbinder, Gitman and Shelepin [14,30], where the discrete symmetries are represented by both outer and inner automorphisms of the Poincaré group.
In 1957, Shirokov pointed out [57] that an universal covering of the inhomogeneous Lorentz group has eight inequivalent realizations. Later on, in the eighties this idea was applied to a general orthogonal group O(p, q) by Dabrowski [18]. As known, the orthogonal group O(p, q) of the real space R p,q is represented by the semidirect product of a connected component O 0 (p, q) and a discrete subgroup {1, P, T, P T }. Further, a double covering of the orthogonal group O(p, q) is a Clifford-Lipschitz group Pin(p, q) which is completely constructed within a Clifford algebra Cℓ p,q . In accordance with squares of elements of the discrete subgroup (a = P 2 , b = T 2 , c = (P T ) 2 ) there exist eight double coverings (Dabrowski groups [18]) of the orthogonal group defining by the signatures (a, b, c), where a, b, c ∈ {−, +}. Such in brief is a standard description scheme of the discrete transformations. However, in this scheme there is one essential flaw. Namely, the Clifford-Lipschitz group is an intrinsic notion of the algebra Cℓ p,q (a set of the all invertible elements of Cℓ p,q ), whereas the discrete subgroup is introduced into the standard scheme in an external way, and the choice of the signature (a, b, c) of the discrete subgroup is not determined by the signature of the space R p,q . Moreover, it is suggest by default that for any signature (p, q) of the vector space there exist the all eight kinds of the discrete subgroups. It is obvious that a consistent description of the double coverings of O(p, q) in terms of the Clifford-Lipschitz groups Pin(p, q) ⊂ Cℓ p,q can be obtained only in the case when the discrete subgroup {1, P, T, P T } is also defined within the algebra Cℓ p,q . Such a description has been given in the works [60,61,62], where the discrete symmetries are represented by fundamental automorphisms of the Clifford algebras. Moreover, this description allows to incorporate the Gel'fand-Minlos-Shapiro automorphism theory into Shirokov-Dabrowski scheme and further to unite them on the basis of the Clifford algebras theory.
In the present paper such an unification is given. First of all, Clifford algebras are understood as 'algebraic coverings' of finite-dimensional representations of the proper Lorentz group G + . In the section 2 Clifford algebras C n over the field F = C are associated with complex finite-dimensional representations C of the group G + . It allows to define a new class of the finite-dimensional representations of G + (quotient representations) corresponded to the type n ≡ 1 (mod 2) of the algebras C n . In its turn, representation spaces of C are the spinspaces S 2 n/2 or the minimal left ideals of the algebras C n . In virtue of this in the section 3 the discrete symmetries representing by spinor representations of the fundamental automorphisms of C n are defined for both complex and real finitedimensional representations of the group G + .
A full system M = M + ⊕M − of the finite-dimensional representations of the group G + allows to define in the section 4 the Atiyah-Bott-Shapiro periodicity [4] on the Lorentz group. In case of the field F = C we have modulo 2 periodicity on the representations C, C ∪ C, that allows to take a new look at the de Broglie-Jordan neutrino theory of light [11,37]. In its turn, over the field F = R we have on the system M the modulo 8 periodicity which relates with octonionic representations of the Lorentz group and the Günaydin-Gürsey construction of the quark structure in terms of an octonion algebra O [31]. In essence, the modulo 8 periodicity on the system M gives an another realization of the well-known Gell-Mann-Ne'emann eightfold way [26]. It should be noted here that a first attempt in this direction was initiated by Coquereaux in 1982 [17].
Other important discrete symmetry is the charge conjugation C. In contrast with the transformations P , T , P T the operation C is not space-time discrete symmetry. This transformation is firstly appearred on the representation spaces of the Lorentz group and its nature is strongly different from other discrete symmetries. By this reason in the section 5 the charge conjugation C is represented by a pseudoautomorphism A → A which is not fundamental automorphism of the Clifford algebra. All spinor representations of the pseudoautomorphism A → A are given in Theorem 3. Quotient representations of the group G + compose the second half M − of the full system M and correspond to the types n ≡ 1 (mod 2) (F = C) and p − q ≡ 1, 5 (mod 8) (F = R). An explicit form of the quotient representations is given in the section 6 (Theorem 4). In the section 7 the first simplest physical field (neutrino field), corresponded to a fundamental representation C 1,0 of the group G + , is studied within a quotient representation χ C 1,0 c ∪ χ C 0,−1 c . Such a description of the neutrino was firstly given in the work [60], but in [60] this description looks like an exotic case, whereas in the present paper it is a direct consequence of all mathematical background developed in the previous sections. It is shown also that the neutrino field (1/2, 0) ∪ (0, 1/2) can be defined in terms of a Dirac-Hestenes spinor field [32,33], and the wave function of this field satisfies the Weyl-Hestenes equations (massless Dirac-Hestenes equations).

Finite-dimensional representations of the Lorentz group and complex Clifford algebras
It is well-known [56,23,7]  . It should be noted that in accordance with [47] any finite-dimensional irreducible representation of the proper Lorentz group G + = O 0 (1, 3) ≃ O 0 (3, 1) ≃ SL(2; C)/Z 2 is equivalent to some spinor representation. Moreover, spinor representations exhaust in essence all the finite-dimensional representations of the group G + . This fact we will widely use below. Let us consider in brief the basic facts concerning the theory of spinor representa-tions of the Lorentz group. The initial point of this theory is a correspondence between transformations of the proper Lorentz group and complex matrices of the second order. Indeed, follows to [29] let us compare the Hermitian matrix of the second order to the vector v of the Minkowski space-time R 1,3 with coordinates x 0 , x 1 , x 2 , x 3 . At this point det X = x 2 0 − x 2 1 − x 2 2 − x 2 3 = S 2 (x). The correspondence between matrices X and vectors v is one-to-one and linear. Any linear transformation X ′ = aXa * in a space of the matrices X may be considered as a linear transformation g a in R 1,3 , where a is a complex matrix of the second order with det a = 1. The correspondence a ∼ g a possesses following properties: 1) 1 0 0 1 ∼ e (identity element); 2) g a 1 g a 2 = g a 1 a 2 (composition); 3) two different matrices a 1 and a 2 correspond to one and the same transformation g a 1 = g a 2 only in the case a 1 = −a 2 . Since every complex matrix is defined by eight real numbers, then from the requirement det a = 1 it follow two conditions Re det a = 1 and Im det a = 0. These conditions leave six independent parameters, that coincides with parameter number of the proper Lorentz group. Further, a set of all complex matrices of the second order forms a full matrix algebra M 2 (C) that is isomorphic to a biquaternion algebra C 2 . In its turn, Pauli matrices form one from a great number of isomorphic spinbasis of the algebra C 2 (by this reason in physics the algebra C 2 ≃ Cℓ + 1,3 ≃ Cℓ 3,0 is called Pauli algebra). Using the basis (2) we can write the matrix (1) in the form The Hermitian matrix (3) is correspond to a spintensor (1, 1) X λν with following coordinates where ξ µ and ξμ are correspondingly coordinates of spinors and cospinors of spinspaces S 2 andṠ 2 . Linear transformations of 'vectors' (spinors and cospinors) of the spinspaces S 2 andṠ 2 have the form Transformations (5) form the group SL(2; C), since σ ∈ M 2 (C) and The expressions (4) and (5) compose a base of the 2-spinor van der Waerden formalism [64,54], in which the spaces S 2 andṠ 2 are called correspondingly spaces of undotted and dotted spinors. The each of the spaces S 2 andṠ 2 is homeomorphic to an extended complex plane C ∪ ∞ representing an absolute (the set of infinitely distant points) of a Lobatchevskii space S 1,2 . At this point a group of fractional linear transformations of the plane C ∪ ∞ is isomorphic to a motion group of S 1,2 [53]. Besides, in accordance with [39] the Lobatchevskii space S 1,2 is an absolute of the Minkowski world R 1,3 and, therefore, the group of fractional linear transformations of the plane C ∪ ∞ (motion group of S 1,2 ) twice covers a 'rotation group' of the space-time R 1,3 , that is the proper Lorentz group.
Theorem 1. Let C 2 be a biquaternion algebra and let σ i be a canonical spinor representations (Pauli matrices) of the units of C 2 , then 2k tensor products of the k matrices σ i form a basis of the full matrix algebra M 2 k (C), which is a spinor representation of a complex Clifford algebra C 2k . The set containing 2k + 1 tensor products of the k matrices σ i is homomorphically mapped onto a set consisting of the same 2k tensor products and forming a basis of the spinor representation of a quotient algebra ǫ C 2k .
where the subset P 0 contains products with the matrix E 2k+1 , and P 1 contains products without the matrix E 2k+1 . Therefore, products E α 1 E α 2 2 · · · E α 2k 2k ⊂ P 1 form a full 2 kdimensional matrix algebra. Further, when we multiply the matrices from the subset P 0 by the matrix U the factors E 2k+1 are mutually annihilate. Thus, matrices of the set UP 0 are also belong to the 2 k -dimensional matrix algebra. Let us denote UP 0 via P 2 . Taking into account that U 2 = E 0 we obtain P 0 = UP 2 and where P 1 , P 2 ∈ M 2 k . Let be an homomorphic mapping of the set (14) containing the matrix E 2k+1 onto the set P 1 + P 2 which does not contain this matrix. The mapping χ preserves addition and multiplication operations. Indeed, let P = P 1 + UP 2 and Q = Q 1 + UQ 2 , then that is, an image of the products equals to the product of factor images in the same order.
In particular, when P = U we obtain P 1 = 0, P 2 = E 0 and In such a way, at the mapping χ all the matrices of the form 2k are mapped into zero. Therefore, a kernel of the homomorphism χ is defined by an expression Ker χ = {P 1 − UP 1 }. We obtain the homomorphic mapping of the set of all the 2 2k+1 matrices E α 1 2k+1 onto the full matrix algebra M 2 k . In the result of this mapping we have a quotient algebra χ M 2 k ≃ P/ Ker χ. As noted previously, 2k tensor products of the k Pauli matrices (or other k matrices defining spinor representation of the biquaternion algebra C 2 ) form a basis of the spinor representation of the algebra C 2k . It is easy to see that there exists one-to-one correspondence between 2 2k+1 matrices of the set P and basis elements of the odd-dimensional Clifford algebra C 2k+1 . It is well-known [52,49] that C 2k+1 is isomorphic to a direct sum of two even-dimensional subalgebras: C 2k+1 ≃ C 2k ⊕ C 2k . Moreover, there exists an homomorphic mapping ǫ : C 2k+1 → C 2k , in the result of which we have a quotient algebra ǫ C 2k ≃ C 2k+1 / Ker ǫ, where Ker ǫ = {A 1 − εωA 1 } is a kernel of the homomorphism ǫ, A 1 is an arbitrary element of the subalgebra C 2k , ω is a volume element of the algebra C 2k+1 . It is easy to see that the homomorphisms ǫ and χ have a similar structure. Thus, hence it immediately follows an isomorphism ǫ C 2k ≃ χ M 2 k . Therefore, a basis of the matrix quotient algebra χ M 2 k is also a basis of the spinor representation of the Clifford quotient algebra ǫ C 2k , that proves the latter assertion of the theorem.
Let us consider now spintensor representations of the proper Lorentz group O 0 (1, 3) ≃ SL(2; C)/Z 2 ≃ Spin + (1, 3)/Z 2 and their relations with the complex Clifford algebras. From each complex Clifford algebra C n = C ⊗ Cℓ p,q (n = p + q) we obtain a spinspace S 2 n/2 , which is a complexification of the minimal left ideal of the algebra Cℓ p,q : S 2 n/2 = C ⊗ I p,q = C ⊗ Cℓ p,q e pq , where e pq is a primitive idempotent of the algebra Cℓ p,q . Further, a spinspace corresponding the Pauli algebra C 2 has a form S 2 = C ⊗ I 2,0 = C ⊗ Cℓ 2,0 e 20 or S 2 = C ⊗ I 1,1 = C ⊗ Cℓ 1,1 e 11 (C ⊗ I 0,2 = C ⊗ Cℓ 0,2 e 02 ). Therefore, the tensor product (12) of the k algebras C 2 induces a tensor product of the k spinspaces S 2 : Vectors of the spinspace S 2 k (or elements of the minimal left ideal of C 2k ) are spintensors of the following form where summation is produced on all the index collections (α 1 . . . α k ), α i = 1, 2. In virtue of (5) for each spinor ξ α i from (18) we have a transformation rule A representation (19) is called undotted spintensor representation of the proper Lorentz group of the rank k.
Further, let * C 2 be a biquaternion algebra, the coefficients of which are complex conjugate. Let us show that the algebra * C 2 is obtained from C 2 under action of the automorphism A → A ⋆ or antiautomorphism A → A. Indeed, in virtue of an isomorphism C 2 ≃ Cℓ 3,0 a general element a ij e ij + a 123 e 123 of the algebra Cℓ 3,0 can be written in the form where ω = e 123 . Since ω belongs to a center of the algebra Cℓ 3,0 (commutes with all the basis elements) and ω 2 = −1, then we can to suppose ω ≡ i. The action of the automorphism ⋆ on the homogeneous element A of a degree k is defined by a formula A ⋆ = (−1) k A. In accordance with this the action of the automorphism A → A ⋆ , where A is the element (20), has a form Therefore, ⋆ : C 2 → − * C 2 . Correspondingly, the action of the antiautomorphism A → A on the homogeneous element A of a degree k is defined by a formula A = (−1) A. Thus, for the element (20) we obtain that is, : C 2 → * C 2 . This allows to define an algebraic analog of the Wigner's representation doubling: Further, from (20) it follows that A = A 1 + ωA 2 = (a 0 e 0 + a 1 e 1 + a 2 e 2 + a 3 e 3 ) + ω(a 123 e 0 + a 23 e 1 + a 31 e 2 + a 12 e 3 ). In general case, by virtue of an isomorphism C 2k ≃ Cℓ p,q , where Cℓ p,q is a real Clifford algebra with a division ring K ≃ C, p − q ≡ 3, 7 (mod 8), we have for a general element of Cℓ p,q an expression A = A 1 + ωA 2 , here ω 2 = e 2 12...p+q = −1 and, therefore, ω ≡ i. Thus, from C 2k under action of the automorphism A → A ⋆ we obtain a general algebraic doubling Correspondingly, a tensor product * C 2 ⊗ * C 2 ⊗ · · · ⊗ * C 2 ≃ * C 2r of r algebras * C 2 induces a tensor product of r spinspacesṠ 2 : Te vectors of the spinspaceṠ 2 r have the form where the each cospinor ξα i from (24) in virtue of (5) is transformed by the rule ′ ξα A representation (25) is called a dotted spintensor representation of the proper Lorentz group of the rank r.
In general case we have a tensor product of k algebras C 2 and r algebras * C 2 : which induces a spinspace In this case we have a natural unification of the representations (19) and (25): So, a representation (27) is called a spintensor representation of the proper Lorentz group of the rank (k, r).
Further, an important notion of the physical field is closely related with finitedimensional representations of the proper Lorentz group G + . In accordance with Wigner interpretation [67], an elementary particle is described by some irreducible finite-dimensional representation of the Poincaré group. The double covering of the proper Poincaré group is isomorphic to a semidirect product SL(2; C) ⊙ T (4), or Spin + (1, 3) ⊙ T (4), where T (4) is the subgroup of four-dimensional translations. Let ψ(x) be a physical field, then at the transformations (a, Λ) of the proper Poincaré group the field ψ(x) is transformed by a following rule where a ∈ T (4), σ ∈ G + , Λ is a Lorentz transformation, and C µν is a representation of the group G + in the space S 2 k+r . Since the group T (4) is Abelian, then all its representations are one-dimensional. Thus, all the finite-dimensional representations of the proper Poincaré group in essence are equivalent to the representations C of the group G + . If the representation C is reducible, then the space S 2 k+r is decomposed into a direct sum of irreducible subspaces, that is, it is possible to choose in S 2 k+r such a basis, in which all the matrices C µν take a block-diagonal form. Then the field ψ(x) is reduced to some number of the fields corresponding to obtained irreducible representations of the group G + , each of which is transformed independently from the other, and the field ψ(x) in this case is a collection of the fields with more simple structure. It is obvious that these more simple fields correspond to irreducible representations C. As known [47,29,56], a system of irreducible finite-dimensional representations of the group G + is realized in the space Sym (k,r) ⊂ S 2 k+r of symmetric spintensors. The dimensionality of Sym (k,r) is equal to (k + 1)(r + 1). A representation of the group G + by such spintensors is irreducible and denoted by the symbol C j,j ′ , where 2j = k, 2j ′ = r, numbers j and j ′ defining the spin are integer or half-integer. Then the field ψ(x) transforming by the formula (28) is, in general case, a field of the type (j, j ′ ). In such a way, all the physical fields are reduced to the fields of this type, the mathematical structure of which requires a knowledge of representation matrices C j,j ′ µν . As a rule, in physics there are two basic types of the fields: 1) The field of type (j, 0). The structure of this field (or the field (0, j)) is described by the representation C j,0 (C 0,j ′ ), which is realized in the space Sym (k,0) ⊂ S 2 k (Sym (0,r) ⊂ S 2 r ). At this point in accordance with Theorem 1 the algebra C 2k ≃ C 2 ⊗ C 2 ⊗ · · · ⊗ C 2 (correspondingly, * is associated with the field of type (j, 0) (correspondingly, (0, j ′ )) The trivial case j = 0 corresponds to a Pauli-Weisskopf field describing the scalar particles. In other particular case, when j = j ′ = 1/2 we have a Weyl field describing the neutrino. At this point the antineutrino is described by a fundamental representation C 1/2,0 = σ of the group G + and the algebra C 2 related with this representation (Theorem 1). Correspondingly, the neutrino is described by a conjugated representation C 0,1/2 and the algebra * C 2 . In relation with this, it is hardly too much to say that the neutrino field is a more fundamental physical field, that is a kind of the basic building block, from which other physical fields built by means of direct sum or tensor product.
2) The field of type (j, 0) ⊕ (0, j). The structure of this field admits a space inversion and, therefore, in accordance with a Wigner's doubling [68] is described by a representation C j,0 ⊕ C 0,j of the group G + . This representation is realized in the space Sym (k,k) ⊂ S 2 2k . In accordance with (23) the Clifford algebra related with this representation is a direct In the simplest case j = 1/2 we have bispinor (electron-positron) Dirac field (1/2, 0) ⊕ (0, 1/2) with the algebra C 2 ⊕ * C 2. It should be noted that the Dirac algebra C 4 considered as a tensor product C 2 ⊗ C 2 (or C 2 ⊗ * C 2 ) in accordance with (18) (or (26)) gives rise to spintensors ξ α 1 α 2 (or ξ α 1α1 ), but it contradicts with the usual definition of the Dirac bispinor as a pair (ξ α 1 , ξα 1 ). Therefore, the Clifford algebra associated with the Dirac field is C 2 ⊕ * C 2 , and a spinspace of this sum in virtue of unique decomposition S 2 ⊕Ṡ 2 = S 4 (S 4 is a spinspace of C 4 ) allows to define γ-matrices in the Weyl basis. The case j = 1 corresponds to Maxwell fields (1, 0) and (0, 1) with the algebras C 2 ⊗ C 2 and * C 2 ⊗ * C 2 . At this point the electromagnetic field is defined by complex linear combinations F = E − iH, * F = E + iH (Helmholtz representation). Besides, the algebra related with Maxwell field is a tensor product of the two algebras C 2 describing the neutrino fields. In this connection it is of interest to recall a neutrino theory of light was proposed by de Broglie and Jordan [11,37]. In the de Broglie-Jordan neutrino theory of light electromagnetic field is constructed from the two neutrino fields (for more details and related papers see [22]). Traditionally, physicists attempt to describe electromagnetic field in the framework of (1, 0) ⊕ (0, 1) representation (see old works [41,54,56] and recent developments based on the Joos-Weinberg formalism [36,66] and its relation with a Bargmann-Wightman-Wigner type quantum field theory [3,20]). However, Weinberg's equations (or Weinberg-like equations) for electromagnetic field obtained within the subspace Sym (k,r) with dimension 2(2j + 1) have acausal (tachionic) solutions [2]. Electromagnetic field in terms of a quotient representation (1, 0) ∪ (0, 1) in the full representation space S 2 k+r will be considered in separate paper.
In this connection it should be noted two important circumstances related with irreducible representations of the group G + and complex Clifford algebras associated with these representations. The first circumstance relates with the Wigner interpretation of an elementary particle. Namely, a relation between finite-dimensional representations of the proper Lorentz group and comlex Clifford algebras (Theorem 1) allows to essentially extend the Wigner interpretation by means of the use of an extraordinary rich and universal structure of the Clifford algebras at the study of space-time (and also intrinsic) symmetries of elementary particles. The second circumstance relates with the spin. Usually, the Clifford algebra is associated with a half-integer spin corresponding to fermionic fields, so-called 'matter fields', whilst the fields with an integer spin (bosonic fields) are eliminated from an algebraic description. However, such a non-symmetric situation is invalid, since the fields with integer spin have a natural description in terms of spintensor representations of the proper Lorentz group with even rank and algebras C 2k and * C 2k associated with these representations, where k is even (for example, Maxwell field). In this connection it should be noted that generalized statistics in terms of Clifford algebras have been recently proposed by Finkelstein and collaborators [25,8].
As known, complex Clifford algebras C n are modulo 2 periodic [4] and, therefore, there exist two types of C n : n ≡ 0 (mod 2) and n ≡ 1 (mod 2). Let us consider these two types in the form of following series: Let us consider the decomposition C 2k+1 ≃ C 2k ⊕ C 2k in more details. This decomposition may be represented by a following scheme Here central idempotents satisfy the relations (λ + ) 2 = λ + , (λ − ) 2 = λ − , λ + λ − = 0. Thus, we have a decomposition of the initial algebra C 2k+1 into a direct sum of two mutually annihilating simple ideals: Each of the ideals λ ± C 2k+1 is isomorphic to the subalgebra C 2k ⊂ C 2k+1 . In accordance with Chisholm and Farwell [16] the idempotents λ + and λ − can be identified with helicity projection operators which distinguish left and right handed spinors. The Chisholm-Farwell notation for λ ± we will widely use below.Therefore, in virtue of the isomorphism C 2k+1 ≃ C 2k ∪ C 2k and the homomorphic mapping ǫ : C 2k+1 → C 2k the second series (type n ≡ 1 (mod 2)) is replaced by a sequence of the quotient algebras ǫ C 2k , that is, Representations corresponded these two series of C n (n ≡ 0, 1 (mod 2)) form a full system M = M 0 ⊕ M 1 of finite-dimensional representations of the proper Lorentz group G + . All the physical fields used in quantum field theory and related representations of the group G + C j,0 (C 2k ), C j,0 ⊕ C 0,j (C 2k ⊗ * C 2k ) are constructed from the upper series (type n ≡ 0 (mod 2)). Whilst the lower series (type n ≡ 1 (mod 2)) is not considered in physics as yet. In accordance with Theorem 1 we have an isomorphism ǫ C 2k ≃ χ M 2 k . Therefore, the quotient algebra ǫ C 2k induces a spinspace ǫ S 2 k that is a space of a quotient representation χ C j,0 of the group G + . Analogously, a quotient representation χ C 0,j ′ is realised in the space ǫṠ 2 r which induced by the quotient algebra ǫ * C 2r . In general case, we have a quotient representation χ C j,j ′ defined by a tensor product ǫ C 2k ⊗ ǫ * C 2r . Thus, the complex type n ≡ 1 (mod 2) corresponds to a full system of irreducible finite-dimensional quotient representations χ C j,j ′ of the proper Lorentz group. Therefore, until now in physics only one half (n ≡ 0 (mod 2)) of all possible finite-dimensional representations of the Lorentz group has been used.
Let us consider now a full system of physical fields with different types. First of all, the field (j, 0) = (1/2, 0) ⊗ (1/2, 0) ⊗ · · · ⊗ (1/2, 0) (29) in accordance with Theorem 1 is a tensor product of the k fields of type (1/2, 0), each of which corresponds to the fundamental representation C 1/2,0 = σ of the group G + and the biquaternion algebra C 2 related with fundamental representation. In its turn, the field is a tensor product of the r fundamental fields of the type (0, 1/2), each of which corresponds to a conjugated representation C 0,1/2 =σ and the conjugated algebra * C 2 obtained in accordance with (21)- (22) under action of the automorphism A → A ⋆ (space inversion), or under action of the antiautomorphism A → A (time reversal). The numbers j and j ′ are integer (bosonic fields) if in the products (29) and (30) there are k, r ≡ 0 (mod 2) (1/2, 0) (or (0, 1/2)) factors, and the numbers j and j ′ are half-integer (fermionic fields) if in the products (29)- (30) there are k, r ≡ 1 (mod 2) factors. Further, the field is a tensor product of the fields (29) and (30). As consequence of the doubling (23) we have the field of type (j, 0) ⊕ (0, j): In general, all the fields (29)-(31) describe multiparticle states. The decompositions of these multiparticle states into single states provided in the full representation space S 2 k+r , where Sym (k,r) and Sym (k,k) with dimensions (2j +1)(2j ′ +1) and 2(2j +1) are subspaces of S 2 k+r (for example, the Clebsh-Gordan decomposition of two spin 1/2 particles into singlet and triplet: ( 0)). In the papers [34,19,59] a multiparticle state is described in the framework of a tensor product Cℓ 3,0 ⊗ Cℓ 3,0 ⊗ · · · ⊗ Cℓ 3,0 . It is easy to see that in virtue of the isomorphism Cℓ 3,0 ≃ C 2 the tensor product of the algebras Cℓ 3,0 is isomorphic to the product (12). Therefore, the Holland approach naturally incorporates into a more general scheme considered here. Finally, for the type n ≡ 1 (mod 2) we have quotient representations χ C j,j ′ of the group G + . The physical fields corresponding to the quotient representations are constructed like the fields (29)- (31). Due to the decomposition C n ≃ C n−1 ∪ C n−1 (n ≡ 1 (mod 2)) we have a field and also we have fields (0, j) ∪ (0, j) and (j, 0) ∪ (0, j) if the quotient algebras ǫ C n−1 admit space inversion or time reversal. The field is analogous to the field (j, 0) ⊕ (0, j), but, in general, the field (j, 0) ∪ (0, j) has a quantity of violated discrete symmetries. An explicit form of the quotient representations and their relations with discrete symmetries will be explored in the following sections.
Further, in the case P 2 = T 2 = (P T ) 2 = ±1 and P T = −T P there is an isomorphism between the group {1, P, T, P T } and automorphism group Aut(Cℓ) = {I, W, E, C}. Spinor representations of the fundamental automorphisms of the algebras C n was first obtained by Rashevskii in 1955 [52]: 1) Involution: A ⋆ = WAW −1 , where W is a matrix of the automorphism ⋆ (matrix representation of the volume element ω); 2) Reversion: C. So, for the Dirac algebra C 4 in the canonical γ-basis there exists a standard (Wigner) representation P = γ 0 and T = γ 1 γ 3 [9], therefore, {1, P, T, On the other hand, the automorphism group of the algebra C 4 for γ-basis has a form Aut In general case, according to Theorem 1 a space of the finite-dimensional representation of the group SL(2; C) is a spinspace S 2 k+r , or a minimal left ideal of the algebra C 2k ⊗ * C 2r . Therefore, in the spinor representation the fundamental automorphisms of the algebra C 2k ⊗ * C 2r (all spinor representations of the fundamental automorphisms have been found in [61]) by virtue of the isomorphism (34) induce discrete transformations on the representation spaces (spinspaces) of the Lorentz group.
Theorem 2. 1) The field F = C. The tensor products C 2 ⊗C 2 ⊗· · ·⊗C 2 , * Then in a spinor representation of the fundamental automorphisms of the algebra C n for the matrix W of the automorphism A → A ⋆ (space inversion) and also for the matrices E and C of the antiautomorphisms A → A (time reversal) and A → A ⋆ (full reflection) the following permutation relations with infinitesimal operators of the group G + take place: where A 23 , A 13 , A 12 are infinitesimal operators of a subgroup of three-dimensional rotations, B 1 , B 2 , B 3 are infinitesimal operators of hyperbolic rotations.
2) The field F = R. The factorization Cℓ s i ,t j ⊗ Cℓ s i ,t j ⊗ · · · ⊗ Cℓ s i ,t j of the real Clifford algebra Cℓ p,q corresponds to a real finite-dimensional representation of the group G + with a pair (l 0 , l 1 ) = p+q 4 , 0 , that is equivalent to a representation of the subgroup SO(3) of three-dimensional rotations (B 1 = B 2 = B 3 = 0). Then there exist two classes of real representations R l 0 0,2 of the group G + corresponding to the algebras Cℓ p,q with a division ring K ≃ R, p − q ≡ 0, 2 (mod 8), and also there exist two classes of quaternionic representations H l 0 4,6 of G + corresponding to the algebras Cℓ p,q with a ring K ≃ H, p − q ≡ 4, 6 (mod 8). For the real representations R l 0 0,2 operators of the discrete subgroup of G + defining by the matrices W, E, C of the fundamental automorphisms of Cℓ p,q with p − q ≡ 0, 2 (mod 8) are always commute with all the infinitesimal operators of the representation. In turn, for the quaternionic representations H l 0 4,6 following relations hold: Proof. 1) Complex representations. As noted previously, a full representation space of the finite-dimensional representation of the proper Lorentz group G + is defined in terms of the minimal left ideal of the algebra where Cℓ p,q is a Clifford algebra over the field F = R with a division ring K ≃ C, p−q ≡ 3, 7 (mod 8), we have for the minimal left ideal of C 2k an expression S = Cℓ p,q f , here is a primitive idempotent of the algebra Cℓ p,q [42], and e α 1 , e α 2 , . . . , e αt are commuting elements with square 1 of the canonical basis of The dimension of the minimal left ideal S is equal to 2 k = 2 p+q−1 2 . Therefore, for the each finite-dimensional representation of the group G + we have 2 t copies of the spinspace S 2 k (full representation space). It should be noted that not all these copies are equivalent to each other, some of them give rise to different reflection groups (see [61]).
In general, all the finite-dimensional representations of group G + in the spinspace S 2 k are reducible. Therefore, there exists a decomposition of the spinspace S 2 k ≃ S 2 ⊗ S 2 ⊗ · · · S 2 into a direct sum of invariant subspaces Sym (k j ,0) of symmetric spintensors with dimensions (k j + 1): (matrices of the infinitesimal operators of G + ) in accordance with Gel'fand-Naimark formulas [29,47] where α m = (l j + m)(l j − m + 1). The formulas (69)-(75) define a finite-dimensional representation of the group G + when l 2 1 = (l 0 + p) 2 , p is some natural number, l 0 is an integer or half-integer number, l 1 is an arbitrary complex number. In the case l 2 1 = (l 0 +p) 2 we have an infinite-dimensional representation of G + . We will deal below only with the finite-dimensional representations, because these representations are most useful in physics.
The relation between the numbers l 0 , l 1 and the number k of the factors C 2 in the product C 2 ⊗ C 2 ⊗ · · · ⊗ C 2 is given by a following formula whence it immediately follows that k = l 0 +l 1 −1. Thus, we have a complex representation C l 0 +l 1 −1,0 of the proper Lorentz group G + in the spinspace S 2 k . Let us calculate now infinitesimal operators of the fundamental representation C 1,0 of G + . The representation C 1,0 is defined by a pair (l 0 , l 1 ) = 1 2 , 3 2 . In accordance with (69)-(75) from (76)-(81) we obtain The operators (82)-(87) satisfy the relations From (85)-(87) it is easy to see that there is an equivalence between infinitesimal operators B 1/2 i and Pauli matrices: In its turn, from (82)-(84) it follows It is obvious that this equivalence takes place also for high dimensions, that is, there exists an equivalence between infinitesimal operators (76)-(81) and tensor products of the Pauli matrices. In such a way, let us suppose that where E i (i = a, b, c) are k-dimensional matrices (the tensor products (6)) and c < a < b.
It is easy to verify that the operators (91)-(92) satisfy the relations (88). Indeed, for the commutator [A 23 , A 13 ] we obtain and so on. Therefore, the operator set (91)-(92) isomorphically defines the set of infinitesimal operators of the group G + . In accordance with Gel'fand-Yaglom approach [27] (see also [29,47]) an operation of space inversion P commutes with all the operators A ik and anticommutes with all the operators B i : Let us consider permutation conditions of the operators (91)-(92) with the matrix W of the automorphism A → A ⋆ (space inversion). Since W = E 1 E 2 · · · E n is a volume element of C n , then E a , E b , E c ∈ W, E 2 i = I, W 2 = I at n ≡ 0 (mod 4) and W 2 = −I at n ≡ 2 (mod 4). Therefore, for the operator A 23 ∼ − 1 2 E a E b we obtain whence it immediately follows a comparison a + b − 2 ≡ 2n − a − b (mod 2) or 2(a + b) ≡ 2(n+1) (mod 2). Thus, W and A 23 are always commute. It is easy to verify that analogous conditions take place for the operators A 13 , A 12 (except the case n = 2). Further, for that is, c − 1 ≡ n − c (mod 2) or n ≡ 2c − 1 (mod 2). Therefore, the matrix W always anticommute with B 1 (correspondingly with B 1 , B 2 ), since n ≡ 0 (mod 2). Thus, in full accordance with Gel'fand-Yaglom relations (93) we have 1 where the matrix W of the automorphism A → A ⋆ is an element of an Abelian automorphism group Aut − (C n ) ≃ Z 2 ⊗ Z 2 with the signature (+, +, +) at n ≡ 0 (mod 4) and also a non-Abelian automorphism group Aut + (C n ) ≃ Q 4 /Z 2 with the signature (−, −, −) at n ≡ 2 (mod 4), here Z 2 ⊗ Z 2 is a Gauss-Klein group and Q 4 is a quaternionic group (see Theorem 9 in [60]). Let us consider now permutation conditions of the operators (91)-(92) with the matrix E of the antiautomorphism A → A (time reversal). Over the field F = C the matrix E has two forms (Theorem 9 in [60] that is, m ≡ 2c−1 (mod 2) and, therefore, the matrix E in this case always anticommutes with B i , since m ≡ 0 (mod 2). Thus, Let us assume now that E a , E b , E c ∈ E, then that is, in this case E and A ik are always commute. For the hyperbolic operators we obtain and since m ≡ 0 (mod 2), then E and B i are also commute. Therefore, Assume now that E a , E b ∈ E and E c ∈ E. Then in accordance with (95) and (99) the matrix E commutes with A 23 and B 1 , and according to (96) anticommutes with B 2 and B 3 . For the operator A 13 we find that is, 2m ≡ 2b − 1 (mod 2) and, therefore, in this case E always anticommutes with A 13 and correspondingly with the operator A 12 which has the analogous structure. Thus, If we take E a ∈ E, E b , E c ∈ E, then for the operator A 23 it follows that that is, a − 2 ≡ 2m − a − 1 (mod 2) or 2m ≡ 2a − 1 (mod 2). Therefore, E anticommutes with A 23 . Further, in virtue of (98) E commutes with A 13 and anticommutes with A 12 in virtue of (101). Correspondingly, from (99) [E, Let us consider now the matrix E of the group Aut (95) it follows that 2(a + b) ≡ 2(m + 1) (mod 2), therefore, in this case E always commutes with A ik . In its turn, from (96) it follows that E always commutes with B i , since m ≡ 1 (mod 2). Thus, we have the relations (100). At E a , E b , E c ∈ E (except the cases n = 2 and n = 4) from (98) it follows that E always commutes with A ik , and from (99) it follows that E always anticommutes with B i . Therefore, for the case E a , E b , E c ∈ E we have the relations (97). Analogously, at E a , E b ∈ E, E c ∈ E in accordance with (95) and (96) E commutes with A 23 and B 1 , B 3 , and in accordance with (101) and (99) anticommutes with A 13 , A 12 and B 1 . Therefore, for this case we have the relations (108). At E a ∈ E, E b , E c ∈ E from (103) it follows that E anticommutes with A 23 . In virtue of (98) E commutes with A 13 and anticommutes with A 12 in virtue of (101). Correspondingly, from (99) and (96) it follows that E anticommutes with B 1 , B 3 and commutes with B 2 . Therefore, for the case E a ∈ E, E b , E c ∈ E we have the relations (106). Further, cyclic permutations of the indices in Let us consider now the permutation conditions of the operators (91)-(92) with the matrix C of the antiautomorphism A → A ⋆ (full reflection). Over the field F = C the matrix C has two different forms (Theorem 9 in [60] , then by analogy with the matrix E = E m+1 E m+2 · · · E n of A → A, m ≡ 0 (mod 2), we have for C the relations of the form (97)-(108). In its turn, the matrix C = E m+1 E m+2 · · · E n is analogous to E = E 1 E 2 · · · E m , m ≡ 1 (mod 2), therefore, in this case we have also the relations (97)-(108) for C. Now we have all possible combinations of permutation relations between the matrices of infinitesimal operators (91)-(92) of the proper Lorentz group G + and matrices of the fundamental automorphisms of the complex Clifford algebra C n associated with the complex representation C l 0 +l 1 −1,0 of G + . It is obvious that the relations (94) take place for any representation we have the relations (97) for E and the relations of the form (100) for C and, therefore, we have the relations (37) and (38) of the present Theorem. Correspondingly, at E a , E b , E c ∈ E (E a , E b , E c ∈ C) we have the relations (100) for E and the relations of the form (97) for C, that is, the relations (41) and (42) we obtain the relations (102) for E and the relations of the form (108) for C ((45)- (46) in Theorem) and so on. Analogously, if C l 0 +l 1 −1,0 with 2(l 0 +l 1 −1) ≡ 2 (mod 4) and if Aut we have the relations (100) for E and the relations of the form (97) for C (relations (38) and (37) in Theorem) and so on.
Further, let us consider the following combinations of A ik and B i (rising and lowering operators): satisfying in virtue of (88) the relations It is easy to see that for W from (94) and (109) we have always the relations (35). Further, (97) and (100) in virtue of (109) it follow the relations (39) and (40). Analogously, at E a , E b , E c ∈ E from (100) and (97) we obtain the relations (43) and (44). In contrast with this, at E a , E b ∈ E, E c ∈ E the combinations (102) and (108) do not form permutation relations with operators H +,−,3 and F +,−,3 , since E and C commute with A 23 and anticommute with A 13 , and B 1 commutes with C and anticommutes with E (inverse relations take place for B 2 ). Other two relations (57)- (60) and (49) It is easy to see that in such a way for the group Aut + (C n ) ≃ Q 4 /Z 2 we obtain the relations (43) In accordance with [29] a representation conjugated to C l 0 +l 1 −1,0 is defined by a pair (l 0 , l 1 ) = − r 2 , r 2 + 1 , that is, this representation has a form C 0,l 0 −l 1 +1 . In its turn, a representation conjugated to fundamental representation C 1,0 is C 0,−1 . Let us find infinitesimal operators of the representation C 0,−1 . At l 0 = −1/2 and l 1 = 3/2 from (69)-(75) and (76)-(81) we obtain It is easy to see that operators (111) differ from (89)-(90) only in the sign at the operators B i of the hyperbolic rotations. This result is a direct consequence of the well-known definition of the group SL(2; C) as a complexification of the special unimodular group SU(2) (see [63]). Indeed, the group SL(2; C) has six parameters a 1 , a 2 , a 3 , ia 1 , ia 2 , ia 3 , where a 1 , a 2 , a 3 ∈ SU(2). It is easy to verify that operators (111) satisfy the relations (88). Therefore, as in case of the the representation C l 0 +l 1 −1,0 infinitesimal operators of the conjugated representation C 0,l 0 −l 1 +1 are defined as follows where E a , E b , E c are tensor products of the form (6). It is not difficult to verify that operators (112)  2) Real representations. As known [29], if an irreducible representation of the proper Lorentz group G + is defined by the pair (l 0 , l 1 ), then a conjugated representation is also irreducible and defined by a pair ±(l 0 , −l 1 ). Hence it follows that the irreducible representation is equivalent to its conjugated representation only in case when this representation is defined by a pair (0, l 1 ) or (l 0 , 0), that is, either of the two numbers l 0 and l 1 is equal to zero. We assume that l 1 = 0. In its turn, for the complex Clifford algebra C n ( * C n ) associated with the representation C l 0 +l 1 −1,0 (C 0,l 0 −l 1 +1 ) of the group G + the equivalence of the representation to its conjugated representation induces a relation C n = * C n , which, obviously, is fulfilled only in case when the algebra C n = C ⊗ Cℓ p,q is reduced into its real subalgebra Cℓ p,q (p + q = n). Thus, a restriction of the complex representation C l 0 +l 1 −1,0 (or C 0,l 0 −l 1 +1 ) of the group G + onto a real representation, (l 0 , l 1 ) → (l 0 , 0), induces a restriction C n → Cℓ p,q . Further, over the field F = R at p + q ≡ 0 (mod 2) there are four types of real algebras Cℓ p,q : two types p − q ≡ 0, 2 (mod 8) with a real division ring K ≃ R and two types p − q ≡ 4, 6 (mod 8) with a quaternionic division ring K ≃ H. Thus, we have four classes of the real representations of the group G + : We will call the representations H l 0 4 and H l 0 6 are quaternionic representations of the group G + . It is not difficult to see that for the real representations with the pair (l 0 , 0) all the coefficients A l = il 0 l 1 /l(l + 1) are equal to zero, since l 1 = 0. Therefore, all the infinitesimal operators B j 1 , B j 2 , B j 3 (see formulas (79)-(81)) of hyperbolic rotations are also equal to zero. Hence it follows that the restriction (l 0 , l 1 ) → (l 0 , 0) induces a restriction of the group G + onto its subgroup SO(3) of three-dimensional rotations. Thus, real representations with the pair (l 0 , 0) are representations of the subgroup SO(3). This result directly follows from the complexification of SU(2) which is equivalent to SL(2; C). Indeed, the parameters a 1 , a 2 , a 3 compose a real part of SL(2; C) which under complex conjugation remain unaltered, whereas the parameters ±ia 1 , ±ia 2 , ±ia 3 of the complex part of SL(2; C) under complex conjugation are mutually annihilate for the representations with the pairs (l 0 , l 1 ) and ±(l 0 , −l 1 ).
Let us find a relation of the number l 0 with dimension of the real algebra Cℓ p,q . If p + q ≡ 0 (mod 2) and ω 2 = e 2 12...p+q = 1, then Cℓ p,q is called positive (Cℓ p,q > 0 at p − q ≡ 0, 4 (mod 8)) and correspondingly negative if ω 2 = −1 (Cℓ p,q < 0 at p − q ≡ 2, 6 (mod 8)). Further, in accordance with Karoubi Theorem [38,Prop. 3.16] it follows that if Cℓ(V, Q) > 0, and dim V is even, then where V is a vector space associated with Cℓ p,q , Q is a quadratic form of V . Using the Karoubi Theorem we obtain for the algebra Cℓ p,q a following factorization where s i , t j ∈ {0, 1, 2}. For example, there are two different factorizations Cℓ 1,1 ⊗ Cℓ 0,2 and Cℓ 1,1 ⊗ Cℓ 2,0 for the spacetime algebra Cℓ 1,3 and Majorana algebra Cℓ 3,1 . It is obvious that l 0 = r/2 and n = 2r = p + q = 4l 0 , therefore, l 0 = (p + q)/4. So, we begin with the representation of the class R l 0 0 . In accordance with Theorem 4 in [61] for the algebra Cℓ p,q of the type p − q ≡ 0 (mod 8) a matrix of the automorphism A → A ⋆ has a form W = E 1 E 2 · · · E p+q and W 2 = I. It is obvious that for E 2 a = E 2 b = E 2 c = I permutation conditions of the matrix W with A ik are analogous to (94), that is, W always commutes with the operators A ik of the subgroup SO(3). It is sufficient to consider permutation conditions of W with the operators A ik of SO(3) only, since B i = 0 for the real representations. In this case the relations (88) take a form Assume now that E 2 a = E 2 b = E 2 c = −I, then it is easy to verify that operators satisfy the relations (115) and commute with the matrix W of A → A ⋆ . It is easy to see that at Further, for the type p − q ≡ 0 (mod 8) (p = q = m) at E = E p+1 E p+2 · · · E p+q and C = E 1 E 2 · · · E p there exist Abelian groups Aut − (Cℓ p,q ) ≃ Z 2 ⊗ Z 2 with the signature (+, +, +) and Aut − (Cℓ p,q ) ≃ Z 4 with (+, −, −) correspondingly at p, q ≡ 0 (mod 4) and p, q ≡ 2 (mod 4), and also at E = E 1 E 2 · · · E p and C = E p+1 E p+2 · · · E p+q there exist non-Abelian groups Aut + (Cℓ p,q ) ≃ D 4 /Z 2 with (+, −, +) and Aut + (Cℓ p,q ) ≃ D 4 /Z 2 with (+, +, −) correspondingly at p, q ≡ 3 (mod 4) and p, q ≡ 1 (mod 4) (Theorem 4 in [61]). Besides, for the algebras Cℓ 8t,0 of the type p − q ≡ 0 (mod 8), t = 1, 2, . . . , the matrices E ∼ I, C ∼ E 1 E 2 · · · E p and W form an Abelian group Aut − (Cℓ p,0 ) ≃ Z 2 ⊗ Z 2 . Correspondingly, for the algebras Cℓ 0,8t of the type p − q ≡ 0 (mod 8) the matrices E ∼ E 1 E 2 · · · E q , C ∼ I and W also form the group Aut − (Cℓ 0,q ) ≃ Z 2 ⊗ Z 2 .
So, let E = E m+1 E m+2 · · · E 2m and C = E 1 E 2 · · · E m be matrices of A → A and A → A ⋆ , p = q = m, m ≡ 0 (mod 2). Then for the operators (91) we obtain since E a , E b , E c ∈ E and a function σ(n) = σ(m − n) has a form Analogously, for the operators (116) we find It is easy to see that in both cases the matrices E and C always commute with the operators A ik . Therefore, for the type p − q ≡ 0 (mod 8) at p, q ≡ 0 (mod 4) and p, q ≡ 2 (mod 4) the elements of the Abelian groups Aut − (Cℓ p,q ) ≃ Z 2 ⊗ Z 2 and Aut − (Cℓ p,q ) ≃ Z 4 with (+, −, −) are always commute with infinitesimal operators A ik of SO(3). Correspondingly, for the algebra Cℓ 8t,0 the elements of Aut − (Cℓ 8t,0 ) ≃ Z 2 ⊗ Z 2 are also commute with A ik . The analogous statement takes place for other degenerate case Aut − (Cℓ 0,8t ) ≃ Z 2 ⊗ Z 2 . In the case of non-Abelian groups Aut + (Cℓ p,q ) ≃ D 4 /Z 2 (signatures (+, −, +) and (+, +, −)) we obtain the same permutation conditions as (117)-(120), that is, in this case the matrices of the fundamental automorphisms always commute with A ik . Thus, for the real representation of the class R l 0 0 operators of the discrete subgroup always commute with all the infinitesimal operators of SO(3).
So, let E = E j 1 E j 2 · · · E j k and C = E i 1 E i 2 · · · E i p+q−k be the matrices of A → A and that is, E and C commute with A 23 (correspondingly with A 13 , A 12 ). It is easy to see that relations (125) and (126) are analogous to the relations (95) and (98) for the field F = C. Therefore, from (125) and (126) we obtain the relations (62) of Theorem. Further, assume that E a , E b , E c ∈ E, that is, all the matrices E i in the operators (91) and (116) are symmetric. Then and analogous relations take place for A 13 , A 12 . It is easy to verify that from (127) and (128) we obtain the same relations (62), since (62) are relations (37) or (41) at B i = 0. Therefore, over the ring K ≃ H (quasicomplex case) the elements of Abelian reflection groups of the quaternionic representations H l 0 4,6 satisfy the relations (37)-(60) over the field F = C at B i = 0. Indeed, at E a , E b ∈ E, E c ∈ E and E c ∈ E, E a , E b ∈ E we have relations (64) which are particular cases of (45) at B i = 0 and so on. It is easy to verify that the same relations take place for non-Abelian reflection groups at E = E i 1 E i 2 · · · E i p+q−k and C = E j 1 E j 2 · · · E j k , k ≡ 1 (mod 2).

Remark.
Theorem exhausts all possible permutation relations between transformations P, T, P T and infinitesimal operators of the group G + . The relations (35)-(36) take place always, that is, at any n ≡ 0 (mod 2) (except the case n = 2). In turn, the relations  (56). Besides, in accordance with [29] for the transformation T there are only two possibilities T = P and T = −P (both these cases correspond to relation (37)). However, from other relations it follows that T = ±P , as it should be take place in general case. The exceptional case n = 2 corresponds to neutrino field and further it will be explored in the following sections within quotient representations of the group G + . Permutations relations with respect to symmetric subspaces Sym (k,r) can be obtained by similar manner.

Atiyah-Bott-Shapiro periodicity on the Lorentz group
In accordance with the section 2 the finite-dimensional representations C, * C, C⊕ * C related with the algebras C 2k , * C 2r , C 2k ⊕ * C 2k of the type n ≡ 0 (mod 2) and the quotient C 2k corresponding to the type n ≡ 1 (mod 2) form a full system M = M 0 ⊕ M 1 of the finite-dimensional representations of the proper Lorentz group G + . This extension of the Lorentz group allows to apply Atiyah-Bott-Shapiro periodicity [4] on the system M, and also it allows to define on M some cyclic relations that give rise to interesting applications in particle physics (in the spirit of Gell-Mann-Ne'eman eightfold way [26]).
Thus, the action of BW C ≃ Z 2 form a cycle of the period 2 on the system M, where the basic 2-period factor is the fundamental representation C 1,0 (C 0,−1 ) of the group G + . This cyclic structure is intimately related with de Broglie-Jordan neutrino theory of light and, moreover, this structure is a natural generalization of BJ-theory. Indeed, in the simplest case we obtain two relations C 2,0 ≃ C 1,0 ⊗ C 1,0 and C 0,−2 ≃ C 0,−1 ⊗ C 0,−1 , which, as it is easy to see, correspond two helicity states of the photon (left-and right-handed polarization) 2 . In such a way, subsequent rotations of the representation 2-cycle give all other higher spin physical fields and follows to de Broglie and Jordan this structure should be called as 'neutrino theory of everything'. Further, when we consider restriction of the complex representation C onto real representations R and H, that corresponds to restriction of the group G + onto its subgroup SO(3) 3 , we come to a more high-graded modulo 8 periodicity over the field F = R. Indeed, the Clifford algebra Cℓ p,q is central simple if p − q ≡ 1, 5 (mod 8). It is known that for the Clifford algebra with odd dimensionality, the isomorphisms are as follows: Cℓ + p,q+1 ≃ Cℓ p,q and Cℓ + p+1,q ≃ Cℓ q,p [52,49]. Thus, Cℓ + p,q+1 and Cℓ + p+1,q are central simple algebras. Further, in accordance with Chevalley Theorem [15] for the graded tensor product there is an isomorphism Cℓ p,q⊗ Cℓ p ′ ,q ′ ≃ Cℓ p+p ′ ,q+q ′ . Two algebras Cℓ p,q and Cℓ p ′ ,q ′ are said to be of the same class if p + q ′ ≡ p ′ + q (mod 8). The graded central simple Clifford algebras over the field F = R form eight similarity classes, which, as it is easy to see, coincide with the eight types of the algebras Cℓ p,q . The set of these 8 types (classes) forms a Brauer-Wall group BW R [65,42] that is isomorphic to a cyclic group Z 8 . Therefore, in virtue of identifications (113) a group action of BW R ≃ Z 8 can be transferred onto the system M = M + ⊕ M − . In its turn, a cyclic structure of the group BW R ≃ Z 8 is defined by a transition Cℓ + p,q h −→ Cℓ p,q , where the type of the algebra Cℓ p,q is defined by a formula q − p = h + 8r, here h ∈ {0, . . . , 7}, r ∈ Z [12,13]. Thus, the action of BW R ≃ Z 8 on M is defined by a transition when D ∈ M + and D r/2 ∪ D r/2 + ≃ χ D r/2 when D ∈ M − , r is a number of tensor products in (114). Therefore, a cyclic structure of the group BW R ≃ Z 8 induces on the system M modulo 8 periodic relations which can be explicitly showed on the following diagram (the round on the diagram is realized by an hour-hand): 1 · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ❯ 3 H l 0 6 p − q ≡ 6 (mod 8) · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 7 · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 5 · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · R l 0 0,2 ∪ R l 0 l 0 = r/2 = 2. On the other hand, in terms of minimal left ideal the modulo 8 periodicity looks like In virtue of the mapping γ 8,0 : Cℓ 8,0 → M 2 (O) [44] (see also excellent review [5]) the latter relation can be written in the form where O is an octonion algebra. Since the algebra Cℓ 8,0 ≃ Cℓ 0,8 admits an octonionic representation, then in virtue of the modulo 8 periodicity the octonionic representations can be defined for all high dimensions and, therefore, on the system M = M + ⊕ M − we have a relation where O is an octonionic representation of the group G + (O ∼ R 2 0 ). Thus, the action of BW R ≃ Z 8 form a cycle of the period 8 on the system M. This is intimately related with an octonionic structure. In 1973, Günaydin and Gürsey showed that an automorphism group of the algebra O is isomorphic to an exceptional Lie group G 2 that contains SU(3) as a subgroup [31]. The Günaydin-Gürsey construction allows to incorporate the quark phenomenology into a general algebraic framework. Moreover, this construction allows to define the quark structure on the system M within octonionic representations of the proper Lorentz group G + . It is obvious that within such a framework the quark structure cannot be considered as a fundamental physical structure underlieing of the world (as it suggested by QCD). This is fully derivative structure firstly appearred in 8-dimension and further reproduced into high dimensions by the round of 8-cycle generated by the group BW R ≃ Z 8 from 8 ad infinitum (growth of quark's flavors with increase of energy). One can say that such a description, included very powerful algebraic tools, opens an another way of understanding of the Gell-Mann-Ne'emann eightfold way in particle physics.

Pseudoautomorphism A −→ A and charge conjugation
As noted previously, an extraction of the minimal left ideal of the complex algebra C n ≃ C 2 ⊗ C 2 ⊗ · · · ⊗ C 2 induces a space of the finite-dimensional spintensor representation of the group G + . Besides, the algebra C n is associated with a complex vector space C n . Let n = p + q, then an extraction operation of the real subspace R p,q in C n forms the foundation of definition of the discrete transformation known in physics as a charge conjugation C. Indeed, let {e 1 , . . . , e n } be an orthobasis in the space C n , e 2 i = 1. Let us remain the first p vectors of this basis unchanged, and other q vectors multiply by the factor i. Then the basis allows to extract the subspace R p,q in C n . Namely, for the vectors R p,q we take the vectors of C n which decompose on the basis (129) with real coefficients. In such a way we obtain a real vector space R p,q endowed (in general case) with a non-degenerate quadratic form where x 1 , . . . , x p+q are coordinates of the vector x in the basis (129). It is easy to see that the extraction of R p,q in C n induces an extraction of a real subalgebra Cℓ p,q in C n . Therefore, any element A ∈ C n can be unambiguously represented in the form where A 1 , A 2 ∈ Cℓ p,q . The one-to-one mapping transforms the algebra C n into itself with preservation of addition and multiplication operations for the elements A; the operation of multiplication of the element A by the number transforms to an operation of multiplication by the complex conjugate number.
Any mapping of C n satisfying these conditions is called a pseudoautomorphism. Thus, the extraction of the subspace R p,q in the space C n induces in the algebra C n a pseudoautomorphism A → A [52]. Let us consider a spinor representation of the pseudoautomorphism A → A of the algebra C n when n ≡ 0 (mod 2). In the spinor representation the every element A ∈ C n should be represented by some matrix A, and the pseudoautomorphism (130) takes a form of the pseudoautomorphism of the full matrix algebra M 2 n/2 : On the other hand, a transformation replacing the matrix A by the complex conjugate matrix, A →Ȧ, is also some pseudoautomorphism of the algebra M 2 n/2 . The composition of the two pseudoautomorpismsȦ → A and A → A,Ȧ → A → A, is an internal automorphismȦ → A of the full matrix algebra M 2 n/2 : where Π is a matrix of the pseudoautomorphism A → A in the spinor representation. The sufficient condition for definition of the pseudoautomorphism A → A is a choice of the matrix Π in such a way that the transformation A → ΠȦΠ −1 transfers into itself the matrices E 1 , . . . , E p , iE p+1 , . . . , iE p+q (the matrices of the spinbasis of Cℓ p,q ), that is, Theorem 3. Let C n be a complex Clifford algebra when n ≡ 0 (mod 2) and let Cℓ p,q ⊂ C n be its subalgebra with a real division ring K ≃ R when p − q ≡ 0, 2 (mod 8) and quaternionic division ring K ≃ H when p − q ≡ 4, 6 (mod 8), n = p + q. Then in dependence on the division ring structure of the real subalgebra Cℓ p,q the matrix Π of the pseudoautomorphism A → A has the following form: The matrix Π for any spinor representation over the ring K ≃ R is proportional to the unit matrix.
where a complex matrices E αt and b real matrices E βs form a basis of the spinor representation of the algebra Cℓ p,q over the ring K ≃ H, where I is the unit matrix.
Proof. The algebra C n (n ≡ 0 (mod 2), n = p + q) in virtue of C n = C ⊗ Cℓ p,q and definition of the division ring K ≃ f Cℓ p,q f (f is a primitive idempotent of the algebra Cℓ p,q ) has four different real subalgebras: p − q ≡ 0, 2 (mod 8) for the real division ring K ≃ R and p − q ≡ 4, 6 (mod 8) for the quaternionic division ring K ≃ H.

1) K ≃ R.
Since for the types p − q ≡ 0, 2 (mod 8) there is an isomorphism Cℓ p,q ≃ M 2 p+q 2 (R) (Wedderburn-Artin Theorem), then all the matrices E i of the spinbasis of Cℓ p,q are real andĖ i = E i . Therefore, in this case the condition (132) can be written as follows whence E i Π = ΠE i . Thus, for the algebras Cℓ p,q of the types p − q ≡ 0, 2 (mod 8) the matrix Π of the pseudoautomorphism A → A commutes with all the matrices E i . It is easy to see that Π ∼ I.

2) K ≃ H.
In turn, for the quaternionic types p − q ≡ 4, 6 (mod 8) there is an isomorphism Cℓ p,q ≃ M 2 p+q 2 (H). Therefore, among the matrices of the spinbasis of the algebra Cℓ p,q there are matrices E α satisfying the conditionĖ α = −E α . Let a be a quantity of the complex matrices, then the spinbasis of Cℓ p,q is divided into two subsets. The first subset {Ė αt = −E αt } contains complex matrices, 0 < t ≤ a, and the second subset {Ė βs = E βs } contains real matrices, 0 < s ≤ p + q − a. In accordance with a spinbasis structure of the algebra Cℓ p,q ≃ M 2 p+q 2 (H) the condition (132) can be written as follows Thus, for the quaternionic types p − q ≡ 4, 6 (mod 8) the matrix Π of the pseudoautomorphism A → A anticommutes with a complex part of the spinbasis of Cℓ p,q and commutes with a real part of the same spinbasis. From (133) it follows that a structure of the matrix Π is analogous to the structure of the matrices E and C of the antiautomorphisms A → A and A → A ⋆ , correspondingly (see Theorem 4 in [61]), that is, the matrix Π of the pseudoautomorphism A → A of the algebra C n is a product of only complex matrices, or only real matrices of the spinbasis of the subalgebra Cℓ p,q . So, let 0 < a < p + q and let Π = E α 1 E α 2 · · · E αa be a matrix of A → A, then permutation conditions of the matrix Π with the matrices E βs of the real part (0 < s ≤ p + q − a) and with the matrices E αt of the complex part (0 < t ≤ a) have the form that is, when a ≡ 0 (mod 2) the matrix Π commutes with the real part and anticommutes with the complex part of the spinbasis of Cℓ p,q . Correspondingly, when a ≡ 1 (mod 2) the matrix Π anticommutes with the real part and commutes with the complex part. Further, let Π = E β 1 E β 2 · · · E β p+q−a be a product of the real matrices, then that is, when p + q − a ≡ 0 (mod 2) the matrix Π anticommutes with the real part and commutes with the complex part of the spinbasis of Cℓ p,q . Correspondingly, when p + q − a ≡ 1 (mod 2) the matrix Π commutes with the real part and anticommutes with the complex part. The comparison of the conditions (134)-(135) with the condition (133) shows that the matrix Π = E α 1 E α 2 · · · E αa exists only at a ≡ 0 (mod 2), that is, Π is a product of the complex matrices E αt of the even number. In its turn, a comparison of (136)-(137) with (133) shows that the matrix Π = E β 1 E β 2 · · · E β p+q−a exists only at p + q − a ≡ 1 (mod 2), that is, Π is a product of the real matrices E βs of the odd number.
Let us calculate now the product ΠΠ. Let Π = E β 1 E β 2 · · · E β p+q−a be a product of the p + q − a real matrices. SinceĖ βs = E βs , thenΠ = Π and ΠΠ = Π 2 . Therefore, Further, let Π = E α 1 E α 2 · · · E αa be a product of the a complex matrices. ThenĖ αt = −E αt andΠ = (−1) a Π = Π, since a ≡ 0 (mod 2). Therefore, Let p + q − a = b be a quantity of the real matrices E βs of the spinbasis of Cℓ p,q , then p + q = a + b. Since p + q is always even number for the quaternionic types p − q ≡ 4, 6 (mod 8), then a and b are simultaneously even or odd numbers. Thus, from (138)  In the present form of quantum field theory complex fields correspond to charged particles. Thus, the extraction of the subalgebra Cℓ p,q with the real ring K ≃ R in C n , p − q ≡ 0, 2 (mod 8), corresponds to physical fields describing truly neutral particles such as photon and neutral mesons (π 0 , η 0 , ρ 0 , ω 0 , ϕ 0 , K 0 ). In turn, the subalgebras Cℓ p,q with the ring K ≃ H, p − q ≡ 4, 6 (mod 8) correspond to charged or neutral fields.
As known [28], the charge conjugation C should be satisfied the following requirement where I ik are infinitesimal operators of the group G + 5 . This requirement is necessary for the definition of the operation C on the representation spaces of C. Let us find permutation conditions of the matrix Π with I ik defined by the relations (91)-(92). It is obvious that in the case of K ≃ R the matrix Π ≃ I commutes with all the operators I ik and, therefore, the relations (140) hold. In the case of K ≃ H and Π = E α 1 E α 2 · · · E αa it is easy to verify that at E a , E b , E c ∈ Π (all E a , E b , E c are complex matrices) the matrix Π commutes with A ik and anticommutes with B i (permutation conditions in this case are analogous to (95) and (96)). In its turn, when E a , E b , E c ∈ Π (all E a , E b , E c are real matrices) the matrix Π commutes with all the operators I ik . It is easy to see that all other cases given by the cyclic permutations E i , E j ∈ Π, E k ∈ Π and E i ∈ Π, E j , E k ∈ Π (i, j, k ∈ {a, b, c}) do not satisfy the relations (140). For example, at E a , E b ∈ Π and E c ∈ Π the matrix Π commutes with A 23 and B 1 and anticommutes with A 13 , A 12 , B 2 , B 3 . Further, in the case of Π = E β 1 E β 2 · · · E β b it is not difficult to see that only at E a , E b , E c ∈ Π (all E a , E b , E c are real matrices) the matrix Π commutes with all I ik . Therefore, in both cases Π = E α 1 E α 2 · · · E αa and Π = E β 1 E β 2 · · · E β b the relations (140) hold when all the matrices E a , E b , E c belonging to (91)-(92) are real.
When we restrict the complex representation C (charged particles) of G + to real representation R (truly neutral particles) and H (neutral particles) we see that in this case the charge conjugation is reduced to an identical transformation I for R and to a particleantiparticle conjugation C ′ for H. Moreover, as follows from Theorem 2 for the real representations B i = 0 and, therefore, the relations (140) take a form Over the ring K ≃ R the relations (141) hold identically. It is easy to verify that over the The same result takes place for the matrix Π = E β 1 E β 2 · · · E β b . All other cases given by the cyclic permutations do not satisfy the relations (141). Therefore, in both cases the relations (141) hold when all the matrices E a , E b , E c in (91) (correspondingly in (116)) are complex or real. Let us consider the action of the pseudoautomorphism A → A on the spinors (5) ('vectors' of the fundamental representation of the group G + ). The matrix Π allows to compare to each spinor ξ α its conjugated spinor ξ α by the following here ξα = (ξ α ) · . In accordance with Theorem 3 for the matrix Π α β we haveΠ = Π −1 oṙ Π = −Π −1 , where Π −1 = Πα β . Then a twice conjugated spinor looks like Therefore, the twice conjugated spinor coincides with the initial spinor in the case of the real subalgebra of C 2 with the ring K ≃ R (the algebras Cℓ 1,1 and Cℓ 2,0 ), and also in the case of K ≃ H (the algebra Cℓ 0,2 ≃ H) at a−b ≡ 0, 1 (mod 4). Since for the algebra Cℓ 0,2 ≃ H we have always a − b ≡ 0 (mod 4), then a property of the reciprocal conjugacy of the spinors ξ α (α = 1, 2) is an invariant fact for the fundamental representation of the group G + (this property is very important in physics, since this is an algebraic expression of the requirement C 2 = 1). Further, since the 'vector' (spintensor) of the finite-dimensional representation of the group G + is defined by the tensor product ξ α 1 α 2 ···α k = ξ α 1 ⊗ ξ α 2 ⊗ · · · ⊗ ξ α k , then its conjugated spintensor takes a form It is obvious that the condition of reciprocal conjugacy ξ α 1 α 2 ···α k = ξ α 1 α 2 ···α k is also fulfilled for (143), since for each matrix Π α i α i in (143) we haveΠ = Π −1 (all the matrices Π α i α i are defined for the algebra C 2 ).
Let us define now permutation conditions of the matrix Π of the pseudoautomorphism A → A (charge conjugation) with the matrix W of the automorphism A → A ⋆ (space inversion). First of all, in accordance with Theorem 3 in the case of Cℓ p,q with the real ring K ≃ R (types p − q ≡ 0, 2 (mod 8)) the matrix Π is proportional to the unit matrix and, therefore, commutes with the matrix W. In the case of K ≃ H (types p − q ≡ 4, 6 (mod 8)) from Theorem 3 it follows two possibilities: Π = E α 1 E α 2 · · · E αa is a product of a complex matrices at a ≡ 0 (mod 2) and Π = E β 1 E β 2 · · · E β b is a product of b real matrices at b ≡ 1 (mod 2). Since a + b = p + q, then the matrix W can be represented by the product E α 1 E α 2 · · · E αa E β 1 E β 2 · · · E β b . Then for Π = E α 1 E α 2 · · · E αa we have Hence it follows that at ab ≡ 0 (mod 2) the matrices Π and W always commute, since a ≡ 0 (mod 2). Taking Π = E β 1 E β 2 · · · E β b we obtain following conditions: Hence it follows that ab ≡ 1 (mod 2), since in this case b ≡ 1 (mod 2), and p + q = a + b is even number, a is odd number. Therefore, at ab ≡ 1 (mod 2) the matrices Π and W always anticommute.
It should be noted one important feature related with the anticommutation of the matrices Π and W, ΠW = −WΠ, that corresponds to relation CP = −P C. The latter relation holds for Bargmann-Wightmann-Wigner type quantum field theories in which bosons and antibosons have mutually opposite intrinsic parities [3]. Thus, in this case the matrix of the operator C is a product of real matrices of odd number.
6 Quotient representations of the Lorentz group Theorem 4. 1) F = C. Let A → A, A → A ⋆ , A → A be the automorphisms of the odd-dimensional complex Clifford algebra C n+1 (n + 1 ≡ 1, 3 (mod 4)) corresponding the discrete transformations C, P, T (charge conjugation, space inversion, time reversal) and let ǫ C n be a quotient algebra obtained in the result of the homomorphic mapping ǫ : C n+1 → C n . Then over the field F = C in dependence on the structure of ǫ C n all the quotient representations of the Lorentz group are divided in the following six classes: 2) F = R. Real quotient representations are divided into four different classes: Proof. 1) Complex representations. Before we proceed to find an explicit form of the quotient representations χ C it is necessary to consider in details a structure of the quotient algebras ǫ C n obtaining in the result of the homomorphic mapping ǫ : C n+1 → C n . The structure of the quotient algebra ǫ C n depends on the transfer of the automorphisms A → A ⋆ , A → A, A → A ⋆ , A → A of the algebra C n+1 under action of the homomorphism ǫ onto its subalgebra C n . As noted previously (see conclusion of Theorem 1), the homomorphisms ǫ and χ have an analogous texture. The action of the homomorphism ǫ is defined as follows where A 1 , A 2 ∈ C n , ω = e 12···n+1 , and ε = 1, if n + 1 ≡ 1 (mod 4), i, if n + 1 ≡ 3 (mod 4); so that (εω) 2 = 1. At this point εω → 1 and the quotient algebra has a form where Ker ǫ = {A 1 − εωA 1 } is a kernel of the homomorphism ǫ.
For the transfer of the antiautomorphism A → A from C n+1 into C n it is necessary that Indeed, since under action of ǫ the elements 1 and εω are equally mapped into the unit, then transformed elements 1 and εω are also should be mapped into 1, but 1 = 1 → 1, and εω = ±εω → ±1 in virtue of ω = (−1) Therefore, under action of the homomorphism ǫ the antiautomorphism A → A is transferred from C n+1 into C n only at n ≡ 0 (mod 4). In its turn, for the transfer of the automorphism A → A ⋆ it is necessary that (εω) ⋆ = εω. However, since the element ω is odd and ω ⋆ = (−1) n+1 ω, then we have always Thus, the automorphism A → A ⋆ is never transferred from C n+1 into C n . Further, for the transfer of the antiautomorphism A → A ⋆ from C n+1 into C n it is necessary that It is easy to see that the condition (147) is satisfied only at n + 1 ≡ 3 (mod 4), since in this case from the second equality of (145) and (146) it follows Therefore, under action of the homomorphism ǫ the antiautomorphism A → A ⋆ is transferred from C n+1 into C n only at n ≡ 2 (mod 4). Let n + 1 = p + q. Defining in C n+1 the basis {e 1 , . . . , e p , ie p+1 , . . . , ie p+q } we extract the real subalgebra Cℓ p,q , where at p − q ≡ 3, 7 (mod 8) we have a complex division ring K ≃ C, and at p − q ≡ 1 (mod 8) and p − q ≡ 5 (mod 8) correspondingly a double real division ring K ≃ R ⊕ R and a double quaternionic division ring K ≃ H ⊕ H. The product e 1 e 2 · · · e p ie p+1 · · · ie p+q = i q ω ∈ C n+1 sets a volume element of the real subalgebra Cℓ p,q . At this point we have a condition (i q ω) = i q ω, that is, (−i) q ω = i q ω, whence When q is even, from (149) it follows ω = ω and, therefore, the pseudoautomorphism A → A is transferred at q ≡ 0 (mod 2), and since p + q is odd number, then we have always p ≡ 1 (mod 2). In more detail, at n + 1 ≡ 3 (mod 4) the pseudoautomorphism A → A is transferred from C n+1 into C n if the real subalgebra Cℓ p,q possesses the complex ring K ≃ C, p − q ≡ 3, 7 (mod 8), and is not transferred (ω = −ω, q ≡ 1 (mod 2), p ≡ 0 (mod 2)) in the case of Cℓ p,q with double rings K ≃ R ⊕ R and K ≃ H ⊕ H, p − q ≡ 1, 5 (mod 8). In its turn, at n+ 1 ≡ 1 (mod 4) the pseudoautomorphism A → A is transferred from C n+1 into C n if the subalgebra Cℓ p,q has the type p − q ≡ 1, 5 (mod 8) and is not transferred in the case of Cℓ p,q with p − q ≡ 3, 7 (mod 8). Besides, in virtue of (146) at n + 1 ≡ 3 (mod 4) with p − q ≡ 1, 5 (mod 8) and at n + 1 ≡ 1 (mod 4) with p − q ≡ 3, 7 (mod 8) a pseudoautomorphism A → A ⋆ (a composition of the pseudoautomorphism Further, in virtue of the second equality of (145) at n + 1 ≡ 3 (mod 4) with p − q ≡ 1, 5 (mod 8) a pseudoantiautomorphism A → A (a composition of the pseudoautomorphism A → A with the antiautomorphism A → A) is transferred from C n+1 into C n , since εω = εω.
Finally, a pseudoantiautomorphism A → A ⋆ (a composition of the pseudoautomorphism A → A with the antiautomorphism A → A ⋆ ), corresponded to CP T -transformation, is transferred from C n+1 into C n at n + 1 ≡ 3 (mod 4) and Cℓ p,q with p − q ≡ 3, 7 (mod 8), since in this case in virtue of (148) and (149) we have (εω ⋆ ) = εω.
The conditions for the transfer of the fundamental automorphisms of the algebra C n+1 into its subalgebra C n under action of the homomorphism ǫ allow to define in evident way an explicit form of the quotient algebras ǫ C n .
As noted previously, in the case n + 1 ≡ 1 (mod 4) the antiautomorphism A → A and pseudoautomorphism A → A are transferred from C n+1 into C n if the subalgebra Cℓ p,q ⊂ C n+1 possesses the double rings K ≃ R ⊕ R, K ≃ H ⊕ H (p − q ≡ 1, 5 (mod 8)), and also the pseudoautomorphism A → A ⋆ and pseudoantiautomorpism A → A ⋆ are transferred if Cℓ p,q has the complex ring K ≃ C (p − q ≡ 3, 7 (mod 8)). It is easy to see that in dependence on the type of Cℓ p,q the structure of the quotient algebras ǫ C n of this type is divided into two different classes: a) The class of quotient algebras ǫ C n containing the antiautomorphism A → A and pseudoautomorphism A → A. It is obvious that in dependence on a division ring structure of the subalgebra Cℓ p,q ⊂ C n+1 this class is divided into two subclasses: a 1 ) ǫ C n with A → A, A → A at Cℓ p,q with the ring K ≃ R ⊕ R, p − q ≡ 1 (mod 8).
a 2 ) ǫ C n with A → A, A → A at Cℓ p,q with the ring K ≃ H ⊕ H, p − q ≡ 5 (mod 8).
b) The class of quotient algebras ǫ C n containing the transformations A → A, A → A ⋆ , A → A ⋆ if the subalgebra Cℓ p,q ⊂ C n+1 has the complex ring K ≃ C, p − q ≡ 3, 7 (mod 8).
In the case n + 1 ≡ 3 (mod 4) the antiautomorphism A → A ⋆ , pseudoautomorphism A → A and pseudoantiautomorphism A → A ⋆ are transferred from C n+1 into C n if the subalgebra Cℓ p,q ⊂ C n+1 possesses the complex ring K ≃ C (p − q ≡ 3, 7 (mod 8)), and also the pseudoautomorphism A → A ⋆ and pseudoantiautomorphism A → A are transferred if Cℓ p,q has the double rings K ≃ R ⊕ R, K ≃ H ⊕ H (p − q ≡ 1, 5 (mod 8)). In dependence on the type of Cℓ p,q ⊂ C n+1 all the quotient algebras ǫ C n of this type are divided into following two classes: c) The class of quotient algebras ǫ C n containing the transformations A → A ⋆ , A → A, A → A ⋆ if the subalgebra Cℓ p,q has the ring K ≃ C, p − q ≡ 3, 7 (mod 8).
d) The class of quotient algebras ǫ C n containing the antiautomorphism A → A ⋆ , pseudoautomorphism A → A ⋆ and pseudoautomorphism A → A. At this point, in dependence on the division ring structure of Cℓ p,q we have two subclasses d 1 ) ǫ C n with A → A ⋆ , A → A ⋆ and A → A at Cℓ p,q with the ring K ≃ R ⊕ R, p − q ≡ 1 (mod 8).
d 2 ) ǫ C n with A → A ⋆ , A → A ⋆ and A → A at Cℓ p,q with the ring K ≃ H ⊕ H, p − q ≡ 5 (mod 8).
Thus, we have 6 different classes of the quotient algebras ǫ C n . Further, in accordance with [60] the automorphism A → A ⋆ corresponds to space inversion P , the antiautomorphisms A → A and A → A ⋆ set correspondingly time reversal T and full reflection P T , and the pseudoautomorphism A → A corresponds to charge conjugation C. Taking into account this relation and Theorem 1 we come to classification presented in Theorem for complex quotient representations.
Let us define real quotient representations of the group G + . First of all, in the case of types p − q ≡ 3, 7 (mod 8) we have the isomorphism (68) and, therefore, these representations are equivalent to complex representations considered in the section 3. Further, when p − q ≡ 1, 5 (mod 8) we have the real algebras Cℓ p,q with the rings K ≃ R ⊕ R, K ≃ H ⊕ H and, therefore, there exist homomorphic mappings ǫ : Cℓ p,q → Cℓ p,q−1 , ǫ : Cℓ p,q → Cℓ q,p−1 .
In its turn, for the transfer of the automorphism A → A ⋆ it is necessary that ω ⋆ = ω. However, since the element ω is odd and ω ⋆ = (−1) p+q ω, then we have always Thus, the automorphism A → A ⋆ is never transferred from Cℓ p,q into Cℓ p,q−1 (Cℓ q,p−1 ) Further, for the transfer of the antiautomorphism A → A ⋆ it is necessary that From (150) and (151) for the types p − q ≡ 1, 5 (mod 8) we obtain Therefore, under action of the homomorphism ǫ the antiautomorphism A → A ⋆ is never transferred from Cℓ p,q into Cℓ p,q−1 (Cℓ q,p−1 ).
As noted previously, for the real representations of G + the pseudoautomorphism A → A is reduced into identical transformation I for R l 0 0,2 and to particle-antiparticle conjugation C ′ for H l 0 4,6 . The volume element ω of Cℓ p,q (types p − q ≡ 1, 5 (mod 8)) can be represented by the product e 1 e 2 · · · e p e ′ p+1 e ′ p+2 · · · e ′ p+q , where e ′ p+j = ie p+j , e 2 j = 1, (e ′ p+j ) 2 = −1. Therefore, for the transfer of A → A from Cℓ p,q into Cℓ p,q−1 (Cℓ q,p−1 ) we have a condition ω = ω, and in accordance with (149) it follows that the pseudoautomorphism A → A is transferred at q ≡ 0 (mod 2). Further, in virtue of the relation (151) the pseudoautomorphism A → A ⋆ is transferred at q ≡ 1 (mod 2), since in this case we have ω ⋆ = ω.
2) The quotient algebras ǫ Cℓ p,q−1 ( ǫ Cℓ q,p−1 ), p − q ≡ 5 (mod 8). In this case the initial algebra Cℓ p,q has the double quaternionic division ring K ≃ H ⊕ H and its subalgebras Cℓ p,q−1 and Cℓ q,p−1 are of the type p − q ≡ 4 (mod 8) or p − q ≡ 6 (mod 8) with the ring K ≃ H. Therefore, in this case the pseudoautomorphism A → A is equivalent to the particle-antiparticle conjugation C ′ . As in the previous case the antiautomorphism A → A is transferred at any p − q ≡ 5 (mod 8). For this type in dependence on the number q there are two different classes: f 1 ) ǫ Cℓ p,q−1 ( ǫ Cℓ q,p−1 ) with A → A, A → A, A → A, p − q ≡ 5 (mod 8), q ≡ 0 (mod 2).
f 2 ) ǫ Cℓ p,q−1 ( ǫ Cℓ q,p−1 ) with A → A, A → A ⋆ , A → A ⋆ , p−q ≡ 5 (mod 8), q ≡ 1 (mod 2). 7 Quotient representation χ C 0,−1 c and neutrino field Analysing the quotient representations of the group G + presented in Theorem 4, we see that only a repsesentation of the class c at j = (l 0 + l 1 − 1)/2 = 1/2 is adequate for description of the neutrino field. This representation admits full reflection P T , charge conjugation C and CP T -transformation (space inversion P is not defined). In contrast with this, the first three classes a 1 , a 2 and b are unsuitable for description of neutrino, since in this case j is an integer number, n ≡ 0 (mod 4) (bosonic fields). In turn, the classes d 1 and d 2 admit CT -transformation that in accordance with CP T -Theorem is equivalent to space inversion P , which, as known, is a forbidden operation for the neutrino field. So, we have an homomorphic mapping ǫ : C 3 → C 2 , where C 3 is a simplest Clifford algebra of the type n + 1 ≡ 3 (mod 4). In accordance with Theorem 4 under action of the homomorphism ǫ : C 3 → C 2 the transformations A → A ⋆ , A → A and A → A ⋆ are transferred from C 3 into C 2 . At this point, the real subalgebra Cℓ 3,0 ⊂ C 3 has the complex ring K ≃ C, p − q ≡ 3 (mod 8), and, therefore, the matrix Π of the pseudoautomorphism A → A is not unit, that according to Theorem 3 corresponds to charged or neutral fields. In accordance with general scheme presented in the section 2 a decomposition of the algebra C 3 looks like
Further, it is obvious that a spinspace of the algebra ǫ C 2 ∪ ǫ