CPT groups for spinor field in de Sitter space

A group structure of the discrete transformations (parity, time reversal and charge conjugation) for spinor field in de Sitter space are studied in terms of extraspecial finite groups. Two $CPT$ groups are introduced, the first group from an analysis of the de Sitter-Dirac wave equation for spinor field, and the second group from a purely algebraic approach based on the automorphism set of Clifford algebras. It is shown that both groups are isomorphic to each other.


Introduction
Quantum field theory in de Sitter spacetime has been extensively studied during the past two decades with the purpose of understanding the generation of cosmic structure from inflation and the problems surrounding the cosmological constant. It is well known that discrete symmetries play an important role in the standard quantum field theory in Minkowski spacetime. In recent paper [1] discrete symmetries for spinor field in de Sitter space with the signature (+, −, −, −, −) have been derived from the analysis of the de Sitter-Dirac wave equation. Discrete symmetries in de Sitter space with the signature (+, +, +, +, −) have been considered in the work [2] within an algebraic approach based on the automorphism set of Clifford algebras.
In the present paper we study a group structure of the discrete transformations in the framework of extraspecial finite groups. In the section 3 we introduce a CP T group for discrete symmetries in the representation of the work [1] (for more details about CP T groups see [3,4,5,6]). It is shown that the discrete transformations form a non-Abelian finite group of order 16. Group isomorphisms and order structure are elucidated for this group. Other realization of the CP T group is given in the section 4. In this section we consider an automorphism set of the Clifford algebra associated with the de Sitter space. It is proven that a CP T group, formed within the automorphism set, is isomorphic the analogous group considered in the section 3.

Preliminaries
Usually, the de Sitter space is understood as a hyperboloid embedded in a five-dimensional Minkowski space R 1,4 where η αβ = diag(1, −1, −1, −1, −1). The spinor wave equation in the de Sitter space-time 1 (1) has been given in the works [8,1], and can be derived via the eigenvalue equation for the second order Casimir operator, where x = η αβ γ α x β and ∂ α = ∂ α + H 2 x α x · ∂. In this case the 4 × 4 matrices γ α are spinor representations of the units of the Clifford algebra Cℓ 1,4 and satisfy the relations An explicit representation 2 for γ α chosen in [1] is where 1 2 is a 2 × 2 unit and σ i are Pauli matrices. Discrete symmetries (parity transformation P , time reversal T and charge conjugation C), obtained from analysis of the equation (2), in spinor notation have the form [1] where η p , η t , η c are arbitrary unobservable phase quantities.

The CP T group
In this section we will show that the transformations (4) form a finite group of order 16, a so called CP T group. Moreover, this group is a subgroup of the more large finite group associated with the algebra Cℓ 1,4 . As is known [10,11,12,13], a structure of the Clifford algebras admits a very elegant description in terms of finite groups. In accordance with a multiplication rule 1 Originally, relativistic wave equations in a five-dimensional pseudoeuclidean space (de Sitter space) were introduced by Dirac in 1935 [7]. They have the form (iγ µ ∂ µ + m)ψ = 0, where five 4 × 4 Dirac matrices form the Clifford algebra Cℓ 4,1 .
basis elements of the Clifford algebra Cℓ p,q (the algebra over the field of real numbers, F = R) form a finite group of order 2 n+1 , The finite group G(1, 4), associated with the algebra Cℓ 1,4 , is a particular case of (7), where γ i have the form (3). It is a finite group of order 64 with an order structure (23,40). Moreover, G(1, 4) is an extraspecial two-group [12,13]. In Salingaros notation the following isomorphism holds: where Q 4 is a quaternion group, D 4 is a dihedral group, Z 2 ⊗ Z 2 is a Gauss-Klein viergroup, • is a central product (Q 4 and D 4 are finite groups of order 8).
As is known, the orthogonal group O(p, q) of the real space R p,q is represented by a semidirect product O 0 (p, q) ⊙ {1, P, T, P T }, where O 0 (p, q) is a connected component, {1, P, T, P T } is a discrete subgroup (reflection group). If we take into account the charge conjugation C, then we come to the product O 0 (p, q) ⊙ {1, P, T, P T, C, CP, CT, CP T }. Universal coverings of the groups O(p, q) are Clifford-Lipschitz groups Pin(p, q) which are completely constructed within the Clifford algebras Cℓ p,q [9]. It has been recently shown [4,5] that there exist 64 universal coverings of the orthogonal group O(p, q): where C a,b,c,d,e,f,g = {±1, ±P, ±T, ±P T, ±C, ±CP, ±CT, ±CP T } is a full CP T -group. C a,b,c,d,e,f,g is a finite group of order 16 (a complete classification of these groups is given in [5]). At this point, the group Ext(Cℓ p,q ) = C a,b,c,d,e,f,g Z 2 is called a generating group. Let us define a CP T group for the spinor field in de Sitter space. The invariance of the dS-Dirac equation (2) with respect to P -, T -, and C-transformations leads to the representation (4). For simplicity we suppose that all the phase quantities are equal to the unit, η p = η t = η c = 1. Thus, we can form a finite group of order 8 {1, P, T, P T, C, CP, CT, CP T } ∼ {1, γ 0 γ 4 , γ 0 , γ 4 , γ 2 , γ 0 γ 2 γ 4 , γ 0 γ 2 , γ 2 γ 4 }.
It is easy to verify that a multiplication table of this group has the form 1 γ 04 γ 0 γ 4 γ 2 γ 024 γ 02 γ 24 Here γ 04 ≡ γ 0 γ 4 , γ 024 ≡ γ 0 γ 2 γ 4 and so on. Hence it follows that the group (8) is a non-Abelian finite group of the order structure (3,4). In more details, it is the group * Z 4 ⊗ Z 2 with the signature (+, +, −, −, −, +, −). Therefore, the CP T group in de Sitter spacetime is It is easy to see that C +,+,−,−,−,+,− is a subgroup of G (1, 4). In this case, the universal covering of the de Sitter group is defined as forms a finite group of order 8. Moreover, there is an isomorphism between Ext(C n ) and a CP T -group of the discrete transformations, Ext(C n ) ≃ {1, P, T, P T, C, CP, CT, CP T }.
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 A → A, A → A ⋆ , pseudoautomorphisms A → A, A → A ⋆ , pseudoantiautomorphisms A → A and A → A ⋆ , respectively (for more details, see [4]). Let us study an automorphism group of the algebra Cℓ 1,4 . First of all, Cℓ 1,4 has the type p − q ≡ 5 (mod 8), therefore, all the eight automorphisms exist. Using the γ-matrices of the basis (3), we will define elements of the group Ext (Cℓ 1,4 ). At first, the matrix of the automorphism A → A ⋆ has the form W = γ 0 γ 1 γ 2 γ 3 γ 4 ≡ γ 01234 . Further, since Hence it follows that E commutes with γ 0 , γ 1 , γ 3 and anticommutes with γ 2 and γ 4 , that is, E = γ 2 γ 4 . From the definition C = EW we find that a matrix of the antiautomorphism A → A ⋆ has the form C = γ 0 γ 1 γ 3 . The basis (3) contains both complex and real matrices: Therefore, from A = ΠA * Π −1 we have From the latter relations we obtain Π = γ 1 γ 3 . Further, in accordance with K = ΠW for the matrix of the pseudoautomorphism A → A ⋆ we have K = γ 0 γ 2 γ 4 . Finally, for the pseudoantiautomorphisms A → A, A → A ⋆ from the definitions S = ΠE, F = ΠC it follows that S = γ 1 γ 2 γ 3 γ 4 , F = γ 0 . Thus, we come to the following automorphism group: The multiplication  Thus, we come to the following result: the finite group (8), derived from the analysis of invariance properties of the dS-Dirac equation with respect to discrete transformations C, P and T , is isomorphic to the automorphism group of the algebra Cℓ 1,4 . This result allows us to study discrete symmetries and their group structure for physical fields without handling to analysis of relativistic wave equations.