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- 1. Quantum ﬁelds on the de Sitter spacetime Ion I. Cotaescu Abstract The theory of external symmetry in curved spacetimes we have proposed few years ago allows us tocorrectly deﬁne the operators of the quantum ﬁeld theory on curved backgrounds. Particularly, despite ofsome doubts appeared in literature, we have shown that a well-deﬁned energy operator can be consideredon the de Sitter manifold. With its help new quantum modes were obtained for the scalar, Dirac and vectorﬁelds on the de Sitter spacetimes. A short review of these results is presented in this report. 1
- 2. Symmetries and conserved observablesThe quantum ﬁeld theory on curved manifolds must be build in local(unholonomic) frames where the spin half can be well-deﬁned.The relativistic covariance and the gauge symmetryLet us consider the manifold (M, g) and a local chart (or natural frame) {x} ofcoordinates xµ ( µ, ν, .. = 0, 1, 2, 3). The general relativistic covariance preservesthe form of the ﬁeld equations under any coordinate transformation x → x = φ(x).The local frames are given by the tetrad ﬁelds eµ(x) and eµ(x), labeled by the local ˆ ˆˆ ˆ ˆindices α, β, ... = 0, 1, 2, 3, which have the usual duality and orthonormalizationproperties eµ eα = δν , eµ eβ = δα , eµ · eν = ηµν , eµ · eν = η µν , ˆ ˆα νˆ µ ˆ ˆ ˆ ˆα µˆ β ˆ ˆ ˆˆ ˆˆ ˆˆ ˆˆ (1)where η =diag(1, −1, −1, −1) is the Minkowski metric. This metric remains invariant ↑under the transformations Λ[A(ω)] of the gauge group G(η) = L+ corresponding to 2
- 3. the transformationsA(ω) ∈ SL(2, C) through the canonical homomorphism. In the ˆˆ αβ ˆˆ βαstandard parametrization, with ω = −ω , we have i ˆ ˆ − 2 ω αβ Sαβ A(ω) = e , ˆˆ (2)where Sαβ are the covariant basis-generators of the sl(2, C) algebra. For small values ˆˆ ˆof ω αβ the matrix elements of Λ can be written as ˆ Λ[A(ω)]µν· = δν + ω·µν· + · · · . ˆ ·ˆ µ ˆ ˆ ˆ ˆ (3)The theory contains matter ﬁelds ψ transforming according to a representation ρ ofthe SL(2, C) group. The entire theory must be gauge invariant in the sense that itremains invariant when one performs a gauge transformation ψ(x) → ψ (x) = ρ[A(x)]ψ(x) (4) ·βˆ eα(x) → eα(x) = ˆ ˆ Λα ·[A(x)]eβ (x) ˆ ˆ (5)produced by the point-dependent transformations A(x) ∈ SL(2, C) and Λ[A(x)] ∈L↑ . +The general relativistic covariance as well as the tetrad-gauge invarianceare not able to give rise to conserved quantities. These are produced byisometries. 3
- 4. The external symmetryIn general, (M, g) can have isometries that form the isometry group I(M ) whoseparameters are denotad by ξa (a, b, ... = 1..N ). Given an isometry φξ ∈ I(M ) then,for each parameter ξa, there exists an associated Killing vector ﬁeld deﬁned as Ka = ∂ξa φξ |ξ=0. (6)Starting with an isometry x → x = φξ (x) we introduced the so called externalsymmetry transformations, (Aξ , φξ ), deﬁned as combined transformations involvinggauge transformations necessary to preserve the gauge [1], ∂φµ(x) ξ Λ[Aξ (x)]αβ· = eα[φξ (x)] ˆ ·ˆ ˆ ˆµ eν (x) , ˆ (7) ∂xν βwith the supplementary condition Aξ=0(x) = 1 ∈ SL(2, C). The transformations(Aξ , φξ ) leave the ﬁeld equation invariant and constitute the group of externalsymmetry, S(M ), which is the universal covering group of I(M ). 4
- 5. The transformations of the group S(M ) are x → x = φξ (x) e(x) → e (x ) = e[φξ (x)] (Aξ , φξ ) : e(x) ˆ → e (x ) ˆ = e[φξ (x)] ˆ (8) ψ(x) → ψ (x ) = ρ[Aξ (x)]ψ(x) .In [11] we presented arguments that S(M ) is the universal covering group of I(M ).For small ξa SL(2, C) parameters of Aξ (x) ≡ A[ωξ (x)] the covariant can be ˆˆ αβ ˆˆ a αβexpanded as ωξ (x) = ξ Ωa (x) + · · · where the functions ˆˆ αβ ˆˆ αβ ∂ωξ ˆˆ Ωa ≡ = eα Ka,ν ˆµ µ ˆ + eα Ka ˆν,µ µ ˆ ν λβ eλ η . ˆ (9) ∂ξ a |ξ=0depend on the Killing vectors (6). 5
- 6. The generators of any representationThe matter ﬁeld ψ transforms according to the operator-valued representation(Aξ , φξ ) → Tξρ which is deﬁned as (Tξρψ)[φξ (x)] = ρ[Aξ (x)]ψ(x) (10) ρ ρ −1and leaves invariant the ﬁeld equation in local frames, Tξ E(Tξ ) = E.The basis-generators of the representations ρ of the s(M ) algebra, ρ ∂Tξρ ρ Xa = i = La + Sa , (11) ∂ξ a |ξ=0 µare formed by orbital parts, La = −iKa ∂µ, and spin terms [1], ρ ∂Aξ (x) 1 αβ ˆˆ Sa (x) =i a = Ωa (x)ρ(Sαβ ) , ˆˆ (12) ∂ξ |ξ=0 2 6
- 7. that depend on the functions (9). These operators can be written in covariant formaccording to the Carter and McLenaghan formula [2] ρ µ ρ 1 µ ν ˆˆ αβ Xa = −iKa Dµ + Ka µ;ν eα eβ ρ(S ) . ˆ ˆ (13) 2 ρwhere Dµ are the covariant derivatives in local frames associated to the representationρ of SL(2, C).However, whenever the mater ﬁelds are vector and tensors, then the basis-generatorsof a tensor representation ρn of the rank n can be written in natural frames, n µ 1 ˜µν Xa = −iKa µ + Ka µ; ν ρn(S ) , (14) 2where µ are the usual covariant derivatives. The point-dependent generators ˜ S(x)are deﬁned as ˆˆ ˆ ˆ ˜µν )στ· = eµ eν eσ [ρv (S αβ )]γ ·eδ = i(g µσ δτ − g νσ δτ ) . ν µ (S · ˆ γ α β ˆ ˆ ˆ ·δ ˆτ (15)We say that the operators (25) and (26) are conserved operators sincethey commute with the operators of the ﬁeld equations. 7
- 8. The de Sitter spacetimeLet (M, g) be the de Sitter (dS) spacetime deﬁned as a hyperboloid in the (1 + 4) -dimensional ﬂat spacetime, (5M,5 η), of coordinates z A, A, B, ... = 0, 1, 2, 3, 5, andmetric 5η = diag(1, −1, −1, −1, −1), 5 ηAB z Az B = −R2 , R = 1/ω = 3/|Λc| . (16)The gauge group of the metric 5η plays the role of isometry group of the dS manifold,G[5η] = I(M ) = SO(1, 4). We use covariant real parameters, 5ω AB = −5ω BA,since in this parametrization the orbital basis-generators of the scalar representation ofG(5η), carried by the spaces of functions over 5M , has the usual form 5 LAB = i 5ηAC z C ∂B − 5ηBC z C ∂A . (17)A local chart {x} on (M, g), of coordinates xµ, µ, ν, ... = 0, 1, 2, 3, is deﬁned µby the functions z A = z A(x). The identiﬁcation 5LAB = −iK(AB)∂µ deﬁnes thecomponents in the chart {x} of the Killing vector ﬁeld K(AB) associated to 5ω AB . 8
- 9. Static charts 1. Cartesian coordinates: {ts, xs}with conformal spherical line element: 2. spherical coordinates: {ts, rs, θ, φ} 1 1 2 ds = dts − drs − 2 sinh2 ωrs (dθ2 + sin2 θ dφ2) , 2 2 (18) cosh2 ωrs ωwith ﬁnite event horizon at ωˆs r = 1: 1. Cartesian coordinates: {ts, xs} ˆ 2. spherical coordinates: {ts, rs, θ, φ} ˆ r2 dˆs ds2 = (1 − ω 2rs )dt2 − ˆ2 s − rs (dθ2 + sin2 θ dφ2) , ˆ2 (19) 1 − ω 2rs ˆ2where ωˆs = tanh ωrs . r (20) 9
- 10. Moving charts 1. Cartesian coordinates: {t, x}with FRW line element, proper time and: 2. spherical coordinates: {t, r, θ, φ} ds2 = dt2 − e2ωtdx · dx = dt2 − e2ωt[dr2 + r2(dθ2 + sin2 θ dφ2)]. (21) 1. Cartesian coordinates: {tc, x}with conformal ﬂat line element and: 2. spherical coordinates: {tc, r, θ, φ} 1 1 ds = 2 2 (dtc − dx · dx) = 2 2 [dt2 − dr2 − r2(dθ2 + sin2 θ dφ2)]. 2 2 (22) ω tc ω tc cwhere ωtc = −e−ωt and: 1 1 1 + ωreωt ts = t − ln 1 − ω 2r2e2ωt , rs = ln ωt , rs = reωt . ˆ (23) 2ω 2ω 1 − ωre 10
- 11. The principal observables on the de Sitter spacetimeIn the case of the dS spacetime, in a chart where we consider the tetrad ﬁelds e ande, we identify ξ a → 5ω AB and a → (AB) such that the generators of an arbitraryˆrepresentation ρ read ρ ρ µ 1 αβ ˆˆ X(AB) = L(AB) + S(AB) = −iK(AB)∂µ + Ω(AB)ρ(Sαβ ) . ˆˆ (24) 2The covariant forms can be written in local frames, ρ µ ρ 1 µ ν ˆˆ αβ X(AB) = −iK(AB)Dµ + K(AB) µ;ν eα eβ ρ(S ) , ˆ ˆ (25) 2for any ﬁeld or even in natural frames but only for the vector and tensor ﬁelds, n µ 1 ˜µν X(AB) = −iK(AB) µ + K(AB) µ; ν ρn(S ) . (26) 2 11
- 12. name {ts, xs} {t, x} {tc, x}deﬁnition any gauge diagonal gauge diagonal gaugeEnergy:H = ωX(05) = i∂ts = i∂t − iωxi∂i −iω(tc∂tc + xi∂i)Momentum:Pi = ω(X(5i) − X(0i)) Ref. [1] = −i∂i id.Angular momentum:Jij = X(ij) sim. = −i(xi∂j − xj ∂i) + ρ(Sij ) id.More three generators:Ni = ω(X(5i) + X(0i)) do not have an immediate physical meaning.The speciﬁc features:1. The momentum and energy operators do not commute amongthemselves, [H, Pi] = iωPi . (27)2. The energy operator H is well-deﬁned only on restricted domains whereK(05) is time-like. For this reason some people asks whether and how theenergy can be measured. 12
- 13. The problem of the energy operatorWhere the Killing vector K(05) is time-like ?chart {ts, xs} ˆ {ts, xs} {t, x} {tc, x}light-cone domain (ωˆs < 1)e.h. r rs < |ts| ωreωt < 1 r < |tc| 1 1 1K(05) (− ω , 0, 0, 0) (− ω , 0, 0, 0) (− ω , x1, x2, x3) (tc, x1, x2, x3) 1g(K(05), K(05)) > 0 1 − ω 2rs > 0 ˆ2 rs > 0 ω2 − r2e2ωt > 0 t2 − r2 > 0 cConclusions:1. The Killing vector K(05) is time-like inside the light-cone of any givenchart.2. The energy operator H is well-deﬁned on the entire domain where anobserver can measure physical events. 13
- 14. Quantum modes on the de Sitter manifoldThe quantum modes are determined by complete systems of commuting operators(c.s.c.o.) formed by the operator of the ﬁeld equation E - of the scalar (S), Proca (V),Maxwell (M) and Dirac (D) - and various generators commuting with E . Among them weconsider the operators H , Pi, Ji = 1 εijk Jjk = Li + ρ(Si) and the helicity operator, 2 W =P ·J. (28) ˆThe operator of the momentum direction, P , is no longer a differential one being deﬁnedas ˆ Pi = Pi P2 . (29)In addition we deﬁne the normalized helicity operator ˆ ˆ W =P ·J. (30)These operators obey the following commutation rules: ˆ ˆ [H, W ] = iωW , [H, Pi] = 0 , [H, W ] = 0 . (31) 14
- 15. The analytically solvable scalar quantum modeschart/c.s.c.o. {ES , H, L2, L3} {ES , P 2, L2, L3} {ES , Pi} ˆ {ES , H, Pi} spherical waves spherical waves plane waves plane waves{ts, rs, θ, φ}. Avis, Isham, Storey, [3] ? no no{t, x}, {tc, x} no no Chernikov, Tagirov [4] Cotaescu [5]The analytically solvable vector quantum modeschart/c.s.c.o. {EV , H, J 2, J3} {EV , P 2, J 2, J3} {EV , Pi, W } ˆ ˆ {EV , H, Pi, W } spherical waves spherical waves plane waves plane waves{ts, rs, θ, φ}. Higuchi [6] ? no no{t, x}, {tc, x} no no Cotaescu [7] (m = 0) Cotaescu [8] 15
- 16. The analytically solvable Dirac quantum modeschart/c.s.c.o. ˆ ˆ {ED , H, J 2, K, J3} {ED , P 2, J 2, K, J3} {ED , Pi, W } {ED , H, Pi, W }gauge spherical waves spherical waves plane waves plane waves{ts, rs, θ, φ}. Otchik [9] ? no nodiagonal gauge{ts, rs, θ, φ}. Cotaescu [10] ? no noCartesian gauge{t, r, θ, φ} Cotaescu [14] Shishkin [12] no noCartesian gauge Cotaescu [13]{t, x}, {tc, x} no no Cotaescu [11] Cotaescu [14]diagonal gaugewhere K is the Dirac angular operator. 16
- 17. The free quantum ﬁelds in moving chartsIn the moving charts, {t, x} and {tc, x}, the principal quantum ﬁelds minimally coupledto gravity can be expanded as mode integrals in the momentum basis.The massive charged scalar ﬁeldThe scalar ﬁeld φ satisfy the Klein-Gordon equation ∂t2 − e−2ωt∆ + 3ω∂t + m2 φ(x) = 0 . (32)The expansion in the momentum basis, φ(x) = φ(+)(x) + φ(−)(x) = d3p fp(x)a(p) + fp (x)b∗(p) , ∗ (33)can be done using the fundamental solutions 1 π 1 −3ωt/2 p −ωt ip·x −πk/2 (1) fp(x) = 3/2 e Zk e e , Zk (s) = e Hik (s) 2 ω (2π) ω (34) 17
- 18. m2where k = ω2 − 9 . These modes satisfy the orthonormalization relations 4 fp, fp = − fp , fp = δ 3(p − p ) , ∗ ∗ ∗ fp , f p = 0 , (35)with respect to the relativistic scalar product ↔ 3 3ωt ∗ φ, φ = i d xe φ (x) ∂t φ (x) , (36)and the completeness condition ↔ 3 ∗ i d p fp (t, x) ∂t fp(t, x ) = e−3ωtδ 3(x − x ) . (37)The massive charged vector ﬁeldThis ﬁeld obeys the Proca-type equation µ2 ∂tc (∂iAi) − ∆A0 + 2 A0 = 0 , (38) tc µ2 ∂t2c Ak − ∆Ak − ∂k (∂c tA0) + ∂k (∂iAi) + 2 Ak = 0 , (39) tc 18
- 19. where µ = m/ω , while the Lorentz condition reads 2 ∂iAi = ∂tc A0 − A0 . (40) tcThe mode expansion in the momentum basis, A = A(+) + A(−) = d3 p {U[p, λ]a(p, λ) + U[p, λ]∗b∗(p, λ)} , (41) λinvolve fundamental solutions, U[p, λ], which satisfy the Proca equation, the Lorenzcondition, the eigenvalue equations P iU[p, λ] = piU[p, λ] , W U[p, λ] = p λU[p, λ] , (42)and the orthonormalization relations U[p, λ]| U[p , λ ] = δλλ δ 3(p − p ) . (43)with respect to the relativistic scalar product ↔ µν µν 3 A| A = −η Aµ, Aν = −iη d x A∗ (tc, x) µ ∂tc Aν (tc, x) . (44) 19
- 20. These solutions are of the form α(tc, p) ei(np, λ) eip·x for λ = ±1 U[p, λ]i(x) = (45) β(tc, p) ei(np, λ) eip·x for λ = 0and 0 for λ = ±1 U[p, λ]0(x) = (46) γ(tc, p) eip·x for λ = 0where np = p/p and 1 − 2 πk 1 (1) α(tc, p) = N1e (−tc) Hik (−ptc) , 2 (47) 1 − 2 πk 3 (1) γ(tc, p) = N2e (−tc) 2 Hik (−ptc) , (48) − 1 πk 1 1 1 (1) β(tc, p) = iN2e 2 ik + (−tc) 2 Hik (−ptc) p 2 3 (1) −(−t) Hik+1(−ptc) , 2 (49) m2 1with the notations k = ω2 − 4 and √ √ π 1 π 1 ωp N1 = 3/2 , N2 = 3/2 m . (50) 2 (2π) 2 (2π) 20
- 21. The polarization vectors e(np, λ) of the helicity basis are longitudinal for λ = 0, i.e.e(np, 0) = np, while for λ = ±1 they are transversal, p · e(np, ±1) = 0. They havec-number components which satisfy e(np, λ)∗ · e(np, λ ) = δλλ , (51) e(np, λ)∗ ∧ e(np, λ) = iλ np , (52) ei(np, λ)∗ ej (np, λ) = δij . (53) λThe massless limit makes sense only if we take β(t, p) = γ(t, p) = 0 which leads tothe Coulomb gauge of the Maxwell free ﬁeld, A0 = 0 and ∂iAi = 0. [8]The Maxwell ﬁeld in Coulomb gauge A0(x) = 0 , Ai(x) = d3 k ei(nk , λ)fk (x)a(k, λ) + [ei(nk , λ)fk (x)]∗a∗(k, λ) (54) , λ=±1is expanded in terms of fundamental solutions of the d’Alambert equation, 1 1 −iktc+ik·x fk (x) = 3/2 √ e , (55) (2π) 2k 21
- 22. The massive Dirac ﬁeldThe Dirac ﬁeld ψ satisﬁes the free equation ED ψ = mψ . In the Cartesian gauge withthe non-vanishing tetrad components 1 i 1 e0 0 = −ωtc , ei j = i −δj ωtc , e0 ˆ0 =− i , ej = −δj ˆ , (56) ωtc ωtcthe Dirac operator reads 0 3iω 0 i ED = −iωtc γ ∂tc + γ ∂i + γ 2 0 −ωt i 3iω 0 = iγ ∂t + ie γ ∂i + γ , (57) 2and the relativistic scalar product is deﬁned as ψ, ψ = d3x e3ωtψ(x)γ 0ψ (x) . (58) DThe mode expansion in momentum basis ψ(t, x) = ψ (+)(t, x) + ψ (−)(t, x) = d3 p Up,σ (x)a(p, σ) + Vp,σ (x)b†(p, σ) . (59) σ 22
- 23. is written in terms of the fundamental spinors of positive and negative frequencies withmomentum p and helicity σ that read 1 πk/2 (1) e Hν− (qe−ωt) ξσ (p) Up,σ (t, x) = iN 2 (1) eip·x−2ωt (60) σ e−πk/2Hν+ (qe−ωt) ξσ (p) (2) −σ e−πk/2Hν− (qe−ωt) ησ (p) Vp,σ (t, x) = iN 1 πk/2 (2) e−ip·x−2ωt , (61) 2 e Hν+ (qe−ωt) ησ (p) 1 √where q = p/ω and N = (2π)3/2 πq .The Pauli spinors ξσ (p) and ησ (p) = iσ2[ξσ (p)]∗ of helicity σ = ±1/2 satisfy σ · p ξσ (p) = 2pσ ξσ (p) , σ · p ησ (p) = −2pσ ησ (p) . (62)Thus we obtain fundamental spinors which have the properties P i Up,σ = pi Up,σ , P i Vp,σ = −pi Vp,σ , (63) W Up,σ = pσUp,σ , W Vp,σ = −pσVp,σ . (64) 23
- 24. Moreover, these spinors are charge-conjugated to each other, Vp,σ = (Up,σ )c = C(U p,σ )T , C = iγ 2γ 0 , (65)satisfy the ortonormalization relations Up,σ , Up ,σ = Vp,σ , Vp ,σ = δσσ δ 3(p − p ) , (66) Up,σ , Vp ,σ = Vp,σ , Up ,σ = 0 , (67)and represent a complete system of solutions, d3 p Up,σ (t, x)Up,σ (t, x ) + Vp,σ (t, x)Vp,σ (t, x ) = e−3ωtδ 3(x − x ) . + + σ (68) 24
- 25. The massless Dirac ﬁeldIn the case of m = 0 the fundamental solutions of the left-handed massless Dirac ﬁeldread 0 1 − γ5 Up,σ (tc, x) = lim Up,σ (tc, x) k→0 2 3/2 −ωtc 1 ( 2 − σ)ξσ (p) = e−iptc+ip·x (69) 2π 0 0 1 − γ5 Vp,σ (tc, x) = lim Vp,σ (tc, x) k→0 2 3/2 −ωtc 1 ( 2 + σ)ησ (p) = eiptc−ip·x , (70) 2π 0are non-vanishing only for positive frequency and σ = −1/2 or negative frequency andσ = 1/2, as in Minkowski spacetime. Obviously, these solutions have similar propertiesas (65)-(64). 25
- 26. Canonical quantizationThe particle (a, a†) and antiparticle (b, b†) operators fulﬁll the standard anticommutationrelations in the momentum representation, {a(p, σ), a†(p , σ )} = {b(p, σ), b†(p , σ )} = δσσ δ 3(p − p ) , (71)since then the equal-time anticommutator takes the canonical form {ψ(t, x), ψ(t, x )} = e−3ωtγ 0δ 3(x − x ) . (72)In general, the propagators are constructed using the partial anticommutator functions, ˜(±)(t, t , x − x ) = i{ψ (±)(t, x), ψ (±)(t , x )} , S (73) ˜ ˜ ˜or the total one S = S (+) + S (−). For example, the Feynman propagator, ˜ SF (t, t , x − x ) = i 0| T [ψ(x)ψ(x )] |0 (74) ˜ ˜ = θ(t − t )S (+)(t, t , x − x ) − θ(t − t)S (−)(t, t , x − x ) , (75)is the causal Green function which obeys ˜ [ED (x) − m]SF (t, t , x − x ) = −e−3ωtδ 4(x − x ) . (76) 26
- 27. The set of commuting operators of the momentum basisThe one-particle operators which are diagonal in the momentum basis are themomentum Pi =: ψ, P iψ := d3 p p i a†(p, σ)a(p, σ) + b†(p, σ)b(p, σ) , (77) σthe helicity (or Pauli-Lubanski) operator, W =: ψ, W ψ := d3p pσ a†(p, σ)a(p, σ) + b†(p, σ)b(p, σ) , (78) σand the charge operator Q = : ψ, ψ : = d3 p a†(p, σ)a(p, σ) − b†(p, σ)b(p, σ) . (79) σThus the momentum basis of the Fock space is formed by the common eigenvectors ofthe set {Q, Pi, W}. 27
- 28. The Hamiltonian operatorThe Hamiltonian operator H =: ψ, Hψ : is conserved but is not diagonal in thisbasis since it does not commute with Pi and W. Its form in momentum representationcan be calculated using the identity i 3 H Up,σ (t, x) = −iω p ∂pi + Up,σ (t, x) , (80) 2and the similar one for Vp,σ , leading to iω 3 i † ↔ † ↔ H= d pp a (p, σ) ∂ pi a(p, σ) + b (p, σ) ∂ pi b(p, σ) (81) 2 σ ↔where the derivatives act as f ∂ h = f ∂h − (∂f )h.These new properties are universal, since similar formulas hold for thescalar and vector ﬁelds. 28
- 29. Concluding remarks1. The energy operator is well-deﬁned wherever measurements can bedone.2. The quantum modes are globally deﬁned by complete systems ofcommuting operators on the whole observer’s domain.3. The charge conjugation is point-independent and, therefore, the vacuumis stable.4. Consequently, the modes created by the same c.s.c.o. in differentcharts can be related among themselves through combined transformationswithout to mix the particle and antiparticle subspaces. 29
- 30. References ˘ [1] I. I. Cotaescu, External Symmetry in General Relativity, J. Phys. A: Math. Gen. 33, 9177 (2000). [2] B. Carter and R. G. McLenaghan, Phys. Rev. D 19, 1093 (1979). [3] S. J. Avis, C. J. Isham and D. Storey, Phys. Rev. D10, 3565 (1978) ´ [4] N. A. Chernikov and E. A. Tagirov, Ann. Inst H. Poincare IX 1147 (1968) ˘ [5] I. I. Cotaescu, C. Crucean and A. Pop, Int. J. Mod. Phys A 23, 2563 (2008), arXiv:0802.1972 [gr-qc] [6] A. Higuchi, Class. Quant. Gravity 4, 712 (1987); D. Bini, G. Esposito and R. V. Montaquila, arXiv:0812.1973. [7] I. I. Cotaescu, GRG in press (2009). ˘ [8] I. I. Cotaescu and C. Crucean arXiv:0806.2515 [gr-qc]. [9] V. S. Otchik, Class. Quant. Grav. 2, 539 (1985). ˘[10] I. I. Cotaescu, The Dirac Particle on de Sitter Background, Mod. Phys. Lett. A 13, 2991-2997 (1998). 30
- 31. ˘[11] I. I. Cotaescu, Polarized Dirac Fermions in de Sitter Spacetime, Phys. Rev. D 65, 084008 (2002).[12] G. V. Shishkin, Class. Quantum Grav. 8, 175 (1991). ˘[13] I. I. Cotaescu, Radu Racoceanu and Cosmin Crucean, Remarks on the spherical waves of the Dirac ﬁeld on de Sitter spacetime, Mod. Phys. Lett. A 21, 1313 (2006) ˘[14] I. I. Cotaescu and Cosmin Crucean, New Dirac quantum modes in moving frames of the de Sitter spacetime Int. J. Mod. Phys. A 23, 3707-3720 (2008). 31

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