aa r X i v : . [ m a t h - ph ] O c t C P T groups of higher spin fields
V. V. Varlamov ∗ Abstract
CP T groups of higher spin fields are defined in the framework of automorphism groupsof Clifford algebras associated with the complex representations of the proper orthochronousLorentz group. Higher spin fields are understood as the fields on the Poincar´e group whichdescribe orientable (extended) objects. A general method for construction of
CP T groupsof the fields of any spin is given.
CP T groups of the fields of spin-1/2, spin-1 and spin-3/2are considered in detail.
CP T groups of the fields of tensor type are discussed. It is shownthat tensor fields correspond to particles of the same spin with different masses.
Keywords : CP T groups, fields on the Poincar´e group, Clifford algebras, automorphism groups,higher spin fieldsPACS numbers:
In 2003,
CP T group was introduced [55] in the context of an extension of automorphism groupsof Clifford algebras. The relationship between
CP T groups and extraspecial groups and universalcoverings of orthogonal groups was established in [55, 57]. In 2004, Socolovsky considered the
CP T group of the spinor field with respect to phase quantities [50] (see also [10, 11, 12, 13, 37]).
CP T groups of spinor fields in the de Sitter spaces of different signatures were studied in theworks [58, 60]. The following logical step in this direction is a definition of the
CP T groups forthe higher spin fields. The formalism developed in the previous works [55, 57] allows us to define
CP T groups for the fields of any spin on the spinspaces associated with representations of thespinor group
Spin + (1 ,
3) (a universal covering of the proper orthochronous Lorentz group).Our consideration based on the concept of generalized wavefunctions introduced by Ginzburgand Tamm in 1947 [22], where the wavefunction depends both coordinates x µ and additional inter-nal variables θ µ which describe spin of the particle, µ = 0 , , ,
3. In 1955, Finkelstein showed [18]that elementary particles models with internal degrees of freedom can be described on manifoldslarger then Minkowski spacetime (homogeneous spaces of the Poincar´e group). The quantum fieldtheories on the Poincar´e group were discussed in the papers [33, 28, 5, 3, 30, 51, 34, 17, 23, 26].A consideration of the field models on the homogeneous spaces leads naturally to a generalizationof the concept of wave function (fields on the Poincar´e group). The general form of these fields isrelated closely with the structure of the Lorentz and Poincar´e group representations [21, 36, 4, 23]and admits the following factorization f ( x, z ) = φ n ( z ) ψ n ( x ), where x ∈ T and φ n ( z ) form a basisin the representation space of the Lorentz group. At this point, four parameters x µ correspond toposition of the point-like object, whereas remaining six parameters z ∈ Spin + (1 ,
3) define orien-tation in quantum description of orientable (extended) object [24, 25] (see also [27]). It is obvious ∗ Siberian State Industrial University, Kirova 42, Novokuznetsk 654007, Russia, e-mail: [email protected] unmeasurable (and hence unphysical)point-like quantities. This is because no physical quantity can be measured in a point, but in aregion, the size of which (or ’diameter’ of the extended object) is constrained by the resolution ofmeasuring equipment [2]. Taking it into account, we come to consideration of physical quantityas an extended object, the generalized wavefunction of which is described by the field ψ ( α ) = h x, g | ψ i on the homogeneous space of some orthogonal group SO( p, q ), where x ∈ T n (position) and g ∈ Spin + ( p, q ) (orientation), n = p + q . So, in [45, 46] Segal and Zhou proved convergence ofquantum field theory, in particular, quantum electrodynamics, on the homogeneous space R × S of the conformal group SO(2 , S is the three-dimensional real sphere.In the present work we describe discrete symmetries of the generalized wavefunctions ψ ( α ) = h x, g | ψ i (fields on the Poincar´e group) in terms of involutive automorphisms of the subgroup Spin + ( p, q ), As is known, the universal covering of the proper Poincar´e group is isomorphic to asemidirect product SL(2; C ) ⊙ T or Spin + (1 , ⊙ T . Since the group T is Abelian, then all itsrepresentations are one-dimensional. Thus, all the finite-dimensional representations of the properPoincar´e group in essence are equivalent to the representations C of the group Spin + (1 , P , time reversal T and their combination P T correspond to an automorphism ⋆ (involution), an antiautomorphism e (reversion) and anantiautomorphism e ⋆ (conjugation), respectively. The fundamental automorphisms of the Cliffordalgebras are compared to elements of the finite group formed by the discrete transformations.In turn, a set of the fundamental automorphisms, added by an identical automorphism, forms afinite group Aut( Cℓ ), for which in virtue of the Wedderburn-Artin Theorem there exists a matrix(spinor) representation. Further, 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 discretesymmetry. This transformation is firstly appeared on the representation spaces of the Lorentzgroup and its nature is strongly different from other discrete symmetries. For that reason thecharge conjugation C is represented by a pseudoautomorphism A → A which is not fundamen-tal automorphism of the Clifford algebra. All spinor representations of the pseudoautomorphism
A → A were given in [55]. An introduction of the transformation
A → A allows us to extendthe automorphism group Aut( Cℓ ) of the Clifford algebra. It was shown [55] that automorphisms A → A ⋆ , A → e A , A → f A ⋆ , A → A , A → A ⋆ , A → e A and A → f A ⋆ form a finite group oforder 8 (an extended automorphism group Ext( Cℓ ) = { Id , ⋆, e , e ⋆, , ⋆, e , e ⋆ } ). The group Ext( Cℓ )is a generating group of the full CP T group {± , ± P, ± T, ± P T, ± C, ± CP, ± CT, ± CP T } . Thereare also other realizations of the discrete symmetries via the automorphisms of the Lorentz andPoincar´e groups, see [21, 35, 29, 49, 9].The present paper is organized as follows. In the section 2 we briefly discuss the basis notionsconcerning Clifford algebras and CP T groups, and also we consider their descriptions withinuniversal coverings of orthogonal groups and spinor representations. In the section 3 we introducethe main objects of our study,
CP T groups of higher spin fields. These groups are defined on thesystem of complex representations of the group
Spin + (1 , CP T groups for the fields (1 / , ⊕ (0 , / , ⊕ (0 ,
1) and (3 / , ⊕ (0 , / CP T groups for the fields of tensor type.2
Algebraic and group theoretical preliminaries
In this section we will consider some basic facts concerning automorphisms of the Clifford algebrasand universal coverings of orthogonal groups.Let F be a field of characteristic 0 ( F = R , F = C ), where R and C are the fields of realand complex numbers, respectively. A Clifford algebra Cℓ over a field F is an algebra with 2 n basis elements: e (unit of the algebra) e , e , . . . , e n and products of the one–index elements e i i ...i k = e i e i . . . e i k . Over the field F = R the Clifford algebra is denoted as Cℓ p,q , where theindices p and q correspond to the indices of the quadratic form Q = x + . . . + x p − . . . − x p + q of a vector space V associated with Cℓ p,q .An arbitrary element A of the algebra Cℓ p,q is represented by a following formal polynomial: A = a e + n X i =1 a i e i + n X i =1 n X j =1 a ij e ij + . . . + n X i =1 · · · n X i k =1 a i ...i k e i ...i k ++ . . . + a ...n e ...n = n X k =0 a i i ...i k e i i ...i k . 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 consistingof homogeneous even elements, respectively. Then the automorphism A → A ⋆ is such that theelement A ′′ is not changed, and the element A ′ changes sign: A ⋆ = −A ′ + A ′′ . If A is a homogeneouselement, then A ⋆ = ( − k A , (1)where k is a degree of the element. It is easy to see that the automorphism A → A ⋆ may beexpressed via the volume element ω = e ...p + q : A ⋆ = ω A ω − , (2)where ω − = ( − ( p + q )( p + q − ω . When k is odd, the basis elements e i i ...i k the sign changes, andwhen k is even, the sign is not changed.3) Reversion : An antiautomorphism
A → e A and e i → e i .The antiautomorphism A → e A is a reversion of the element A , that is the substitution of eachbasis element e i i ...i k ∈ A by the element e i k i k − ...i : e i k i k − ...i = ( − k ( k − e i i ...i k . Therefore, for any
A ∈ Cℓ p,q we have e A = ( − k ( k − A . (3)3) Conjugation : An antiautomorphism
A → f A ⋆ and e i → − e i .This antiautomorphism is a composition of the antiautomorphism A → e A with the automorphism A → A ⋆ . In the case of a homogeneous element from the formulae (1) and (3), it follows f A ⋆ = ( − k ( k +1)2 A . (4)As is known, the complex 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 ofthe discrete transformation known in physics as a charge conjugation C . Indeed, let { e , . . . , e n } be an orthobasis in the space C n , e 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 { e , . . . , e p , i e p +1 , . . . , i e p + q } (5)allows one 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 (5) with real coefficients. In such a way we obtain a real vectorspace R p,q endowed (in general case) with a non–degenerate quadratic form Q ( x ) = x + x + . . . + x p − x p +1 − x p +2 − . . . − x p + q , where x , . . . , x p + q are coordinates of the vector x in the basis (5). It is easy to see that theextraction of R p,q in C n induces an extraction of a real subalgebra Cℓ p,q in C n . Therefore, anyelement A ∈ C n can be unambiguously represented in the form A = A + i A , where A , A ∈ Cℓ p,q . The one-to-one mapping A −→ A = A − i A (6)transforms the algebra C n into itself with preservation of addition and multiplication operationsfor the elements A ; the operation of multiplication of the element A by the number transforms toan operation of multiplication by the complex conjugate number. Any mapping of C n satisfyingthese conditions is called a pseudoautomorphism . Thus, the extraction of the subspace R p,q in thespace C n induces in the algebra C n a pseudoautomorphism A → A [39, 40].An introduction of the pseudoautomorphism
A → A allows us to extend the automorphismset of the complex Clifford algebra C n . Namely, we add to the four fundamental automorphisms A → A , A → A ⋆ , A → e A , A → f A ⋆ the pseudoautomorphism A → A and following threecombinations:1) A pseudoautomorphism
A → A ⋆ . This transformation is a composition of the pseudoautomor-phism A → A with the automorphism
A → A ⋆ .2) A pseudoantiautomorphism A → e A . This transformation is a composition of A → A with theantiautomorphism
A → e A .3) A pseudoantiautomorphism A → f A ⋆ (a composition of A → A with the antiautomorphism
A → f A ⋆ ).Thus, we obtain an automorphism set of C n consisting of the eight transformations. Let usshow that the set { Id , ⋆, e , e ⋆, , ⋆, e , e ⋆ } forms a finite group of order 8 and let us give a physicalinterpretation of this group. Proposition 1 ([55]) . Let C n be a Clifford algebra over the field F = C and let Ext( C n ) = { Id , ⋆, e , e ⋆, , ⋆, e , e ⋆ } be an extended automorphism group of the algebra C n . Then there is n isomorphism between Ext( C n ) and CP T / Z group of the discrete transformations, Ext( C n ) ≃{ , P, T, P T, C, CP, CT, CP T } ≃ Z ⊗ Z ⊗ Z . In this case, space inversion P , time rever-sal T , full reflection P T , charge conjugation C , transformations CP , CT and the full CP T –transformation correspond to the automorphism
A → A ⋆ , antiautomorphisms A → e A , A → f A ⋆ ,pseudoautomorphisms A → A , A → A ⋆ , pseudoantiautomorphisms A → e A and A → f A ⋆ , respec-tively.Proof. The group { , P, T, P T, C, CP, CT, CP T } at the conditions P = T = ( P T ) = C =( CP ) = ( CT ) = ( CP T ) = 1 and commutativity of all the elements forms an Abelian groupof order 8, which is isomorphic to a cyclic group Z ⊗ Z ⊗ Z . The multiplication table of thisgroup shown in Tab. 1. 1 P T P T C CP CT CP T
P T P T C CP CT CP TP P P T T CP C CP T CTT T P T P CT CP T C CPP T P T T P CP T CT CP CC C CP CT CP T P T P TCP CP C CP T CT P P T TCT CT CP T C CP T P T PCP T CP T CT CP C P T T P Tab. 1:
The multiplication table of the
CP T / Z group. In turn, for the extended automorphism group { Id , ⋆, e , e ⋆, , ⋆, e , e ⋆ } in virtue of commu-tativity g ( A ⋆ ) = (cid:16) e A (cid:17) ⋆ , ( A ⋆ ) = (cid:0) A (cid:1) ⋆ , (cid:16) e A (cid:17) = g(cid:0) A (cid:1) , (cid:16) f A ⋆ (cid:17) = g(cid:0) A (cid:1) ⋆ and an involution property ⋆⋆ = e e = = Id we have the multiplication table shown in Tab. 2. The identity of multipli-Id ⋆ e e ⋆ ⋆ e e ⋆ Id Id ⋆ e e ⋆ ⋆ e e ⋆⋆ ⋆ Id e ⋆ e ⋆ e ⋆ ee e ⋆ Id ⋆ e e ⋆ ⋆ e ⋆ e ⋆ e ⋆ Id e ⋆ e ⋆⋆ e e ⋆ Id ⋆ e e ⋆⋆ ⋆ e ⋆ e ⋆ Id e ⋆ ee e e ⋆ ⋆ e e ⋆ Id ⋆ e ⋆ e ⋆ e ⋆ e ⋆ e ⋆ Id Tab. 2:
The multiplication table of the extended automorphism group. { , P, T, P T, C, CP, CT, CP T } ≃ { Id , ⋆, e , e ⋆, , ⋆, e , e ⋆ } ≃ Z ⊗ Z ⊗ Z . Further, in the case of P = T = . . . = ( CP T ) = ± CP T / Z group and a group Ext ( C n ). The elementsof Ext ( C n ) are spinor representations of the automorphisms of the algebra C n . As mentionedpreviously, the Wedderburn-Artin Theorem allows us to define any spinor representaions for theautomorphisms of C n . We list these transformations and their spinor representations (for moredetails see [55]): A −→ A ⋆ , A ⋆ = WAW − , (7) A −→ e A , e A = EA T E − , (8) A −→ f A ⋆ , f A ⋆ = CA T C − , C = EW , (9) A −→ A , A = Π A ∗ Π − , (10) A −→ A ⋆ , A ⋆ = KA ∗ K − , K = Π W , (11) A −→ e A , e A = S (cid:0) A T (cid:1) ∗ S − , S = Π E , (12) A −→ f A ⋆ , f A ⋆ = F ( A ∗ ) T F − , F = Π C , (13)where the symbol T means a transposition, and ∗ is a complex conjugation. The detailed clas-sification of the extended automorphism groups Ext ( C n ) was given in [55]. First of all, since forthe subalgebras Cℓ p,q over the ring K ≃ R the group Ext ( C n ) is reduced to Aut ± ( C n ) (reflectiongroup [53]), then all the essentially different groups Ext ( C n ) correspond to subalgebras Cℓ p,q withthe quaternionic ring K ≃ H , p − q ≡ , Ext ( C n ) isgiven with respect to the subgroups Aut ± ( Cℓ p,q ). Taking into account the structure of Aut ± ( Cℓ p,q ),we have at p − q ≡ , Ext ( C n ) = { I , W , E , C , Π , K , S , F } the followingrealizations [55]: Ext ( C n ) = (cid:8) I , E ··· p + q , E j j ··· j k , E i i ··· i p + q − k , E α α ··· α a , E β β ··· β b , E c c ··· c s , E d d ··· d g (cid:9) , Ext ( C n ) = (cid:8) I , E ··· p + q , E j j ··· j k , E i i ··· i p + q − k , E β β ··· β b , E α α ··· α a , E d d ··· d g , E c c ··· c s (cid:9) , Ext ( C n ) = (cid:8) I , E ··· p + q , E i i ··· i p + q − k , E j j ··· j k , E α α ··· α a , E β β ··· β b , E d d ··· d g , E c c ··· c s (cid:9) , Ext ( C n ) = (cid:8) I , E ··· p + q , E i i ··· i p + q − k , E j j ··· j k , E β β ··· β b , E α α ··· α a , E c c ··· c s , E d d ··· d g (cid:9) . The groups
Ext ( C n ) and Ext ( C n ) have Abelian subgroups Aut − ( Cℓ p,q ) ( Z ⊗ Z or Z ). In turn,the groups Ext ( C n ) and Ext ( C n ) have non-Abelian subgroups Aut + ( Cℓ p,q ) ( Q / Z or D / Z ).The full number of different realizations of Ext ( C n ) is 64.As is known, the Lipschitz group Γ p,q , also called the Clifford group, introduced by Lipschitzin 1886 [31], may be defined as the subgroup of invertible elements s of the algebra Cℓ p,q : Γ p,q = (cid:8) s ∈ Cℓ + p,q ∪ Cℓ − p,q | ∀ x ∈ R p,q , s x s − ∈ R p,q (cid:9) . The set Γ + p,q = Γ p,q ∩ Cℓ + p,q is called special Lipschitz group [14].Let N : Cℓ p,q → Cℓ p,q , N ( x ) = x e x . If x ∈ R p,q , then N ( x ) = x ( − x ) = − x = − Q ( x ).Further, the group Γ p,q has a subgroup Pin ( p, q ) = { s ∈ Γ p,q | N ( s ) = ± } . (14)6nalogously, a spinor group Spin ( p, q ) is defined by the set Spin ( p, q ) = (cid:8) s ∈ Γ + p,q | N ( s ) = ± (cid:9) . (15)It is obvious that Spin ( p, q ) = Pin ( p, q ) ∩ Cℓ + p,q . The group Spin ( p, q ) contains a subgroup Spin + ( p, q ) = { s ∈ Spin ( p, q ) | N ( s ) = 1 } . (16)The groups O( p, q ) , SO( p, q ) and SO + ( p, q ) are isomorphic, respectively, to the following quotientgroups O( p, q ) ≃ Pin ( p, q ) / Z , SO( p, q ) ≃ Spin ( p, q ) / Z , SO + ( p, q ) ≃ Spin + ( p, q ) / Z , where the kernel Z = { , − } . Thus, the groups Pin ( p, q ), Spin ( p, q ) and Spin + ( p, q ) are theuniversal coverings of the groups O( p, q ) , SO( p, q ) and SO + ( p, q ), respectively.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 , (17)and C a,b,c,d,e,f,g = {± , ± P, ± T, ± P T, ± C, ± CP, ± CT, ± CP T } is a full CP T group [55, 57]. C a,b,c,d,e,f,g is a finite group of order 16. The groupExt( Cℓ p,q ) = C a,b,c,d,e,f,g Z ≃ CP T / Z is called the generating group . In essence, C a,b,c,d,e,f,g are five double coverings of the group Z ⊗ Z ⊗ Z (extraspecial Salingaros groups, see [43, 6]). All the possible double coverings C a,b,c,d,e,f,g are given in the Table 3. The group (17) with non-Abelian C a,b,c,d,e,f,g is called Cliffordian group a b c d e f g C a,b,c,d,e,f,g
Type+ + + + + + + Z ⊗ Z ⊗ Z ⊗ Z Abelianthree ‘+’ and four ‘ − ’ Z ⊗ Z ⊗ Z one ‘+’ and six ‘ − ’ Q ⊗ Z Non–Abelianfive ‘+’ and two ‘ − ’ D ⊗ Z three ‘+’ and four ‘ − ’ ∗ Z ⊗ Z ⊗ Z Tab. 3:
Extraspecial finite groups C a,b,c,d,e,f,g of order 16. and respectively non-Cliffordian group when C a,b,c,d,e,f,g is Abelian. It is easy to see that in thecase of the algebra Cℓ p,q (or subalgebra Cℓ p,q ⊂ C n ) with the real division ring K ≃ R , p − q ≡ , CP T -structures, defined by the groups (17), are reduced to the eight Shirokov-D¸abrowski
P T -structures [47, 48, 16]. 7
C P T groups on the representation spaces of Spin + (1 , Let us consider the field ψ ( α ) = h x, g | ψ i , (18)where x ∈ T , g ∈ Spin + (1 , Spin + (1 , ≃ SU(2) ⊗ SU(2) is a universalcovering of the proper orthochronous Lorentz group SO (1 , x ∈ T and g ∈ Spin + (1 ,
3) describe position and orientation of the extended object defined by the field (18)(the field on the Poincar´e group). The basic idea is to define discrete symmetries of the field (18)within the group
Pin a,b,c,d,e,f,g (1 , ≃ Spin + (1 , ⊙ C a,b,c,d,e,f,g Z . The automorphisms (discrete symmetries) of
Pin a,b,c,d,e,f,g (1 ,
3) are outer automorphisms withrespect to transformations of the group
Spin + (1 , CP T groups C a,b,c,d,e,f,g of physicalfields of any spin on the representation spaces of Spin + (1 , Theorem 1.
Let
Pin a,b,c,d,e,f,g (1 , ≃ Spin + (1 , ⊙ C a,b,c,d,e,f,g Z be the universal covering of the proper Lorentz group SO(1 , , where C a,b,c,d,e,f,g = {± , ± P, ± T, ± P T, ± C, ± CP, ± CT, ± CP T } is a CP T group of some physical field defined inthe framework of finite-dimensional representation of the group
Spin + (1 , . At this point, thereexits a correspondence P ∼ W , T ∼ E , P T ∼ C , C ∼ Π , CP ∼ K , CT ∼ S , CP T ∼ F , where { I , W , E , C , Π , K , S , F } ≃ Ext( C n ) is an automorphism group of the algebra C n . Then CP T groupof the field ( l, ⊕ (0 , ˙ l ) is constructed in the framework of the finite-dimensional representation C l + l − , ⊕ C ,l − l +1 of Spin + (1 , defined on the spinspace S k ⊗ S r with the algebra C ⊗ C ⊗ · · · ⊗ C | {z } k times M ∗ C ⊗ ∗ C ⊗ · · · ⊗ ∗ C | {z } r times , where ( l , l ) = (cid:0) k , k + 1 (cid:1) , ( − l , l ) = (cid:0) − r , r + 1 (cid:1) . In turn, a CP T group of the field ( l ′ , l ′′ ) ⊕ ( ˙ l ′′ , ˙ l ′ ) is constructed in the framework of representation C l + l − ,l − l +1 ⊕ C l − l +1 ,l + l − of Spin + (1 , defined on the spinspace S k + r ⊕ S k + r with the algebra C ⊗ C ⊗ · · · ⊗ C O ∗ C ⊗ ∗ C ⊗ · · · ⊗ ∗ C | {z } k + r times M ∗ C ⊗ ∗ C ⊗ · · · ⊗ ∗ C O C ⊗ C ⊗ · · · ⊗ C | {z } r + k times , where ( l , l ) = (cid:0) k − r , k + r + 1 (cid:1) .Proof. As is known, when Cℓ p,q is simple, then the map Cℓ p,q γ −→ End K ( S ) , u −→ γ ( u ) , γ ( u ) ψ = uψ (19)gives an irreducible and faithful representation of Cℓ p,q in the spinspace S m ( K ) ≃ I p,q = Cℓ p,q f ,where ψ ∈ S m , m = p + q .On the other hand, when Cℓ p,q is semi-simple, then the map Cℓ p,q γ −→ End K ⊕ ˆ K ( S ⊕ ˆ S ) , u −→ γ ( u ) , γ ( u ) ψ = uψ (20)gives a faithful but reducible representation of Cℓ p,q in the double spinspace S ⊕ ˆ S , where ˆ S = { ˆ ψ | ψ ∈ S } . In this case, the ideal S ⊕ ˆ S possesses a right K ⊕ ˆ K -linear structure, ˆ K = { ˆ λ | λ ∈ K } ,8nd K ⊕ ˆ K is isomorphic to the double division ring R ⊕ R when p − q ≡ H ⊕ H when p − q ≡ γ in (19) and (20) defines the so called left-regular spinorrepresentation of Cℓ ( Q ) in S and S ⊕ ˆ S , respectively. Furthermore, γ is faithful which means that γ is an algebra monomorphism. In (19), γ is irreducible which means that S possesses no proper(that is, = 0 , S ) invariant subspaces under the left action of γ ( u ), u ∈ Cℓ p,q . Representation γ in (20) is therefore reducible since { ( ψ, | ψ ∈ S } and { (0 , ˆ ψ ) | ˆ ψ ∈ ˆ S } are two proper subspaces of S ⊕ ˆ S invariant under γ ( u ) (see [32, 15, 38]).Since the spacetime algebra Cℓ , is the simple algebra, then the map (19) gives an irreduciblerepresentation of Cℓ , in the spinspace S ( H ). In turn, representations of the group Spin + (1 , ∈ Cℓ +1 , ≃ Cℓ , are defined in the spinspace S ( C ).Let us consider now spintensor representations of the group G + ≃ SL(2; C ) which, as is known,form the base of all the finite-dimensional representations of the Lorentz group, and also weconsider their relationship with the complex Clifford algebras. From each complex Clifford algebra C n = C ⊗ Cℓ p,q ( n = p + q ) we obtain the spinspace S n/ which is a complexification of theminimal left ideal of the algebra Cℓ p,q : S n/ = C ⊗ I p,q = C ⊗ Cℓ p,q f pq , where f pq is the primitiveidempotent of the algebra Cℓ p,q . Further, a spinspace related with the Pauli algebra C has theform S = C ⊗ I , = C ⊗ Cℓ , f or S = C ⊗ I , = C ⊗ Cℓ , f ( C ⊗ I , = C ⊗ Cℓ , f ).Therefore, the tensor product of the k algebras C induces a tensor product of the k spinspaces S : S ⊗ S ⊗ · · · ⊗ S = S k . Vectors of the spinspace S k (or elements of the minimal left ideal of C k ) are spintensors of thefollowing form: s α α ··· α k = X s α ⊗ s α ⊗ · · · ⊗ s α k , (21)where summation is produced on all the index collections ( α . . . α k ), α i = 1 ,
2. For the eachspinor s α i from (21) we have ′ s α ′ i = σ α ′ i α i s α i . Therefore, in general case we obtain ′ s α ′ α ′ ··· α ′ k = X σ α ′ α σ α ′ α · · · σ α ′ k α k s α α ··· α k . (22)A representation (22) is called undotted spintensor representation of the proper Lorentz group ofthe rank k .Further, let ∗ C be the Pauli algebra with the coefficients which are complex conjugate tothe coefficients of C . Let us show that the algebra ∗ C is derived from C under action of theautomorphism A → A ⋆ or antiautomorphism A → e A . Indeed, in virtue of an isomorphism C ≃ Cℓ , a general element A = a e + X i =1 a i e i + X i =1 3 X j =1 a ij e ij + a e of the algebra Cℓ , can be written in the form A = ( a + ωa ) e + ( a + ωa ) e + ( a + ωa ) e + ( a + ωa ) e , (23)where ω = e . Since ω belongs to a center of the algebra Cℓ , ( ω commutes with all the basiselements) and ω = −
1, then we can to suppose ω ≡ i . The action of the automorphism ⋆ on thehomogeneous element A of the degree k is defined by the formula A ⋆ = ( − k A . In accordancewith this the action of the automorphism A → A ⋆ , where A is the element (23), has the form A −→ A ⋆ = − ( a − ωa ) e − ( a − ωa ) e − ( a − ωa ) e − ( a − ωa ) e . (24)9herefore, ⋆ : C → − ∗ C . Correspondingly, the action of the antiautomorphism A → e A on thehomogeneous element A of the degree k is defined by the formula e A = ( − k ( k − A . Thus, for theelement (23) we obtain A −→ e A = ( a − ωa ) e + ( a − ωa ) e + ( a − ωa ) e + ( a − ωa ) e , (25)that is, e : C → ∗ C . This allows us to define an algebraic analogue of the Wigner’s representationdoubling: C ⊕ ∗ C . Further, from (23) it follows that A = A + ω A = ( a e + a e + a e + a e ) + ω ( a e + a e + a e + a e ). In general case, by virtue of an isomorphism C k ≃ Cℓ p,q ,where Cℓ p,q is a real Clifford algebra with a division ring K ≃ C , p − q ≡ , Cℓ p,q an expression A = A + ω A , here ω = e ...p + q = − ω ≡ i . Thus, from C k under action of the automorphism A → A ⋆ we obtain a general algebraicdoubling C k ⊕ ∗ C k . (26)The tensor product ∗ C ⊗ ∗ C ⊗ · · · ⊗ ∗ C ≃ ∗ C r of the r algebras ∗ C induces the tensor productof the r spinspaces ˙ S : ˙ S ⊗ ˙ S ⊗ · · · ⊗ ˙ S = ˙ S r . Vectors of the spinspace ˙ S r has the form s ˙ α ˙ α ··· ˙ α r = X s ˙ α ⊗ s ˙ α ⊗ · · · ⊗ s ˙ α r , (27)where the each cospinor s ˙ α i from (27) is transformed by the rule ′ s ˙ α ′ i = σ ˙ α ′ i ˙ α i s ˙ α i . Therefore, ′ s ˙ α ′ ˙ α ′ ··· ˙ α ′ r = X σ ˙ α ′ ˙ α σ ˙ α ′ ˙ α · · · σ ˙ α ′ r ˙ α r s ˙ α ˙ α ··· ˙ α r . (28)The representation (28) is called a dotted spintensor representation of the proper Lorentz group ofthe rank r .In general case we have a tensor product of the k algebras C and the r algebras ∗ C : C ⊗ C ⊗ · · · ⊗ C O ∗ C ⊗ ∗ C ⊗ · · · ⊗ ∗ C ≃ C k ⊗ ∗ C r , which induces a spinspace S ⊗ S ⊗ · · · ⊗ S O ˙ S ⊗ ˙ S ⊗ · · · ⊗ ˙ S = S k + r with the vectors s α α ··· α k ˙ α ˙ α ··· ˙ α r = X s α ⊗ s α ⊗ · · · ⊗ s α k ⊗ s ˙ α ⊗ s ˙ α ⊗ · · · ⊗ s ˙ α r . (29)In this case we have a natural unification of the representations (22) and (28): ′ s α ′ α ′ ··· α ′ k ˙ α ′ ˙ α ′ ··· ˙ α ′ r = X σ α ′ α σ α ′ α · · · σ α ′ k α k σ ˙ α ′ ˙ α σ ˙ α ′ ˙ α · · · σ ˙ α ′ r ˙ α r s α α ··· α k ˙ α ˙ α ··· ˙ α r . (30)So, a representation (30) is called a spintensor representation of the proper Lorentz group of therank ( k, r ).Further, let g → T g be an arbitrary linear representation of the proper orthochronous Lorentzgroup G + = SO (1 ,
3) and let A i ( t ) = T a i ( t ) be an infinitesimal operator corresponding to the10otation a i ( t ) ∈ G + . Analogously, let B i ( t ) = T b i ( t ) , where b i ( t ) ∈ G + is the hyperbolic rotation.The operators A i and B i satisfy to the following relations:[ A , A ] = A , [ A , A ] = A , [ A , A ] = A , [ B , B ] = − A , [ B , B ] = − A , [ B , B ] = − A , [ A , B ] = 0 , [ A , B ] = 0 , [ A , B ] = 0 , [ A , B ] = B , [ A , B ] = − B , [ A , B ] = B , [ A , B ] = − B , [ A , B ] = B , [ A , B ] = − B . (31)Denoting I = A , I = A , I = A , and I = B , I = B , I = B we write the relations (31)in a more compact form: (cid:2) I µν , I λρ (cid:3) = δ µρ I λν + δ νλ I µρ − δ νρ I µλ − δ µλ I νρ . As is known [21], finite-dimensional (spinor) representations of the group SO (1 ,
3) in the spaceof symmetrical polynomials Sym ( k,r ) have the following form: T g q ( ξ, ξ ) = ( γξ + δ ) l + l − ( γξ + δ ) l − l +1 q (cid:18) αξ + βγξ + δ ; αξ + βγξ + δ (cid:19) , (32)where k = l + l − r = l − l + 1, and the pair ( l , l ) defines some representation of the groupSO (1 ,
3) in the Gel’fand-Naimark basis: H ξ kν = mξ kν ,H + ξ kν = p ( k + ν + 1)( k − ν ) ξ k,ν +1 ,H − ξ kν = p ( k + ν )( k − ν + 1) ξ k,ν − ,F ξ kν = C l √ k − ν ξ k − ,ν − A l νξ k,ν − C k +1 p ( k + 1) − ν ξ k +1 ,ν ,F + ξ kν = C k p ( k − ν )( k − ν − ξ k − ,ν +1 − A k p ( k − ν )( k + ν + 1) ξ k,ν +1 ++ C k +1 p ( k + ν + 1)( k + ν + 2) ξ k +1 ,ν +1 ,F − ξ kν = − C k p ( k + ν )( k + ν − ξ k − ,ν − − A k p ( k + ν )( k − ν + 1) ξ k,ν − −− C k +1 p ( k − ν + 1)( k − ν + 2) ξ k +1 ,ν − ,A k = i l l k ( k + 1) , C k = i k s ( k − l )( k − l )4 k − , (33) ν = − k, − k + 1 , . . . , k − , k,k = l , l + 1 , . . . , where l is positive integer or half-integer number, l is an arbitrary complex number. Theseformulae define a finite–dimensional representation of the group SO (1 ,
3) when l = ( l + p ) , p is some natural number. In the case l = ( l + p ) we have an infinite-dimensional representationof SO (1 , H , H + , H − , F , F + , F − are H + = i A − A , H − = i A + A , H = i A ,F + = i B − B , F − = i B + B , F = i B . X l = 12 i ( A l + i B l ) , Y l = 12 i ( A l − i B l ) , (34)( l = 1 , , . Using the relations (31), we find that[ X k , X l ] = i ε klm X m , [ Y l , Y m ] = i ε lmn Y n , [ X l , Y m ] = 0 . (35)Further, introducing generators of the form X + = X + i X , X − = X − i X , Y + = Y + i Y , Y − = Y − i Y , (cid:27) (36)we see that in virtue of commutativity of the relations (35) a space of an irreducible finite–dimensional representation of the group SL(2 , C ) can be spanned on the totality of (2 l + 1)(2 ˙ l + 1)basis vectors | l, m ; ˙ l, ˙ m i , where l, m, ˙ l, ˙ m are integer or half–integer numbers, − l ≤ m ≤ l , − ˙ l ≤ ˙ m ≤ ˙ l . Therefore, X − | l, m ; ˙ l, ˙ m i = p ( l + m )( l − m + 1) | l, m − , ˙ l, ˙ m i ( m > − l ) , X + | l, m ; ˙ l, ˙ m i = p ( l − m )( l + m + 1) | l, m + 1; ˙ l, ˙ m i ( m < l ) , X | l, m ; ˙ l, ˙ m i = m | l, m ; ˙ l, ˙ m i , Y − | l, m ; ˙ l, ˙ m i = q ( ˙ l + ˙ m )( ˙ l − ˙ m + 1) | l, m ; ˙ l, ˙ m − i ( ˙ m > − ˙ l ) , Y + | l, m ; ˙ l, ˙ m i = q ( ˙ l − ˙ m )( ˙ l + ˙ m + 1) | l, m ; ˙ l, ˙ m + 1 i ( ˙ m < ˙ l ) , Y | l, m ; ˙ l, ˙ m i = ˙ m | l, m ; ˙ l, ˙ m i . (37)From the relations (35) it follows that each of the sets of infinitesimal operators X and Y generatesthe group SU(2) and these two groups commute with each other. Thus, from the relations (35) and(37) it follows that the group SL(2 , C ), in essence, is equivalent locally to the group SU(2) ⊗ SU(2).In contrast to the Gel’fand–Naimark representation for the Lorentz group [21, 36], which does notfind a broad application in physics, a representation (37) is a most useful in theoretical physics(see, for example, [1, 44, 41, 42]). This representation for the Lorentz group was first given byVan der Waerden in [62]. It should be noted here that the representation basis, defined by theformulae (34)–(37), has an evident physical meaning. For example, in the case of (1 , ⊕ (0 , X and Y correspond to the right and left polarization states of the photon. The following relationsbetween generators Y ± , X ± , Y , X and H ± , F ± , H , F define a relationship between the Van derWaerden and Gel’fand-Naimark bases: Y + = −
12 ( F + + i H + ) , Y − = −
12 ( F − + i H − ) , Y = −
12 ( F + i H ) , X + = 12 ( F + − i H + ) , X − = 12 ( F − − i H − ) , X = 12 ( F − i H ) . The relation between the numbers l , l and the number l (the weight of representation in thebasis (37)) is given by the following formula:( l , l ) = ( l, l + 1) . l = l + l − . (38)As is known [21], if an irreducible representation of the proper Lorentz group SO (1 ,
3) is definedby the pair ( l , l ), then a conjugated representation is also irreducible and is defined by a pair ± ( l , − l ). Therefore, ( l , l ) = (cid:16) − ˙ l, ˙ l + 1 (cid:17) . Thus, ˙ l = l − l + 12 . (39)Further, representations τ s ,s and τ s ′ ,s ′ are called interlocking irreducible representations ofthe Lorentz group , that is, such representations that s ′ = s ± , s ′ = s ± [20]. The two mostfull schemes of the interlocking irreducible representations of the Lorentz group (Gel’fand-Yaglomchains) for integer and half-integer spins are shown on the Fig. 1 and Fig. 2. As follows from( s, · · · ...(2 , · · · (cid:18) s + 22 , s − (cid:19) · · · (1 , (cid:18) , (cid:19) · · · (cid:18) s + 12 , s − (cid:19) · · · (0 , (cid:18) , (cid:19) (1 , · · · (cid:16) s , s (cid:17) · · · (0 , (cid:18) , (cid:19) · · · (cid:18) s − , s + 12 (cid:19) · · · (0 , · · · (cid:18) s − , s + 22 (cid:19) · · · ...(0 , s ) · · · Fig. 1:
Interlocking representation scheme for the fields of integer spin (Bose-scheme).
Fig. 1 the simplest field is the scalar field (0 , . s, · · · ... (cid:18) , (cid:19) · · · (cid:18) s + 34 , s − (cid:19) · · · (cid:18) , (cid:19) (cid:18) , (cid:19) · · · (cid:18) s + 14 , s − (cid:19) · · · (cid:18) , (cid:19) (cid:18) , (cid:19) · · · (cid:18) s − , s + 14 (cid:19) · · · (cid:18) , (cid:19) · · · (cid:18) s − , s + 34 (cid:19) · · · ...(0 , s ) · · · Fig. 2:
Interlocking representation scheme for the fields of half-integer spin (Fermi-scheme).
This field is described by the Fock-Klein-Gordon equation. In its turn, the simplest field fromthe Fermi-scheme (Fig. 2) is the electron-positron (spinor) field corresponding to the followinginterlocking scheme: (cid:18) , (cid:19) (cid:18) , (cid:19) u w . This field is described by the Dirac equation. Further, the next field from the Bose-scheme (Fig. 1)is a photon field (Maxwell field) defined within the interlocking scheme(1 , (cid:18) , (cid:19) (0 , u w u w . This interlocking scheme leads to the Maxwell equations. The fields (1 / , ⊕ (0 , /
2) and (1 , ⊕ (0 ,
1) (Dirac and Maxwell fields) are particular cases of fields of the type ( l, ⊕ (0 , l ). Waveequations for such fields and their general solutions were found in the works [54, 56, 59].It is easy to see that the interlocking scheme, corresponded to the Maxwell field, contains thefield of tensor type: (cid:18) , (cid:19) . (cid:18) , (cid:19) (cid:18) , (cid:19) (cid:18) , (cid:19) (cid:18) , (cid:19) u w u w u w , corresponding to the Pauli-Fierz equations [19], contains a chain of the type (cid:18) , (cid:19) (cid:18) , (cid:19) u w . In such a way we come to wave equations for the fields ψ ( α ) = h x, g | ψ i of tensor type ( l , l ) ⊕ ( l , l ). Wave equations for such fields and their general solutions were found in the work [61].A relation between the numbers l , l of the Gel’fand-Naimark representation (33) and thenumber k of the factors C in the product C ⊗ C ⊗ · · · ⊗ C is given by the following formula:( l , l ) = (cid:18) k , k (cid:19) , Hence it immediately follows that k = l + l −
1. Thus, we have a complex representation C l + l − , of the group Spin + (1 , in the spinspace S k . If the representation C l + l − , is reducible, thenthe space S k is decomposed into a direct sum of irreducible subspaces, that is, it is possible tochoose in S k such a basis, in which all the matrices take a block-diagonal form. Then the field ψ ( α ) is reduced to some number of the fields corresponding to irreducible representations of thegroup Spin + (1 , ψ ( α )in this case is a collection of the fields with more simple structure. It is obvious that these moresimple fields correspond to irreducible representations C .Analogously, a relation between the numbers l , l of the Gel’fand-Naimark representation (33)and the number r of the factors ∗ C in the product ∗ C ⊗ ∗ C ⊗ · · · ⊗ ∗ C is given by the followingformula: ( − l , l ) = (cid:16) − r , r (cid:17) . Hence it immediately follows that r = l − l +1. Thus, we have a complex representation C ,l − l +1 of Spin + (1 ,
3) in the spinspace S r .As is known [36, 21, 41], a system of irreducible finite-dimensional representations of the group G + is realized in the space Sym ( k,r ) ⊂ S k + r of symmetric spintensors. The dimensionality ofSym ( k,r ) is equal to ( k + 1)( r + 1). A representation of the group G + , defined by such spintensors,is irreducible and denoted by the symbol D ( l, ˙ l ) ( σ ), where 2 l = k, l = r , the numbers l and ˙ l are integer or half-integer. In general case, the field ψ ( α ) is the field of type ( l, ˙ l ). As a rule, inphysics there are two basic types of the fields:1) The field of type ( l, , ˙ l )) is described by therepresentation D ( l, ( σ ) ( D (0 , ˙ l ) ( σ )), which is realized in the space S k ( S r ). At this point, thealgebra C k ≃ C ⊗ C ⊗ · · · ⊗ C (correspondingly, ∗ C k ≃ ∗ C ⊗ ∗ C ⊗ · · · ⊗ ∗ C ) is associatedwith the field of the type ( l,
0) (correspondingly, (0 , ˙ l )). The trivial case l = 0 correspondsto a Pauli-Weisskopf field describing the scalar particles. Further, at l = ˙ l = 1 / aWeyl field describing the neutrino. At this point the antineutrino is described by a fundamentalrepresentation D (1 / , ( σ ) = σ of the group G + and the algebra C . Correspondingly, the neutrinois described by a conjugated representation D (0 , / ( σ ) and the algebra ∗ C . In essence, one can15ay that the algebra C ( ∗ C ) is the basic building block, from which other physical fields built bymeans of direct sum or tensor product. One can say that this situation looks like the de Brogliefusion method [8]2) The field of type ( l, ⊕ (0 , ˙ l ). The structure of this field admits a space inversion and, therefore,in accordance with a Wigner’s doubling [63] is described by a representation D ( l, ⊕ D (0 , ˙ l ) of thegroup G + . This representation is realized in the space S k . The Clifford algebra, related withthis representation, is a direct sum C k ⊕ ∗ C k ≃ C ⊗ C ⊗ · · · ⊗ C L ∗ C ⊗ ∗ C ⊗ · · · ⊗ ∗ C . In thesimplest case l = 1 / bispinor (electron–positron) Dirac field (1 / , ⊕ (0 , /
2) with thealgebra C ⊕ ∗ C . It should be noted that the Dirac algebra C , considered as a tensor product C ⊗ C (or C ⊗ ∗ C ) in accordance with (21) (or (29)) gives rise to spintensors s α α (or s α ˙ α ),but it contradicts with the usual definition of the Dirac bispinor as a pair ( s α , s ˙ α ). Therefore,the Clifford algebra, associated with the Dirac field, is C ⊕ ∗ C , and a spinspace of this sum invirtue of unique decomposition S ⊕ ˙ S = S is a spinspace of C .Spinor representations of the units of C n we will define in the Brauer-Weyl representation [7]: E = σ ⊗ ⊗ · · · ⊗ ⊗ ⊗ , E = σ ⊗ σ ⊗ ⊗ · · · ⊗ ⊗ , E = σ ⊗ σ ⊗ σ ⊗ ⊗ · · · ⊗ ,. . . . . . . . . . . . . . . . . . . . . . . . . . . . E m = σ ⊗ σ ⊗ · · · ⊗ σ ⊗ σ , E m +1 = σ ⊗ ⊗ · · · ⊗ , E m +2 = σ ⊗ σ ⊗ ⊗ · · · ⊗ ,. . . . . . . . . . . . . . . . . . . . . . . . . . . . E m = σ ⊗ σ ⊗ · · · ⊗ σ ⊗ σ , (40)where σ = (cid:18) (cid:19) , σ = (cid:18) − ii (cid:19) , σ = (cid:18) i − i (cid:19) are spinor representations of the units of C , is the unit 2 × l ′ , l ′′ ) ⊕ ( ˙ l ′′ , ˙ l ′ ). The fields ( l ′ , l ′′ ) and ( ˙ l ′′ , ˙ l ′ ) are defined within the arbitrary spinchains (see Fig. 1 and Fig. 2). Universal coverings of these spin chains are constructed within therepresentations C l + l − ,l − l +1 and C l − l +1 ,l + l − of Spin + (1 ,
3) in the spinspaces S k + r associatedwith the algebra C ⊗ C ⊗ · · · ⊗ C N ∗ C ⊗ ∗ C ⊗ · · · ⊗ ∗ C . A relation between the numbers l , l of the Gel’fand-Naimark basis (33) and the numbers k and r of the factors C and ∗ C is given bythe following formula: ( l , l ) = (cid:18) k − r , k + r (cid:19) . Finally, extended automorphisms groups Ext( C k ⊕ ∗ C k ) and Ext( C k ⊗ ∗ C k ) (correspondingly, CP T groups) can be derived via the same procedure that described in detail in our previous work[55]. 16
The
C P T group of the spin- / field In accordance with the general Fermi-scheme (Fig. 1) of the interlocking representations of G + thefield (1 / , ⊕ (0 , /
2) is defined within the following chain: (cid:18) , (cid:19) (cid:18) , (cid:19) u w . A double covering of the representation associated with the field (1 / , ⊕ (0 , /
2) is realized inthe spinspace S ⊕ ˙ S . This spinspace is a space of the representation C , ⊕ C , − of Spin + (1 , C ⊕ ∗ C corresponds to C , ⊕ C , − and the automorphisms of this algebra arerealized within the representations of Pin (1 , Spin + (1 , S ⊕ ˙ S , are constructed via the Brauer-Weyl representation (40). Thespinbasis of the algebra C ⊕ ∗ C is defined by the following 4 × E = σ ⊗ = (cid:18) (cid:19) , E = σ ⊗ σ = (cid:18) iσ − iσ (cid:19) , E = σ ⊗ = (cid:18) − i i (cid:19) , E = σ ⊗ σ = (cid:18) iσ − iσ (cid:19) . (41)In accordance with (7) we have for the matrix of the automorphism A → A ⋆ the followingexpression: W = E E E E = E ∼ P. Further, it is easy to see that among the matrices of the basis (41) there are symmetric andskewsymmetric matrices: E T = E , E T = E , E T = −E , E T = −E . In accordance with e A = EA T E − (see (8)) we have E = E E E − , E = E E E − , E = − E E E − , E = − E E E − . Hence it follows that E commutes with E and E and anticommutes with E and E , that is, E = E E ∼ T . From the definition C = EW (see (9)) we find that the matrix of the antiautomorphism A → f A ⋆ has the form C = E E ∼ P T . The basis (41) contains both complex and real matrices: E ∗ = E , E ∗ = −E , E ∗ = −E , E ∗ = E . Therefore, from A = Π A ∗ Π − (see (10)) we have E = Π E Π − , E = − Π E Π − , E = − Π E Π − , E = Π E Π − . From the latter relations we obtain Π = E E ∼ C . Further, in accordance with K = Π W (thedefinition (11)) for the matrix of the pseudoautomorphism A → A ⋆ we have K = E E ∼ CP .Finally, for the pseudoantiautomorphisms A → e A and A → f A ⋆ from the definitions S = Π E and F = Π C (see (12) and (13)) it follows that S = E E E E = E E ∼ CT and F = E E E E = E E ∼ CP T . Thus, we come to the following automorphism group:Ext( C ) = { I , W , E , C , Π , K , S , F } ≃ { , P, T, P T, C, CP, CT, CP T } ≃{ , E E E E , E E , E E , E E , E E , E E , E E } . The multiplication table of this group is shown in Tab. 4. From this table it follows that Ext( C ) ≃ D , and for the CP T group we have the following isomorphism: C + , + , + , + , + , − , − ≃ D ⊗ Z .17 E E E E E E E E E E E E E E E E E E E E E E E E −E E −E −E −E −E E E E E −E −E −E −E E E E E E E E E E E E E E E E E E E E E E −E −E − −E E E E E E −E −E −E − Tab. 4:
The multiplication table of the
CP T / Z group of the field (1 / , ⊕ (0 , / C P T group of the spin- field In accordance with the general Bose-scheme of the interlocking representations of G + (see Fig. 1),the field (1 , ⊕ (0 ,
1) is defined within the following interlocking scheme:(1 , (cid:18) , (cid:19) (0 , u w u w . A double covering of the representation, associated with the field (1 , ⊕ (0 , S ⊗ S M ˙ S ⊗ ˙ S , (42)This spinspace is a space of the representation C , ⊕ C , − of the group Spin + (1 , C ⊗ C M ∗ C ⊗ ∗ C . (43)is associated with C , ⊕ C , − . The automorphisms of this algebra are realized within representa-tions of the group Pin (1 , Spin + (1 , × E = σ ⊗ ⊗ =
00 0 0 , E = σ ⊗ σ ⊗ = i i − i − i , = σ ⊗ σ ⊗ σ = − σ σ σ
00 0 0 − σ , E = σ ⊗ ⊗ = − i
00 0 0 − i i i , E = σ ⊗ σ ⊗ = − − , E = σ ⊗ σ ⊗ σ = − σ σ σ
00 0 0 − σ . Using these matrices, we construct
CP T group for the field (1 , ⊕ (0 , A → A ⋆ has the form W = E E E E E E = E ∼ P. Further, since E T = E , E T = E , E T = E , E T = −E , E T = −E , E T = −E , then in accordance with e A = EA T E − we have E = E E E − , E = E E E − , E = E E E − , E = − E E E − , E = − E E E − , E = − E E E − . Hence it follows that E commutes with E , E , E and anticommutes with E , E , E , that is, E = E ∼ T . From the definition C = EW we find that a matrix of the antiautomorphism A → f A ⋆ has the form C = E ∼ P T . The basis {E , E , E , E , E , E } contains both complex andreal matrices: E ∗ = E , E ∗ = −E , E ∗ = E , E ∗ = −E , E ∗ = E , E ∗ = −E . Therefore, from A = Π A ∗ Π − we have E = Π E Π − , E = − Π E Π − , E = Π E Π − , E = − Π E Π − , E = Π E Π − , E = − Π E Π − . From the latter relations we obtain Π = E ∼ C . Further, in accordance with K = Π W forthe matrix of the pseudoautomorphism A → A ⋆ we have K = E ∼ CP . Finally, for thepseudoantiautomorphisms A → e A , A → f A ⋆ from the definitions S = Π E , F = Π C it follows that S = E ∼ CT , F = E ∼ CP T . Thus, we come to the following automorphism group:Ext( C ) ≃ { I , W , E , C , Π , K , S , F } ≃ { , P, T, P T, C, CP, CT, CP T } ≃{ , E , E , E , E , E , E , E } . The multiplication table of this group is given in Tab. 5. From this table it follows that Ext( C ) ≃ D , and for the CP T group we have the following isomorphism: C − , + , + , + , + , − , + ≃ D ⊗ Z .19 E E E E E E E E E E E E E E E E − E −E −E E −E E E E −E −E −E E −E E E E E E E E E E E E E E E E E E E E −E E −E −E −E E E E −E E −E −E E − E E E E E E E E E Tab. 5:
The multiplication table of the
CP T / Z group of the field (1 , ⊕ (0 , C P T group of the spin- / field In accordance with the general Fermi-scheme of the interlocking representations of G + (see Fig. 2),the field (3 / , ⊕ (0 , /
2) is defined within the following interlocking scheme: (cid:18) , (cid:19) (cid:18) , (cid:19) (cid:18) , (cid:19) (cid:18) , (cid:19) u w u w u w . A double covering of the representation, associated with the field (3 / , ⊕ (0 , / S ⊗ S ⊗ S M ˙ S ⊗ ˙ S ⊗ ˙ S , (44)This spinspace is a space of the representation C , ⊕ C , − of the group Spin + (1 , C ⊗ C ⊗ C M ∗ C ⊗ ∗ C ⊗ ∗ C (45)is associated with the representation C , ⊕ C , − . Spinor representations of the automorphisms,defined on the spinspace (44), are constructed via the Brauer-Weyl representation (40). A spinbasisof the algebra (45) is defined by the following 16 ×
16 matrices: E = σ ⊗ ⊗ ⊗ =
00 0 0 0 0 0 0 , = σ ⊗ σ ⊗ ⊗ = i i i i − i
00 0 0 0 0 0 0 − i − i − i , E = σ ⊗ σ ⊗ σ ⊗ = − − − − , E = σ ⊗ σ ⊗ σ ⊗ σ = − iσ iσ iσ − iσ iσ − iσ − iσ
00 0 0 0 0 0 0 iσ , E = σ ⊗ ⊗ ⊗ = − i − i − i
00 0 0 0 0 0 0 − i i i i i , E = σ ⊗ σ ⊗ ⊗ = i i − i − i − i
00 0 0 0 0 0 0 − i i i , = σ ⊗ σ ⊗ σ ⊗ = i − i − i i − i i i − i , E = σ ⊗ σ ⊗ σ ⊗ σ = − iσ iσ iσ − iσ iσ − iσ − iσ
00 0 0 0 0 0 0 iσ . Using this spinbasis, we construct
CP T group for the field (3 / , ⊕ (0 , / A → A ⋆ has the form W = E E E E E E E E = E ∼ P. Further, since E T = E , E T = E , E T = E , E T = E , E T = −E , E T = −E , E T = −E , E T = −E , then in accordance with e A = EA T E − we have E = E E E − , E = E E E − , E = E E E − , E = E E E − , E = − E E E − , E = − E E E − , E = − E E E − , E = − E E E − . Hence it follows that E commutes with E , E , E , E and anticommutes with E , E , E , E , thatis, E = E ∼ T . From the definition C = EW we find that a matrix of the antiautomorphism A → f A ⋆ has the form C = E ∼ P T . The basis {E , E , E , E , E , E , E , E } contains bothcomplex and real matrices: E ∗ = E , E ∗ = −E , E ∗ = E , E ∗ = −E , E ∗ = −E , E ∗ = −E , E ∗ = −E , E ∗ = E . Therefore, from A = Π A ∗ Π − we have E = Π E Π − , E = − Π E Π − , E = Π E Π − , E = − Π E Π − , E = − Π E Π − , E = − Π E Π − , E = − Π E Π − , E = Π E Π − . From the latter relations we obtain Π = E ∼ C . Further, in accordance with K = Π W for the matrix of the pseudoautomorphism A → A ⋆ we have K = E ∼ CP . Finally, for thepseudoantiautomorphisms A → e A and A → f A ⋆ from the definitions S = Π E and F = Π C itfollows that S = E ∼ CT , F = E ∼ CP T . Thus, we come to the following automorphismgroup:Ext( C ) ≃ { I , W , E , C , Π , K , S , F } ≃ { , P, T, P T, C, CP, CT, CP T } ≃{ , E , E , E , E , E , E , E } . The multiplication table of this group is given in Tab. 6. From this table it follows that Ext( C ) ≃ D , and for the CP T group we have the following isomorphism: C − , − , + , + , + , + , + ≃ D ⊗ Z .22 W E E E E E E W E E E E E E W W − −E E E −E −E −E E E −E − W E −E −E E E E E W E E E E E E −E −E E − W −E E E E E E E W E E E E E E E E E W E E −E −E E E −E − W Tab. 6:
The multiplication table of the
CP T / Z group of the field (3 / , ⊕ (0 , / C P T groups of the tensor fields
As it is shown in the section 3 double coverings of the representations associated with the tensorfields are constructed within the product C ⊗ C ⊗ · · · ⊗ C N ∗ C ⊗ ∗ C ⊗ · · · ⊗ ∗ C , where we have k algebras C and r algebras ∗ C . A relation between the number l (a weight of the representationin the Van der Waerden basis (37)) and the numbers k and r is given by the formula l = k − r . (46)It is easy to see that a central row in the scheme shown on the Fig. 1,(0 , (cid:18) , (cid:19) (1 , · · · (cid:16) s , s (cid:17) · · · (47)in virtue of (46) is equivalent to the following row:[0 ,
0] [0 ,
0] [0 , · · · [0 , · · · Analogously, the row shown on the Fig. 2, (cid:18) , (cid:19) (cid:18) , (cid:19) · · · (cid:18) s + 14 , s − (cid:19) · · · (48)is equivalent to (cid:20) , (cid:21) (cid:20) , (cid:21) · · · (cid:20) , (cid:21) · · · Therefore, all the representations of
Spin + (1 ,
3) can be divided on the equivalent rows which weshow on the Fig. 3 and Fig. 4. On the other hand, the row (47) corresponds to the following chain23 s, · · · ...[2 , · · · [2 , · · · [1 ,
0] [1 , · · · [1 , · · · [0 ,
0] [0 ,
0] [0 , · · · [0 , · · · [0 ,
1] [0 , · · · [0 , · · · [0 , · · · [0 , · · · ...[0 , s ] · · · Fig. 3:
Integer spin representations of the group
Spin + (1 , of the algebras: −→ C ⊗ ∗ C −→ C ⊗ C O ∗ C ⊗ ∗ C −→ . . . −→−→ C ⊗ C ⊗ · · · ⊗ C | {z } s times O ∗ C ⊗ ∗ C ⊗ · · · ⊗ ∗ C | {z } s times −→ . . . In its turn, the row (48) corresponds to the chain C −→ C ⊗ C O ∗ C −→ . . . −→ C ⊗ C ⊗ · · · ⊗ C | {z } (2 s +1) / O ∗ C ⊗ ∗ C ⊗ · · · ⊗ ∗ C | {z } (2 s − / −→ . . . Moreover, these chains induces the following chains of the spinspaces: S −→ S −→ S −→ . . . −→ S s −→ . . . and S −→ S −→ . . . −→ S s −→ . . . Thus, the row (47) (or (48)) induces a sequence of the fields of the spin 0 (or 1 /
2) realized inthe spinspaces of different dimensions. In general case presented on the Fig. 3 and Fig. 4 we havesequences of the fields of the same spin realized in the different representation spaces of
Spin + (1 , −→ electron −→ . . . (spin 1 / µ ( l ) = κl + = 2 κ l + 1 , (49)24 s, · · · ... (cid:20) , (cid:21) · · · (cid:20) , (cid:21) · · · (cid:20) , (cid:21) (cid:20) , (cid:21) · · · (cid:20) , (cid:21) · · · (cid:20) , (cid:21) (cid:20) , (cid:21) · · · (cid:20) , (cid:21) · · · (cid:20) , (cid:21) · · · (cid:20) , (cid:21) · · · ...[0 , s ] · · · Fig. 4:
Half-integer spin representations of the group
Spin + (1 , where the mass µ ( l ) corresponds the spin l , κ is a constant. It is easy to see that the denominator2 l + 1 in (49) is equal to a dimensionality of the representation space Sym ( k, corresponding tothe field ψ ( α ) of type ( l,
0) (or (0 , ˙ l ) and Sym (0 ,r ) ). For the tensor fields ψ ( α ) of type ( l ˙ l ) we have µ ( s ) = κ ( k + 1)( r + 1) , (50)where s = | k − r | / ψ ( α ). In this case, the denominator in (50) is equalto a dimensionality of the representation space Sym ( k,r ) corresponding to the tensor field. Massspectrum formulas (49) and (50) give a relationship between dimensions of the representationspaces of Spin + (1 ,
3) and particle masses. From the formula (50) it follows directly that on theparallel rows presented on the Fig. 3 and Fig. 4 we have particles of the same spin with differentmasses. When l → ∞ (or ( k + 1)( r + 1) → ∞ ) we come to particles with zero mass (like aphoton). In this case, finite-dimensional representation spaces Sym ( k, and Sym ( k,r ) should bereplaced by a Hilbert space, and such (massless) particles should be described within principalseries of infinite-dimensional representations of the group Spin + (1 , P T groups of the tensor fields are constructed via the same procedure that considered inthe sections 4–6. For example, the tensor field of the spin 1 / (cid:18) , (cid:19) (cid:18) , (cid:19) u w (which is equivalent to (1 / , ⊕ (0 , / C ⊗ C ⊗ C O ∗ C M ∗ C ⊗ ∗ C O C ⊗ C ⊗ C . (51)This algebra induces the spinspace S ⊗ S ⊗ S O ˙ S ⊗ ˙ S M ˙ S ⊗ ˙ S O S ⊗ S ⊗ S ≃ S . The spinbasis of the algebra (51) is defined by the following 64 ×
64 matrices: E = σ ⊗ ⊗ ⊗ ⊗ ⊗ , E = σ ⊗ σ ⊗ ⊗ ⊗ ⊗ , E = σ ⊗ σ ⊗ σ ⊗ ⊗ ⊗ , E = σ ⊗ σ ⊗ σ ⊗ σ ⊗ ⊗ , E = σ ⊗ σ ⊗ σ ⊗ σ ⊗ σ ⊗ , E = σ ⊗ σ ⊗ σ ⊗ σ ⊗ σ ⊗ σ , E = σ ⊗ ⊗ ⊗ ⊗ ⊗ , E = σ ⊗ σ ⊗ ⊗ ⊗ ⊗ , E = σ ⊗ σ ⊗ σ ⊗ ⊗ ⊗ , E = σ ⊗ σ ⊗ σ ⊗ σ ⊗ ⊗ , E = σ ⊗ σ ⊗ σ ⊗ σ ⊗ σ ⊗ , E = σ ⊗ σ ⊗ σ ⊗ σ ⊗ σ ⊗ σ . The extended automorphism group Ext( C ) can be derived from this spinbasis via the samecalculations that presented in the sections 4–6. We have presented a group theoretical method for description of discrete symmetries of the fields ψ ( α ) = h x, g | ψ i , where x ∈ T and g ∈ Spin + (1 , Spin + (1 , ≃ SU(2) ⊗ SU(2). We have shown that an extended automorphism groupExt( C n ), where C n is a complex Clifford algebra, lead to CP T groups of the fields ψ ( α ) = h x, g | ψ i of any spin defined on the representation spaces (spinspaces) of Spin + (1 , CP T groups for the fields of the type ( l, ⊕ (0 , l ) (for example, (1 / , ⊕ (0 , / , ⊕ (0 ,
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