{'fraction_non_alphanumeric': 0.06797904383948837, 'fraction_numerical': 0.02021384419922628, 'mean_word_length': 3.4468990033644387, 'pattern_counts': {'":': 0, '<': 11, '<?xml version=': 0, '>': 1, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 217, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Group Theoretical Interpretation of the CPT-theorem

An algebraic description of basic discrete symmetries (space inversion P, time reversal T, charge conjugation C and their combinations PT, CP, CT, CPT) is studied. Discrete subgroups {1,P,T,PT} of orthogonal groups of multidimensional spaces over the fields of real and complex numbers are considered in terms of fundamental automorphisms of Clifford algebras. The fundamental automorphisms form a finite group of order 4. The charge conjugation is represented by a pseudoautomorphism of the Clifford algebra. Such a description allows one to extend the automorphism group. It is shown that an extended automorphism group (CPT-group) forms a finite group of order 8. The group structure and isomorphisms between the extended automorphism groups and finite groups are studied in detail. It is proved that there exist 64 different realizations of CPT-group. An extension of universal coverings (Clifford-Lipschitz groups) of the orthogonal groups is given in terms of CPT-structures which include well-known Shirokov-Dabrowski PT-structures as a particular case.


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. A fundamental notion of antimatter immediately relates with the charge conjugation C. The requirement of invariance concerning the each of the discrete transformations gives rise to certain consequences which can be verified in experience. So, from the invariance concerning the CP T -transformation it follows an identity of the masses and full life times of particles and antiparticles, from the invariance concerning the time reversal T we have certain relations between the forward and reverse reaction cross-sections. In turn, from the invariance concerning the charge conjugation C it follows an absence of the reactions forbidden by the conservation law of charge parity, and from the invariance concerning the space inversion P and time reversal T it follows an absence of the electric dipol moment of particles. As follows from experience, for the processes defined by strong and electromagnetic interactions there exists an invariance with respect to the all discrete transformations. In contrast to strong and electromagnetic interactions, for the weak interaction, as shown by experiment, there is no invariance concerning space inversion P , but there is invariance with respect to CP -transformation. Moreover, there are experimental evidences confirming CP -violation (a decay of the neutral K-mesons). It is clear that the analysis of the discrete symmetries allows us to reveal the most profound structural characteristics of the matter.
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 unclear 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 [20], Bargmann-Wigner [1] and Gel'fand-Yaglom [10] multispinor theories, and also Joos-Weinberg 2(2j + 1)-component formalism [14,31] 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 [11]. 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, see [17,15,27]). At present, the Gel'fand-Minlos-Shapiro ideas have been found further development in the works of Buchbinder, Gitman and Shelepin [5,12], where the discrete symmetries are represented by both outer and inner automorphisms of the Poincaré group.
Discrete symmetries P and T transform (reflect) space and time (two the most fundamental notions in physics), but in the Minkowski 4-dimensional space-time continuum [18] space and time are not separate and independent. By this reason a transformation of one (space or time) induces a transformation of another. Therefore, discrete symmetries should be expressed by such trnsformations of the continuum, that transform all its structure totally with a full preservation of discrete nature. The only possible candidates on the role of such transformations are automorphisms. In such a way, the idea of representation of the discrete symmetries via the automorphisms of the Lorentz group ('rotation' group of the 4-dimensional continuum) is appearred in the Gel'fand-Minlos-Shapiro approach, or via the automorphisms of the Poincaré group (motion group of the 4-dimensional continuum) in the Buchbinder-Gitman-Shelepin approach.
In 1909, Minkowski showed [18] that a causal structure of the world is described by a 4-dimensional pseudo-Euclidean geometry. In accordance with [18] the quadratic form x 2 + y 2 + z 2 − c 2 t 2 remains invariant under the action of linear transformations of the four variables x, y, z and t, which form a general Lorentz group G. As is known, the general Lorentz group G consists of a proper orthochronous Lorentz group G 0 and three reflections (discrete transformations) P, T, P T , where P and T are space and time reversal, and P T is a so-called full reflection. The discrete transformations P, T and P T added to an identical transformation form a finite group. Thus, the general Lorentz group may be represented by a semidirect product G 0 ⊙ {1, P, T, P T }.
In 1958, Shirokov pointed out [25,26] 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 [9]. As is 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 [9]) 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 [28,29,30], where the discrete symmetries are represented by fundamental automorphisms of the Clifford algebras. So, the space inversion P , time reversal T and their combination P T correspond to an automorphism ⋆ (involution), an antiautomorphism (reversion) and an antiautomorphism ⋆ (conjugation), respectively. Group theoretical structure of the discrete transformations is a central point in this work. The fundamental automorphisms of the Clifford algebras are compared to elements of the finite group formed by the discrete transformations. In its turn, a set of the fundamental automorphisms, added by an identical automorphism, forms a finite group Aut(Cℓ), for which in virtue of the Wedderburn-Artin Theorem there exists a matrix representation. In such a way, an isomorphism {1, P, T, P T } ≃ Aut(Cℓ) plays a central role and allows us to use methods of the Clifford algebra theory at the study of a group theoretical structure of the discrete transformations. First of all, it allows us to classify the discrete groups into Abelian Z 2 ⊗ Z 2 , Z 4 and non-Abelian D 4 , Q 4 finite groups, and also we establish a dependence between the finite groups and signature of the spaces in case of real numbers. It is shown that the division ring structure of Cℓ p,q imposes hard restrictions on existence and choice of the discrete subgroup, and the signature (a, b, c) depends upon the signature of the underlying space R p,q . Moreover, this description allows us to incorporate the Gel'fand-Minlos-Shapiro automorphism theory into Shirokov-Dabrowski scheme and further to unite them on the basis of the Clifford algebra theory.
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 first 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 3 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 9. An introduction of the transformation A → A allows us to extend the automorphism group Aut(Cℓ) of the Clifford algebra. It is shown that automor-

Clifford-Lipschitz groups
Theory of spinors and spinor representations (universal coverings of the groups) is closely related with the foundations of quantum mechanics. As is known, any quantized system S corresponds to a complex separabel Hilbert space H. At this point, a physical state is represented by a vector | x ∈ H and x | x = 1. In turn, any physical observable a (for example, energy, electric charge and so on) corresponds to a self-conjugated operator A in the space H. The spectrum of A coincides with a set of all possible values of a. Quantum mechanics does not give a definite value of the magnitude a in the state | x . We have here only a mathematical expectation of a: where P x is a Hermitean projection operator (P + x = P x ) on the state | x . The unit eigenvectors of P x are distinguished by the scalar phase factor. All these vectors lead to identical physical predictions and, therefore, they describe one and the same state. In turn, the operators P x in itself are observables. Indeed, the magnitude is a location probability of the physical system S in the state | x (or | y ). For the each quantized system S there exists an invariance group G. It means that G acts on the system S and there exists an isomorphism between S and g(S), g ∈ G. Let P gx i be the state obtained from P x i via the transformation g ∈ G. Then, the group G is the invariance group if all probabilities in (1) are invariant: It means that G acts on H isometrically. As it shown in [32,33,2], the mapping | x →| gx is either unitary operator U(g) or antiunitary operator V (g) on the Hilbert space H. Let V(H) be a group of both unitary and antiunitary operators on H, and let U(H) be a subgroup of unitary operators. The group U(H) is a subgroup of index 2 of the group V(H), and by this reason U(H) is an invariant subgroup of V(H). The transformations U(g) for g ∈ G generate a subgroup E(G) of the group V(H), which is an extension of G by means of the group U (multiplication of the vectors of H by the phase factor e iα , where α is a real number; at this point the state remains unaltered), that is, where a kernel of the mapping Ker f is an invariant subgroup G + ⊂ G of index 2, which acts as the group of unitary transformations, and a nontrivial element of the mapping image Imf is the complex conjugation. Therefore, one can say that G + acts as a linear unitary projective representation. For example, if G + is a rotation group SO(3), then its projective representations are coincide with linear irreducible unitary representations of the group SU(2) (the group SU (2) is an universal covering of SO (3)). In turn, universal coverings of the orthogonal groups are completely defined within Clifford-Lipschitz groups.
The Lipschitz group Γ p,q , also called the Clifford group, introduced by Lipschitz in 1886 [16], may be defined as the subgroup of invertible elements s of the algebra Cℓ p,q : The set Γ + p,q = Γ p,q ∩ Cℓ + p,q is called special Lipschitz group [7].
Further, the group Γ p,q has a subgroup Analogously, a spinor group Spin(p, q) is defined by the set It is obvious that Spin(p, q) = Pin(p, q) ∩ Cℓ + p,q . The group Spin(p, q) contains a subgroup It is easy to see that the groups O(p, q), SO(p, q) and SO + (p, q) are isomorphic, respectively, to the following quotient groups where the kernel Z 2 = {1, −1}. Thus, the groups Pin(p, q), Spin(p, q) and Spin + (p, q) are the double coverings of the groups O(p, q), SO(p, q) and SO + (p, q), respectively. Further, since Cℓ + p,q ≃ Cℓ + q,p , then Spin(p, q) ≃ Spin(q, p).
In turn, over the field F = C there exists a complex Clifford-Lipschitz group Γ n = s ∈ C + n ∪ C − n | ∀x ∈ C n , sxs −1 ∈ C n . The group Γ n has a subgroup Pin(n, C) is an universal covering of the complex orthogonal group O(n, C). When n ≡ 1 (mod 2) we have On the other hand, there exists a more detailed version of the Pin-group (2) proposed by Dabrowski in 1988 [9]. In general, there are eight double coverings of the orthogonal group O(p, q) [9,4]: where a, b, c ∈ {+, −}. As is known, the group O(p, q) consists of four connected components: identity connected component O 0 (p, q), and three components corresponding to space inversion P , time reversal T , and the combination of these two P T , i.e., O(p, q) = (O 0 (p, q)) ∪ P (Q 0 (p, q)) ∪ T (O 0 (p, q)) ∪ P T (O 0 (p, q)). Further, since the four-element group (reflection group) {1, P, T, P T } is isomorphic to the finite group Z 2 ⊗ Z 2 (Gauss-Klein viergruppe [23,24]), then O(p, q) may be represented by a semidirect product O(p, q) . The signs of a, b, c correspond to the signs of the squares of the elements in Pin a,b,c (p, q) which cover space inversion P , time reversal T and a combination of these two P T (a = −P 2 , b = T 2 , c = −(P T ) 2 in Dabrowski's notation [9] and a = P 2 , b = T 2 , c = (P T ) 2 in Chamblin's notation [6] which we will use below). An explicit form of the group Pin a,b,c (p, q) is given by the following semidirect product where C a,b,c are the four double coverings of Z 2 ⊗ Z 2 . All the eight double coverings of the orthogonal group O(p, q) are given in the following table: a b c C a,b,c Remark Here Z 4 , Q 4 , and D 4 are complex, quaternion, and dihedral groups, respectively. According to [9] the group Pin a,b,c (p, q) satisfying the condition P T = −T P is called Cliffordian, and respectively non-Cliffordian when P T = T P .

Discrete symmetries and Clifford algebras
In Clifford algebra Cℓ there exist four fundamental automorphisms.
1) Identity: An automorphism A → A and e i → e i . This automorphism is an identical automorphism of the algebra Cℓ. A is an arbitrary element of Cℓ.
2) Involution: An automorphism A → A ⋆ and e i → −e i . In more details, for an arbitrary element A ∈ Cℓ there exists a decomposition A = A ′ + A ′′ , where A ′ is an element consisting of homogeneous odd elements, and A ′′ is an element consisting of homogeneous even elements, respectively. Then the automorphism A → A ⋆ is such that the element A ′′ is not changed, and the element A ′ changes sign: If A is a homogeneous element, then where k is a degree of the element. It is easy to see that the automorphism A → A ⋆ may be expressed via the volume element ω = e 12...p+q : where When k is odd, the basis elements e i 1 i 2 ...i k the sign changes, and when k is even, the sign is not changed.
3) Reversion: An antiautomorphism A → A and e i → e i . The antiautomorphism A → A is a reversion of the element A, that is the substitution of each basis element e i 1 i 2 ...i k ∈ A by the element e i k i k−1 ...i 1 : Therefore, for any A ∈ Cℓ p,q we have 4) Conjugation: An antiautomorpism A → A ⋆ and e i → −e i . This antiautomorphism is a composition of the antiautomorphism A → A with the automorphism A → A ⋆ . In the case of a homogeneous element from the formulae (11) and (13), it follows One of the most fundamental theorems in the theory of associative algebras is Theorem 4 (Wedderburn-Artin). Any finite-dimensional associative simple algebra A over the field F is isomorphic to a full matrix algebra M n (K), where a natural number n defined unambiguously, and a division ring K defined with an accuracy of isomorphism.
In accordance with this theorem all properties of the initial algebra A are isomorphically transferred to the matrix algebra M n (K). Later on we will widely use this theorem. In its turn, for the Clifford algebra Cℓ p,q over the field F = R we have an isomorphism Cℓ p,q ≃ End K (I p,q ) ≃ M 2 m (K), where m = p+q 2 , I p,q = Cℓ p,q f is a minimal left ideal of Cℓ p,q , and K = f Cℓ p,q f is a division ring of Cℓ p,q . The primitive idempotent of the algebra Cℓ p,q has a form f = 1 2 (1 ± e α 1 ) where e α 1 , e α 2 , . . . , e α k are commuting elements with square 1 of the canonical basis of Cℓ p,q generating a group of order 2 k . The values of k are defined by a formula k = q − r q−p , where r i are the Radon-Hurwitz numbers [19,13], values of which form a cycle of the period 8: The all Clifford algebras Cℓ p,q over the field F = R are divided into eight different types with a following division ring structure: I. Central simple algebras.

4)
The type p − q ≡ 1 (mod 8) with a double division ring K ≃ R ⊕ R.

5)
The type p − q ≡ 5 (mod 8) with a double quaternionic division ring K ≃ H ⊕ H.
Over the field F = C there is an isomorphism C n ≃ M 2 n/2 (C) and there are two different types of complex Clifford algebras C n : n ≡ 0 (mod 2) and n ≡ 1 (mod 2). In virtue of the Wedderburn-Artin theorem the all fundamental automorphisms of Cℓ are transferred to the matrix algebra. Matrix representations of the fundamental automorphisms of C n were first obtained by Rashevskii in 1955 [21]: 1) Involution: A ⋆ = WAW −1 , where W is a matrix of the automorphism ⋆ (matrix representation of the volume element ω); 2) Reversion: C. In the recent paper [28] it has been shown that space reversal P , time reversal T and combination P T are correspond respectively to the fundamental automorphisms A → A ⋆ , A → A and A → A ⋆ . Proposition 1. Let Cℓ p,q (p + q = 2m) be a Clifford algebra over the field F = R and let Pin(p, q) be a double covering of the orthogonal group O(p, q) = O 0 (p, q) ⊙ {1, P, T, P T } ≃ O 0 (p, q) ⊙ (Z 2 ⊗ Z 2 ) of transformations of the space R p,q , where {1, P, T, P T } ≃ Z 2 ⊗ Z 2 is a group of discrete transformations of R p,q , Z 2 ⊗ Z 2 is the Gauss-Klein group. Then there is an isomorphism between the group {1, P, T, P T } and an automorphism group {Id, ⋆, , ⋆} of the algebra Cℓ p,q . In this case, space inversion P , time reversal T and combination P T are correspond to the fundamental automorphisms 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 an automorphism group Aut(Cℓ) = {I, W, E, 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 [3], therefore, {1, P, T, P T } = {1, γ 0 , γ 1 γ 3 , γ 0 γ 1 γ 3 }. On the other hand, in the γ-basis an automorphism group of C 4 has a form Aut(C 4 ) = {I, W, E, C} = {I, γ 0 γ 1 γ 2 γ 3 , γ 1 γ 3 , γ 0 γ 2 }. It has been shown [28] that {1, P, T, where Z 4 is a complex group with the signature (+, −, −). Generalizations of these results onto the algebras C n are contained in the following two theorems: Theorem 5 ( [28]). Let Aut = {I, W, E, C} be the automorphism group of the algebra . Let Aut − and Aut + be the automorphism groups, in which the all elements correspondingly commute (m ≡ 0 (mod 2)) and anticommute (m ≡ 1 (mod 2)). Then over the field F = C there are only two non-isomorphic groups: Aut − ≃ Z 2 ⊗ Z 2 for the signature (+, +, +) if n ≡ 0, 1 (mod 4) and Aut + ≃ Q 4 /Z 2 for the signature (−, −, −) if n ≡ 2, 3 (mod 4).
Theorem 6 ( [28]). Let Pin a,b,c (p, q) be a double covering of the complex orthogonal group O(n, C) of the space C n associated with the complex algebra C n . Squares of the symbols a, b, c ∈ {−, +} are correspond to squares of the elements of the finite group A = {I, W, E, C} : where W, E and C are correspondingly the matrices of the fundamental automorphisms A → A ⋆ , A → A and A → A ⋆ of C n . Then over the field F = C for the algebra C n there exist two non-isomorphic double coverings of the group O(n, C): 1) Non-Cliffordian groups if n ≡ 0 (mod 4) and Pin +,+,+ (n, C) ≃ Pin +,+,+ (n − 1, C) e 12...n Pin +,+,+ (n − 1, C), if n ≡ 1 (mod 4). A consideration of the discrete symmetries over the field of real numbers is a much more complicated problem. First of all, in contrast to the field of complex numbers over the field F = R there exist eight different types of the Clifford algebras and five division rings, which, in virtue of the Wedderburn-Artin Theorem, impose hard restrictions on existence and choice of the matrix representations for the fundamental automorphisms.
The types p − q ≡ 3, 7 (mod 8) admit the Abelian group Aut − (Cℓ p,q ) ≃ Z 2 ⊗ Z 2 with the signature (+, +, +) if p ≡ 0 (mod 2) and q ≡ 1 (mod 2), and also non-Abelian group Theorem 8. Let Pin a,b,c (p, q) be a double covering of the orthogonal group O(p, q) of the real space R p,q associated with the algebra Cℓ p,q . The squares of symbols a, b, c ∈ {−, +} correspond to the squares of the elements of a finite group Aut(Cℓ p,q ) = {I, W, E, C} : a = W 2 , b = E 2 , c = C 2 , where W, E and C are the matrices of the fundamental automorphisms A → A ⋆ , A → A and A → A ⋆ of the algebra Cℓ p,q , respectively. Then over the field F = R in dependence on a division ring structure of the algebra Cℓ p,q , there exist eight double coverings of the orthogonal group O(p, q):

Pseudoautomorphism A −→ A and charge conjugation
As is known, 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 us 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 (15) with real coefficients. In this 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 (15). 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 [21,22]. 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 (16) 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 automorphisṁ A → 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 9. Let C n be a complex Clifford algebra for 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 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.
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 (18) can be written as follows 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 (18) 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 (19) 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 7), 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 (20)- (21) with the condition (19) 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 turn, a comparison of (22)-(23) with (19) 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 (24) and (25) it follows which required to be proved.
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.

Extended automorphism groups
An introduction of the pseudoautomorphism A → A allows us to extend the automorphism set of the complex Clifford algebra C n . Namely, we add to the four fundamental automor- Thus, we obtain an automorphism set of C n consisting of the eight transformations. Let us show that the set {Id, ⋆, , ⋆, , ⋆, , ⋆} forms a finite group of order 8 and let us give a physical interpretation of this group. Proposition 2. Let C n be a Clifford algebra over the field F = C and let Ext(C n ) = {Id, ⋆, , ⋆, , ⋆, , ⋆} be an extended automorphism group of the algebra C n . Then there is an isomorphism between Ext(C n ) and the full CP T -group of the discrete transformations, In this case, space inversion P , time reversal T , full reflection P T , charge conjugation C, transformations CP , CT and the full CP T -transformation correspond to the automorphism A → A ⋆ , antiautomorphisms Proof. The group {1, P, T, P T, C, CP, CT, CP T } at the conditions P 2 = T 2 = (P T ) 2 = C 2 = (CP ) 2 = (CT ) 2 = (CP T ) 2 = 1 and commutativity of all the elements forms an Abelian group of order 8, which is isomorphic to a cyclic group Z 2 ⊗ Z 2 ⊗ Z 2 . The Cayley tableau of this group has a form The identity of multiplication tables proves the group isomorphism Further, in the case of P 2 = T 2 = . . . = (CP T ) 2 = ±1 and anticommutativity of the elements we have an isomorphism between the CP T -group and a group Ext(C n ). The elements of Ext(C n ) are spinor representations of the automorphisms of the algebra C n . The Wedderburn-Artin Theorem allows us to define any spinor representaions for the automorphisms of C n .

Pseudoautomorphism
Otherwise, we have (A ⋆ ) = ΠẆȦẆ −1 Π −1 . Let us assume that W is a complex matrix, theṅ W = −W and, therefore, Thus, the relation (26) is always fulfilled. Let K = ΠW be a matrix of the pseudoautomorphism A → A ⋆ . Then (26) can be written as follows Since under action of the pseudoautomorphism A → A ⋆ we have e i → −e i , in the spinor representation we must demand E i → −E i also, or In the case of real subalgebras Cℓ p,q with the ring K ≃ R we haveĖ i = E i and the relation (28) takes a form that is, the matrix K is always anticommutes with the matrices of the spinbasis. However, for the ring K ≃ R the matrix Π of A → A is proportional to the unit matrix, Π ∼ I (Theorem 9). Therefore, in this case we have K ∼ W.
In the case of real subalgebras Cℓ p,q with the quaternionic ring K ≃ H the spinbasis is divided into two parts: a complex part Ė αt = −E αt , (0 < t ≤ a), where a is a number of the complex matrices of the spinbasis, and a real part Ė βs = E βs , where p + q − a is a number of the real matrices, (0 < s ≤ p + q − a). Then, in accordance with the spinbasis structure of the algebra Cℓ p,q ≃ M 2 p+q 2 (H), the relation (28) can be written as follows Whence Thus, for the quaternionic types p − q ≡ 4, 6 (mod 8) the matrix K of the pseudoautomorphism A → A ⋆ commutes with the complex part and anticommutes with the real part of the spinbasis of Cℓ p,q . Hence it follows that a structure of the matrix K is analogous to the structure of the matrix Π of the pseudoautomorphism A → A (see Theorem 9), that is, the matrix K of A → A ⋆ is a product of only complex or only real matrices. So, let 0 < a ≤ p+q and let K = E α 1 E α 2 · · · E αa be the matrix of the pseudoautomorphism A → A ⋆ , then permutation conditions of the matrix K with the matrices E βs of the real part (0 < s ≤ p + q − a) and the matrices E αt of the complex part (0 < t ≤ a) have the form that is, at a ≡ 0 (mod 2) K commutes with the real part and anticommutes with the complex part of the spinbasis. Correspondingly, at a ≡ 1 (mod 2) K anticommutes with the real and commutes with the complex part. Further, let K = E β 1 E β 2 · · · E β p+q−a be a product of the real matrices of the spinbasis, then that is, at p+q−a ≡ 0 (mod 2) the matrix K anticommutes with the real part and commutes with the complex part of the spinbasis. Correspondingly, at p + q − a ≡ 1 (mod 2) K commutes with the real and anticommutes with the complex part. A comparison of the conditions (30)-(31) with (29) shows that the matrix K = E α 1 E α 2 · · · E αa exists only if a ≡ 1 (mod 2). In turn, a comparison of the conditions (32)- (33) with (29) shows that the matrix K = E β 1 E β 2 · · · E β p+q−a exists only if p + q − a ≡ 0 (mod 2).
Let us find now squares of the matrix K. In accordance with obtained conditions there exist two possibilities: 1) K = E α 1 E α 2 · · · E αa , a ≡ 1 (mod 2).
where a + and a − are numbers of matrices with '+'-and '−'-squares in the product where b + and b − are numbers of matrices with '+'-and '−'-squares in the product E β 1 E β 2 · · · E β p+q−a , respectively.

Pseudoantiautomorphism
in the case whenĖ = −E is a complex matrix. Let S = ΠE be a matrix of the pseudoantiautomorphism A → A in the spinor representation. Then (34) can be rewritten as follows Since under action of the transformation A → A we have e i → e i , in the spinor representation we must demand E i → E i also, or In the case of real subalgebras Cℓ p,q with the ring K ≃ R we haveĖ i = E i and, therefore, the relation (36) takes a form Let {E γ i } be a set of symmetric matrices E T γ i = E γ i and let E δ j be a set of skewsymmetric matrices E T δ j = −E δ j of the spinbasis of the algebra Cℓ p,q . Then from the relation (37) it follows that is, the matrix S of the pseudoantiautomorphism A → A in the case of K ≃ R commutes with the symmetric part and anticommutes with the skewsymmetric part of the spinbasis of Cℓ p,q . In virtue of Theorem 9, over the ring K ≃ R the matrix Π of the pseudoautomorphism A → A is proportional to the unit matrix, Π ∼ I. Therefore, in this case we have S ∼ E and an explicit form of S coincides with E. Further, in case of the quaternionic ring K ≃ H, p − q ≡ 4, 6 (mod 8), a spinbasis of Cℓ p,q contains both complex matrices E αt and real matrices E βs , among which there are symmetric and skewsymmetric matrices. It is obvious that the sets of complex and real matrices do not coincide with the sets of symmetric and skewsymmetric matrices. Let {E αt } be a complex part of the spinbasis, then the relation (36) takes a form Correspondingly, let E αγ and {E α δ } be the sets of symmetric and skewsymmetric matrices of the complex part. Then the relation (38) can be written as follows Therefore, the matrix S of the pseudoantiautomorphism A → A anticommutes with the complex symmetric matrices and commutes with the complex skewsymmetric matrices of the spinbasis of Cℓ p,q . Let us consider now the real part {E βs } of the spinbasis of Cℓ p,q , p − q ≡ 4, 6 (mod 8). In this case the relation (36) takes a form Let E βγ and {E β δ } be the sets of real symmetric and real skewsymmetric matrices, respectively. Then the relation (40) can be written as follows Thus, the matrix S of the transformation A → A commutes with the real symmetric matrices and anticommutes with the real skewsymmetric matrices of the spinbasis of Cℓ p,q . Let us find now an explicit form of the matrix S = ΠE. In accordance with Theorem 9 for the quaternionic types p − q ≡ 4, 6 (mod 8) the matrix Π takes the two different forms: 1) Π = E α 1 E α 2 · · · E αa is the product of complex matrices at a ≡ 0 (mod 2); 2) Π = E β 1 E β 2 · · · E β b is the product of real matrices at b ≡ 1 (mod 2). In turn, for the matrix E of the antiautomorphism A → A over the ring K ≃ H (see Theorem 7) we have the following two forms: 1) E = E j 1 E j 2 · · · E j k is the product of skewsymmetric matrices at k ≡ 0 (mod 2); 2) E = E i 1 E i 2 · · · E i p+q−k is the product of symmetric matrices at k ≡ 1 (mod 2). Thus, in accordance with definition S = ΠE we have four different products: It is obvious that in the given products there are identical matrices.
Let us examine the first product S = E α 1 α 2 ···αa E j 1 j 2 ···j k . Since in this case Π contains all the complex matrices of the spinbasis, among which there are symmetric and skewsymmetric matrices, and E contains all the skewsymmetric matrices of the spinbasis, then Π and E contain a quantity of identical matrices (complex skewsymmetric matrices). Let m be a number of the complex skewsymmetric matrices of the spinbasis of the algebra Cℓ p,q , p − q ≡ 4, 6 (mod 8). Then the product S = ΠE takes a form where the indices c 1 , . . . , c s present itself a totality of the indices α 1 , . . . , α a , j 1 , . . . , j k obtained after removal of the indices occured twice; N is a number of inversions.
First of all, let us remark that E c 1 E c 2 · · · E cs is an even product, since the original product ΠE is the even product also. Besides, the product E c 1 E c 2 · · · E cs contains all the complex symmetric matrices and all the real skewsymmetric matrices of the spinbasis.
Let us find now permutation conditions between the matrix S = E c 1 E c 2 · · · E cs and the units of the spinbasis of Cℓ p,q , p − q ≡ 4, 6 (mod 8). For the complex symmetric matrices E αγ and complex skewsymmetric matrices E α δ we have that is, the matrix S always anticommutes with the complex symmetric matrices and always commutes with the complex skewsymmetric matrices, since s ≡ 0 (mod 2). Further, for the real symmetric matrices E βγ and real skewsymmetric matrices E β δ we have Therefore, S always commutes with the real symmetric matrices and always anticommutes with the real skewsymmetric matrices. A comparison of the obtained conditions (42)-(45) with (39) and (41) shows that S = E c 1 E c 2 · · · E cs automatically satisfies the conditions, which define the matrix of the pseudoantiautomorphism A → A.
Let examine the second product S = E α 1 α 2 ···αa E i 1 i 2 ···i p+q−k . In this case Π contains all the complex matrices of the spinbasis, among which there are symmetric and skewsymmetric matrices, and E contains all the symmetric matrices of the spinbasis. Thus, Π and E contain a quantity of identical complex symmetric matrices. Let l be a number of the complex symmetric matrices, then for the product S = ΠE we obtain where d 1 , . . . , d g present itself a totality of the indices α 1 , . . . , α a , i 1 , . . . , i p+q−k obtained after removal of the indices occured twice. The product S = E d 1 E d 2 · · · E dg is odd, since the original product ΠE is odd also. Besides, E d 1 E d 2 · · · E dg contains all the complex skewsymmetric matrices and all the real symmetric matrices of the spinbasis.
Let us find permutation conditions of the matrix S = E d 1 E d 2 · · · E dg with the units of the spinbasis of Cℓ p,q . For the complex part of the spinbasis we have that is, the matrix S anticommutes with the complex symmetric matrices and commutes with the complex skewsymmetric matrices, since g ≡ 1 (mod 2). For the real part of the spinbasis of Cℓ p,q we obtain Therefore, S commutes with the real symmetric matrices and anticommutes with the real skewsymmetric matrices. Comparing the obtained conditions (46)-(49) with the conditions (39) and (41) we see that S = E d 1 E d 2 · · · E dg automatically satisfies the conditions which define the matrix of the transformation A → A.
Let us consider now the third product S = E β 1 β 2 ···β b E j 1 j 2 ···k . In this product the matrix Π of the pseudoautomorphism A → A contains all the real matrices of the spinbasis, among which there are symmetric and skewsymmetric matrices, and the matrix E of A → A contains all the skewsymmetric matrices of the spinbasis, among which there are both the real and complex matrices. Therefore, S contains a quantity of identical real skewsymmetric matrices. Let u be a number of the real skewsymmetric matrices of the spinbasis of the algebra Cℓ p,q , p − q ≡ 4, 6 (mod 8), then for the product S we obtain where the indices e 1 , e 2 , . . . , e n present itself a totality of the indices β 1 , . . . , β b , j 1 , . . . , j k obtained after removal of the indices occurred twice. The product S = E e 1 E e 2 · · · E e h is odd, since the original product ΠE is odd also. It is easy to see that E e 1 E e 2 · · · E e h contains all the real symmetric matrices and all the complex skewsymmetric matrices of the spinbasis. Therefore, the matrix S = E e 1 E e 2 · · · E e h is similar to the matrix S = E d 1 E d 2 · · · E dg , and its permutation conditions with the units of the spinbasis of Cℓ p,q are equivalent to the relations (46)-(49).
Finally, let us examine the fourth product S = E β 1 β 2 ···β b E i 1 i 2 ···i p+q−k . In turn, this product contains a quantity of identical real symmetric matrices. Let v be a number of the real symmetric matrices of the spinbasis of Cℓ p,q , then where f 1 , . . . , f w present itself a totality of the indices β 1 , . . . , β b , i 1 , . . . , i p+q−k obtained after removal of the indices occurred twice. The product S = E f 1 E f 2 · · · E fw is even, since the original product ΠE is even also. It is easy to see that E f 1 E f 2 · · · E fw contains all the real skewsymmetric matrices and all the complex symmetric matrices of the spinbasis. Therefore, the matrix S = E f 1 E f 2 · · · E fw is similar to the matrix S = E c 1 E c 2 · · · E cs , and its permutation conditions with the units of the spinbasis are equivalent to the relations (42)-(45).
Thus, from the four products we have only two non-equivalent products. The squares of the non-equivalent matrices S = E c 1 E c 2 · · · E cs (s ≡ 0 (mod 2)) and S = E d 1 E d 2 · · · E dg (g ≡ 1 (mod 2)) are

Pseudoantiautomorphism
Let F = ΠC (or F = ΠEW) be a matrix of the pseudoantiautomorphism A → A ⋆ . Then the relation (52) can be written as follows Since under action of the transformation A → A ⋆ we have e i → −e i , in the spinor representation we must demand E i → −E i also, or In case of the real subalgebras Cℓ p,q with the ring K ≃ R, p − q ≡ 0, 2 (mod 8), we havė E i = E i for all matrices of the spinbasis and, therefore, the relation (54) takes a form Let {E γ i } ∪ E δ j be a spinbasis of the algebra Cℓ p,q over the ring K ≃ R, . Then the relation (55) can be written in the form that is, the matrix F of the pseudoantiautomorphism A → A ⋆ in case of the ring K ≃ R anticommutes with the symmetric part of the spinbasis of Cℓ p,q and commutes with the skewsymmetric part. In virtue of Theorem 9 the matrix Π of the pseudoautomorphism A → A over the ring K ≃ R is proportional to the unit matrix, Π ≃ I. Therefore, in this case F ∼ C (F ∼ EW) and an explicit form of the matrix F coincides with C (see Theorem 7). In case of the quaternionic ring K ≃ H, p − q ≡ 4, 6 (mod 8), the spinbasis of Cℓ p,q contains both complex matrices E αt and real matrices E βr . For the complex part the relation (54) takes a form E αt −→ E αt = FE T αt F −1 , Or, taking into account complex symmetric and complex skewsymmetric components of the spinbasis, we obtain For the real part of the spinbasis of Cℓ p,q from (54) we obtain or, taking into account symmetric and skewsymmetric components of the real part of the spinbasis, we find Therefore, the matrix F of the pseudoantiautomorphism A → A ⋆ commutes with the complex symmetric and real skewsymmetric matrices, and also F anticommutes with the complex skewsymmetric and real symmetric matrices of the spinbasis of Cℓ p,q , p − q ≡ 4, 6 (mod 8).
Let us find an explicit form of the matrix F = ΠC. It is easy to see that in virtue of F = ΠC = (ΠE)W = SW the matrix F is a dual with respect to the matrix S of the pseudoantiautomorphism A → A. In accordance with Theorem 9 for the quaternionic types p − q ≡ 4, 6 (mod 8) the matrix Π has two different forms: a ≡ 0 (mod 2); Π = E β 1 E β 2 · · · E β b (Ė βr = E βr ), b ≡ 1 (mod 2). In turn, for the quaternionic types the matrix C of the antiautomorphism A → A ⋆ has the two forms (see Theorem 7): 1) C = E i 1 E i 2 · · · E i p+q−k is the product of all symmetric matrices of the spinbasis of Cℓ p,q at p + q − k ≡ 0 (mod 2); 2) C = E j 1 E j 2 · · · E j k is the product of all skewsymmetric matrices of the spinbasis at k ≡ 1 (mod 2). Thus, in accordance with definition F = ΠC we have four products: Let us examine the first product F = E α 1 α 2 ···αa E i 1 i 2 ···i p+q−k . In this case Π contains all the complex matrices of the spinbasis, among which there are symmetric and skewsymmetric matrices. In turn, C contains all the symmetric matrices, among which there are both complex and real matrices. It is obvious that in this case ΠC contains a quantity of identical complex symmetric matrices. Therefore, the product F consists of all the complex skewsymmetric matrices and all the real symmetric matrices of the spinbasis. The product F is even, since the original product ΠC is even also. It is easy to see that F coincides with the product Let us find permutation conditions of the matrix F = E d 1 E d 2 · · · E dg with the units of the spinbasis of Cℓ p,q , p − q ≡ 4, 6 (mod 8). For the complex and real parts we obtain Therefore, since g ≡ 0 (mod 2) the matrix F always commutes with the complex symmetric and real skewsymmetric matrices and always anticommutes with the complex skewsymmetric and real symmetric matrices of the spinbasis. A comparison of the permutation conditions (58)-(61) with the conditions (56)-(57) shows that F = E d 1 E d 2 · · · E dg at g ≡ 0 (mod 2) automatically satisfies the conditions which define the matrix of the pseudoantiautomorphism Let examine the second product F = E α 1 α 2 ···αa E j 1 j 2 ···j k . This product contains all the complex part of the spinbasis and all the skewsymmetric matrices. Therefore, in the product ΠC there is a quantity of identical complex skewsymmetric matrices. The product F is odd and consists of all the complex symmetric and real skewsymmetric matrices of the spinbasis. It is easy to see that in this case F coincides with the product E c 1 E c 2 · · · E cs at s ≡ 1 (mod 2). Permutation conditions of the matrix F = E c 1 E c 2 · · · E cs with the units of the spinbasis are Therefore, since s ≡ 1 (mod 2) the matrix F always commutes with the complex symmetric and real skewsymmetric matrices and always anticommutes with the complex skewsymmetric and real symmetric matrices of the spinbasis. Comparing the conditions (62)-(65) with the conditions (56) and (57) we see that F = E c 1 E c 2 · · · E cs at s ≡ 1 (mod 2) identically satisfies the conditions which define the matrix of the transformation A → A ⋆ . The third product F = ΠC = E β 1 β 2 ···β b E i 1 i 2 ···i p+q−k contains all the real part and all the symmetric matrices of the spinbasis. Therefore, in the product ΠC there is a quantity of identical real symmetric matrices. Thus, the product F is odd and consists of all the real skewsymmetric and complex symmetric matrices of the spinbasis. It is easy to see that we came again to the matrix F = E c 1 E c 2 · · · E cs (s ≡ 1 (mod 2)) with the permutation conditions (62)-(65).
Finally, the fourth product F = E β 1 β 2 ···β b E j 1 j 2 ···j k contains all the real part and all the skewsymmetric matrices of the spinbasis. Therefore, in the product ΠC there is a quantity of identical real skewsymmetric matrices. This product is equivalent to the matrix F = E d 1 E d 2 · · · E dg (g ≡ 0 (mod 2)) with the permutation conditions (58)-(61).
As with the pseudoantiautomorphism A → A, from the four products we have only two non-equivalent products. Let us find squares of the non-equivalent matrices F = E d 1 E d 2 · · · E dg (g ≡ 0 (mod 2)) and F = E c 1 E c 2 · · · E cs (s ≡ 1 (mod 2)): 6 The structure of Ext(C n ) As noted previously, the group Ext(C n ) is a finite group of order eight. This group contains as a subgroup the automorphism group Aut ± (C n ) (reflection group). Moreover, in the case of Π ∼ I (the subalgebra Cℓ p,q has the ring K ≃ R, p − q ≡ 0, 2 (mod 8)) the group Ext(C n ) is reduced to its subgroup Aut ± (C n ). The structure of the groups Aut ± (C n ), Aut ± (Cℓ p,q ) is studied in detail (see Theorems 5 and 7). There are six finite groups of order eight (see Table 1). One is cyclic and two are direct group products of cyclic groups, hence these three are Abelian. The remaining three groups are the quaternion group Q 4 with elements {±1, ±i, ±j, ±k}, the dihedral group D 4 , and the group * Z 4 ⊗ Z 2 . All these groups are non-Abelian. As is known, an important property of each finite group is its order structure. The order of a particular element α in the group is the smallest integer p for which α p = 1. The following table lists the number of distinct elements in each group which have order 2, 4, or 8 (the identity 1 is the only element of order 1). Example 1. Let us consider a Dirac algebra C 4 . In the algebra C 4 we can evolve four different real subalgebras Cℓ 1,3 , Cℓ 3,1 , Cℓ 4,0 , Cℓ 0,4 . Let us evolve the spacetime algebra Cℓ 1,3 . The algebra Cℓ 1,3 has the quaternionic division ring K ≃ H (p − q ≡ 6 (mod 8)) and, therefore, admits the following spinor representation (the well known γ-basis): The famous Dirac equation in the γ-basis looks like The invariance of the Dirac equation with respect to P -, T -, and C-transformations leads to the following representation (see, for example, [3] and also many other textbooks on quantum field theory): Thus, we can form a finite group of order 8 (CP T -group) {1, P, T, P T, C, CP, CT, CP T } ∼ ∼ {1, γ 0 , γ 1 γ 3 , γ 0 γ 1 γ 3 , γ 2 γ 0 , γ 2 , γ 2 γ 0 γ 1 γ 3 , γ 2 γ 1 γ 3 } . (70) It is easy to verify that a Cayley tableau of this group has a form 1 γ 0 γ 13 γ 013 γ 20 γ 2 γ 2013 γ 213 Hence it follows that the CP T -group (70) is a non-Abelian finite group of the order structure (3,4). In more details, it is the group * Then over the field F = C in dependence on a division ring structure of the real subalgebras Cℓ p,q ⊂ C n there exist following isomorphisms between finite groups and groups Ext(C n ): In this case the matrix Π of the pseudoautomorphism A → A is proportional to the unit matrix (identical transformation) and the extended automorphism group Ext(C n ) is reduced to the group of fundamental automorphisms Aut ± (C n ).
In dependence on a spinbasis structure of the subalgebra Cℓ p,q there exist the following group isomorphisms: Proof. First of all, it is necessary to define permutation relations between the elements of the group Ext. We start with the matrix of the pseudoautomorphism A → A (permutation relations between the elements W, E and C are found in Theorem 7). As is known, for the types p − q ≡ 4, 6 (mod 8) the matrix Π exists in the two different forms: 1) Π = E α 1 E α 2 · · · E αa is the product of all complex matrices of the spinbasis at a ≡ 0 (mod 2); 2) Π = E β 1 E β 2 · · · E β b is the product of all real matrices of the spinbasis at b ≡ 1 (mod 2).
Let us consider permutation relations of Π with the matrix K of the pseudoautomorphism A → A ⋆ . The matrix K also exists in the two different forms: K = E α 1 E α 2 · · · E αa at a ≡ 1 (mod 2) and K = E β 1 E β 2 · · · E β b at b ≡ 0 (mod 2). In virtue of the definition K = ΠW, where W = E 1 E 2 · · · E p+q is the spinor representation of the automorphism A → A ⋆ , the matrix Π = E α 1 E α 2 · · · E αa corresponds to a matrix K = E β 1 E β 2 · · · E β b , since n = p + q is always even for the types p − q ≡ 4, 6 (mod 8). Correspondingly, for the matrix Π = E β 1 E β 2 · · · E β b we obtain K = E α 1 E α 2 · · · E αa , where a, b ≡ 1 (mod 2). It is easy to see that in both cases we have a relation ΠK = (−1) ab KΠ, that is, at a, b ≡ 0 (mod 2) the matrices Π and K always commute and at a, b ≡ 1 (mod 2) always anticommute. Let us find now permutation relations of Π with the matrix S of the pseudoantiautomorphism A → A. As is known, the matrix S exists in the two non-equivalent forms: 1) S = E c 1 E c 2 · · · E cs is the product of all complex symmetric and real skewsymmetric matrices at s ≡ 0 (mod 2); 2) S = E d 1 E d 2 · · · E dg is the product of all complex skewsymmetric and real symmetric matrices at g ≡ 1 (mod 2). From S = ΠE it follows that Π = E α 1 E α 2 · · · E αa corresponds to Thus, taking into account that S = ΠE, we obtain for the matrices Π = E α 1 E α 2 · · · E αa and S = E c 1 E c 2 · · · E cs , where m is the number of complex skewsymmetric matrices of the spinbasis of Cℓ p,q , p − q ≡ 4, 6 (mod 8). Since a comparison ak ≡ 0 (mod 2) holds always, then the matrices Π and S commute at m ≡ 0 (mod 2) and anticommute at m ≡ 1 (mod 2). Correspondingly, for the matrices Π = E α 1 E α 2 · · · E αa and S = E d 1 E d 2 · · · E dg , where l is the number of complex symmetric matrices of the spinbasis. Since a(p + q − k) ≡ 0 (mod 2) (a ≡ 0 (mod 2), p + q − k ≡ 1 (mod 2)), then in this case the matrices Π and S commute at l ≡ 0 (mod 2) and anticommute at l ≡ 1 (mod 2). Further, we have for the matrices Π = E β 1 E β 2 · · · E β b and S = E d 1 E d 2 · · · E dg , where u is the number of real skewsymmetric matrices of the spinbasis. Since bk ≡ 0 (mod 2) (b ≡ 1 (mod 2), k ≡ 0 (mod 2)), then Π and S commute at u ≡ 0 (mod 2) and anticommute at u ≡ 1 (mod 2). Finally, for the matrices Π = E β 1 E β 2 · · · E β b and S = E c 1 E c 2 · · · E cs , where v is the number of real symmetric matrices. Therefore, permutation conditions of the matrices Π and S in this case have the form b(p + q − k) ≡ v (mod 2), that is, Π and S commute at v ≡ 1 (mod 2) and anticommute at v ≡ 0 (mod 2), since b, p + q − k ≡ 1 (mod 2). Now we find permutation conditions of Π with the matrix F of the pseudoantiautomorphism A → A ⋆ . In turn, the matrix F exists also in the two non-equivalent forms: Thus, taking into account that F = ΠC, we obtain for the matrices Π = E α 1 E α 2 · · · E αa and F = E d 1 E d 2 · · · E dg . It is easy to see that Π and F commute at l ≡ 0 (mod 2) and anticommute at l ≡ 1 (mod 2), since a, p + q − k ≡ 0 (mod 2). Analogously, for Π = E α 1 E α 2 · · · E αa and F = E c 1 E c 2 · · · E cs . Therefore, Π and F commute at m ≡ 0 (mod 2) and anticommute at m ≡ 1 (mod 2), since a ≡ 0 (mod 2), k ≡ 1 (mod 2). Further, we have for the matrices Π = E β 1 E β 1 · · · E β b and F = E c 1 E c 2 · · · E cs . In this case, Π and F commute at v ≡ 0 (mod 2) and anticommute at v ≡ 1 (mod 2), since b ≡ 1 (mod 2), p + q − k ≡ 0 (mod 2). Finally, Therefore, in this case permutation conditions of the matrices Π and F have the form bk ≡ u (mod 2), that is, Π and F commute at u ≡ 1 (mod 2) and anticommute at u ≡ 0 (mod 2), since b, k ≡ 1 (mod 2). Let us define now permutation conditions of Π with the matrices W, E and C of the transformations A → A ⋆ , A → A and A → A ⋆ , respectively. First of all, according to Theorem 9 in the case of subalgebras 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, Π commutes with W, E and C. In case of the ring K ≃ H (types p − q ≡ 4, 6 (mod 8)) the matrix Π exists in the two forms: Π a at a ≡ 0 (mod 2) and Π b at b ≡ 1 (mod 2). Since a + b = p + q, then the matrix W can be represented by a product E α 1 E α 2 · · · E αa E β 1 E β 2 · · · E β b and for Π = E α 1 E α 2 · · · E αa we have Hence it follows that Π and W commute at ab ≡ 0 (mod 2) and anticommute at ab ≡ 1 (mod 2), but a ≡ 0 (mod 2) and, therefore, the matrices Π and W always commute in this case. Taking Π = E β 1 E β 2 · · · E β b , we find the following conditions Hence it follows that ab ≡ 1 (mod 2), since in this case a, b ≡ 1 (mod 2) (p + q = a + b is an even number). Therefore, at b ≡ 1 (mod 2) the matrices Π and W always anticommute. Let us find now permutation conditions between Π and the matrix E of the antiautomorphism A → A. As is known (see Theorem 7), the matrix E exists in the two non-equivalent forms. First of all, if Π = E α 1 E α 2 · · · E αa and E = E j 1 E j 2 · · · E j k , then it is obvious that Π and E contain m identical complex skewsymmetric matrices. We can represent the matrices Π and E in the form of the following products: Π = E α 1 E α 2 · · · E α l E i 1 E i 2 · · · E im and E = E i 1 E i 2 · · · E im E j 1 E j 2 · · · E ju , where l and u are the numbers of complex symmetric and real skewsymmetric matrices, respectively. Therefore, that is, Π and E commute at m(u + l) ≡ 0 (mod 2) and anticommute at m(u + l) ≡ 1 (mod 2). Analogously, if Π = E α 1 E α 2 · · · E αa and E = E i 1 E i 2 · · · E i p+q−k , then it is easy to see that in this case Π and E contain l identical complex symmetric matrices.
iv (v is the number of all real symmetric matrices of the spinbasis) and Therefore, in this case Π and E commute at l(m + v) ≡ 0 (mod 2) and anticommute at l(m + v) ≡ 1 (mod 2). In turn, the matrices Π = E β 1 E β 2 · · · E β b and E = E j 1 E j 2 · · · E j k contain u identical real skewsymmetric matrices. Therefore, Hence it follows that Π and E commute at u(m + v) ≡ 0 (mod 2) and anticommute at u(m + v) ≡ 1 (mod 2). Finally, the matrices Π = E β 1 E β 2 · · · E β b and E = E i 1 E i 2 · · · E i p+q−k contain v identical real symmetric matrices. Therefore, in this case that is, in this case Π and E commute at v(u+l) ≡ 0 (mod 2) and anticommute at v(u+l) ≡ 1 (mod 2). It is easy to see that permutation conditions of Π with the matrix C of the antiautomorphism A → A ⋆ are analogous to the conditions (82)-(85). Indeed, at Π = E α 1 E α 2 · · · E αa and C = E i 1 E i 2 · · · E i p+q−k these conditions are analogous to (83), that is, l(m+v) ≡ 0, 1 (mod 2). For the matrices Π = E α 1 E α 2 · · · E αa and C = E j 1 E j 2 · · · E j k we obtain the condition (82), that is, m(u+l) ≡ 0, 1 (mod 2). In turn, for Π = E β 1 E β 2 · · · E β b and C = E i 1 E i 2 · · · E i p+q−k we have the condition (85), that is, v(u + l) ≡ 0, 1 (mod 2). Finally, the matrices Let us consider now permutation conditions of the matrix K of A → A ⋆ with other elements of the group Ext(C n ). As it has been shown previously, the structure of K is analogous to the structure of Π, that is, K = E α 1 E α 2 · · · E αa at a ≡ 1 (mod 2) and K = E β 1 E β 2 · · · E β b at b ≡ 0 (mod 2). Therefore, permutation conditions for K are similar to the conditions for Π. Permutation conditions between K and Π are defined by the relation (71).
Let consider permutation conditions of the matrix S of the transformation A → A with other elements of the group Ext(C n ). Permutation conditions of S with the matrices Π and K have been defined previously (see (72)-(75)). Now we define permutation conditions of S with the matrix F of the pseudoantiautomorphism A → A ⋆ and elements of the subgroup Aut ± (Cℓ p,q ). So, let S = E c 1 E c 2 · · · E cs and F = E d 1 E d 2 · · · E dg , where s ≡ 0 (mod 2) and g ≡ 0 (mod 2). As is known, in this case the product S contains all complex symmetric matrices and all real skewsymmetric matrices of the spinbasis. In turn, the product F contains all complex skewsymmetric and real symmetric matrices. Therefore, the product SF does not contain identical matrices. Then that is, in this case S and F always commute, since s, g ≡ 0 (mod 2). If S = E d 1 E d 2 · · · E dg and F = E c 1 E c 2 · · · E cs , where s, g ≡ 1 (mod 2), then from (86) it follows that S and F always anticommute. Further, if S = E c 1 E c 2 · · · E cs , then permutation conditions of S with the matrix W of the automorphism A → A ⋆ have the form: that is, in this case S always commutes with W, since s, g ≡ 0 (mod 2). In turn, the matrix S = E d 1 E d 2 · · · E dg always anticommutes with W, since s, g ≡ 1 (mod 2). Let us define now permutation conditions between S and E. If S = E c 1 E c 2 · · · E cs and E = E j 1 E j 2 · · · E j k , then the product SE contains u identical real skewsymmetric matrices.
that is, S and E commute at u(l + m) ≡ 0 (mod 2) and anticommute at u(l + m) ≡ 1 (mod 2). In turn, the products S = E d 1 E d 2 · · · E dg and E = E j 1 E j 2 · · · E j k contain m identical complex skewsymmetric matrices. Therefore, that is, permutation conditions between S and E in this case have the form m(v + u) ≡ 0, 1 (mod 2). Further, the products that is, permutation conditions between S and E are v(m + l) ≡ 0, 1 (mod 2). Finally, the products S = E c 1 E c 2 · · · E cs and E = E i 1 E i 2 · · · E i p+q−k contain l identical complex symmetric matrices and, therefore, and permutation conditions for S and E in this case have the form l(u + v) ≡ 0, 1 (mod 2). It is easy to see that permutation conditions between S and C are analogous to (88)-(91). Indeed, if S = E c 1 E c 2 · · · E cs and C = E i 1 E i 2 · · · E i p+q−k , then from (91) it follows the comparison l(u + v) ≡ 0, 1 (mod 2). In turn, if S = E d 1 E d 2 · · · E dg and C = E i 1 E i 2 · · · E i p+q−k , then from (90) we obtain v(m + l) ≡ 0, 1 (mod 2). Further, for S = E d 1 E d 2 · · · E dg and C = E j 1 E j 2 · · · E j k from (89) it follows that m(v + u) ≡ 0, 1 (mod 2). Analogously, for S = E c 1 E c 2 · · · E cs and C = E j 1 E j 2 · · · E j k from (88) we have u(l + m) ≡ 0, 1 (mod 2).
Finally, let us consider permutation conditions of the matrix F of A → A ⋆ with other elements of the group Ext(C n ). Permutation conditions of F with the matrices Π, K and S have been found previously (see (76)-(79), (86)). Now we define permutation conditions between F and the elements of the subgroups Aut ± (Cℓ p,q ). It is easy to see that permutation conditions between F and W are equivalent to (87), that is, F = E d 1 E d 2 · · · E dg always commute with W, since s, g ≡ 0 (mod 2), and F = E c 1 E c 2 · · · E cs always anticommute with W, since in this case s, g ≡ 1 (mod 2). In turn, permutation conditions between F and E are equivalent to (88)-(91). (90) it follows v(m+l) ≡ 0, 1 (mod 2). It is easy to see that permutation conditions between F and C are equivalent to (88)-(91). Namely, for F = E d 1 E d 2 · · · E dg , C = E i 1 E i 2 · · · E i p+q−k from (90) we obtain v(m + l) ≡ 0, 1 (mod 2), for F = E c 1 E c 2 · · · E cs , C = E i 1 E i 2 · · · E i p+q−k from (91) it follows l(u + v) ≡ 0, 1 (mod 2). Correspondingly, for F = E c 1 E c 2 · · · E cs , C = E j 1 E j 2 · · · E j k from (88) we find u(l + m) ≡ 0, 1 (mod 2), and for F = E d 1 E d 2 · · · E dg , C = E j 1 E j 2 · · · E j k from (89) we see that m(v + u) ≡ 0, 1 (mod 2). Now, we are in a position to define a detailed classification for extended automorphism groups Ext(C n ). First of all, since for the subalgebras Cℓ p,q over the ring K ≃ R the group Ext(C n ) is reduced to Aut ± (C n ), then all essentially different groups Ext(C n ) correspond to the subalgebras Cℓ p,q with the quaternionic ring K ≃ H, p − q ≡ 4, 6 (mod 8). Let us classify the groups Ext(C n ) with respect to their subgroups Aut ± (Cℓ p,q ). Taking into account the structure of Aut ± (Cℓ p,q ) at p − q ≡ 4, 6 (mod 8) (see Theorem 7) we obtain for the group Ext(C n ) = {I, W, E, C, Π, K, S, F} the following possible realizations: The groups Ext 1 (C n ) and Ext 2 (C n ) have Abelian subgroups Aut − (Cℓ p,q ) (Z 2 ⊗ Z 2 or Z 4 ). In turn, the groups Ext 3 (C n ) and Ext 4 (C n ) have only non-Abelian subgroups Aut + (Cℓ p,q ) (Q 4 /Z 2 or D 4 /Z 2 ).
Finally, let us consider the group Ext 4 (C n ). These groups contain non-Abelian subgroups Aut + (Cℓ p,q ) and, therefore, all Ext 4 are non-Abelian. The isomorphism structure of Ext 4 is similar to Ext 3 .
Example 2. Let us study an extended automorphism group of the Dirac algebra C 4 . We evolve in C 4 the real subalgebra with the quaternionic ring. Let it be the spacetime algebra Cℓ p,q with a spinbasis defined by the matrices (68). We define now elements of the group Ext(C 4 ). First of all, the matrix of the automorphism A → A ⋆ has a form: W = γ 0 γ 1 γ 2 γ 3 . Further, since γ T 0 = γ 0 , γ T 1 = −γ 1 , γ T 2 = −γ 2 , γ T 3 = −γ 3 , then in accordance with Theorem 7 the matrix E of the antiautomorphism A → A is an even product of skewsymmetric matrices of the spinbasis (68), that is, E = γ 1 γ 3 . From the definition C = EW we find that the matrix of the antiautomorphism A → A ⋆ has a form C = γ 0 γ 2 . γ-basis contains three real matrices γ 0 , γ 1 and γ 3 , therefore, for the matrix of the pseudoautomorphism A → A we obtain Π = γ 0 γ 1 γ 3 (see Theorem 9). Further, in accordance with K = ΠW for the matrix of the pseudoautomorphism A → A ⋆ we have K = γ 2 . Finally, for the pseudoantiautomorphisms A → A, A → A ⋆ from the definitions S = ΠE, F = ΠC it follows S = γ 0 , F = γ 1 γ 2 γ 3 . Thus, we come to the following extended automorphism group: Ext(C 4 ) ≃ {I, W, E, C, Π, K, S, F} ≃ {I, γ 0 γ 1 γ 2 γ 3 , γ 1 γ 3 , γ 0 γ 2 , γ 0 γ 1 γ 3 , γ 2 , γ 0 , γ 1 γ 2 γ 3 }. (92) The As follows from this tableau, the group Ext(C 4 ) is non-Abelian. Ext + (C 4 ) contains Abelian group of spacetime reflections, Aut − (Cℓ 1,3 ) ≃ Z 4 , as a subgroup. It is easy to see that the group (92) is a group of the form Ext 2 + with order structure (3,4). More precisely, the group (92) is a finite group * Z 4 ⊗ Z 2 with the signature (−, −, +, −, −, +, +). Coming back to example 1, we see that the groups (70)  Moreover, the subgroups of spacetime reflections of these groups are also isomorphic: Thus, we come to the following result: the finite group (70), derived from the analysis of invariance properties of the Dirac equation with respect to discrete transformations C, P and T , is isomorphic to an extended automorphism group of the Dirac algebra C 4 . This result allows us to study discrete symmetries and their group structure for physical fields of any spin (without handling to analysis of relativistic wave equations).

CP T -structures
As it has been shown previously, there exist 64 different signatures (a, b, c, d, e, f, g) for the extended automorphism group Ext(C n ) of the complex Clifford algebra C n . At this point, the group of fundamental automorphisms, Aut ± (Cℓ p,q ), which has 8 different signa- tures (a, b, c), is defined as a subgroup of Ext(C n ). As is known, the Clifford-Lipschitz group Pin(n, C) (an universal covering of the complex orthogonal group O(n, C)) is completely constructed within the algebra C n (see definition (8)). If we take into account spacetime reflections, then according to [25,26,9] there exist 8 types of universal covering (P T -structures) described by the group Pin a,b,c (n, C). As it shown in [28,29], the group Pin a,b,c (n, C) (correspondingly Pin a,b,c (p, q) over the field F = R) is completely defined within the algebra C n (correspondingly Cℓ p,q ) by means of identification of the reflection subgroup {1, P, T, P T } with the automorphism group {Id, ⋆, , ⋆} of C n (correspondingly Cℓ p,q ). In turn, the pseudoautomorphism A → A of C n and the extended automorphism group {Id, ⋆, , ⋆, , ⋆, , ⋆} ≃ Z 2 ⊗ Z 2 ⊗ Z 2 allows us to give a further generalization of the Clifford-Lipschitz group (8) (correspondingly (2)). We claim that there exist 64 types of the universal covering (CP T -structures) for the complex orthogonal group O(n, C): Pin a,b,c,d,e,f,g (n, C) ≃ (Spin + (n, C) ⊙ C a,b,c,d,e,f,g ) where C a,b,c,d,e,f,g are five double coverings of the group Z 2 ⊗ Z 2 ⊗ Z 2 . All the posiible double coverings C a,b,c,d,e,f,g are given in the following table: The group (93) with non-Abelian C a,b,c,d,e,f,g we will call Cliffordian and respectively non-Cliffordian when C a,b,c,d,e,f,g is Abelian.
Analogously, over the field F = R there exist 64 universal coverings of the real orthogonal group O(p, q): ρ a,b,c,d,e,f,g : Pin a,b,c,d,e,f,g −→ O(p, q), where Pin a,b,c,d,e,f,g (p, q) ≃ (Spin + (p, q) ⊙ C a,b,c,d,e,f,g ) Z 2 .
It is easy to see that in case of the algebra Cℓ p,q (or subalgebra Cℓ p,q ⊂ C n ) with the real division ring K ≃ R, p − q ≡ 0, 2 (mod 8), CP T -structures, defined by the groups (93) and (94), are reduced to the eight Shirokov-Dabrowski P T -structures. Further, using the well-known isomorphism (9), we obtain for the group O(n, C) with odd dimensionality the following universal covering: Pin a,b,c,d,e,f,g (n, C) ≃ Pin a,b,c,d,e,f,g (n − 1, C) e 12···n Pin a,b,c,d,e,f,g (n − 1, C).