"Massless'' Spin-$\frac{3}{2}$ Fields in the de Sitter Space

In this paper, the ``massless"spin-$\frac{3}{2}$ fields in the de Sitter space are considered. This work is in the continuation of a previous paper devoted to the quantization of the de Sitter ``massive"spin-$\frac{3}{2}$ fields. Due to the appearance of gauge invariance and indefinite metric, the covariant quantization of the ``massless"spin-$\frac{3}{2}$ fields requires an indecomposable representation of the de Sitter group. The gauge fixing corresponding to the simplest Gupta-Bleuler structure is used, and gauge invariant field is discussed. The field equation is obtained by using the Casimir operator of the de Sitter group. The solutions are written in terms of the coordinate-independent de Sitter plane waves. Finally, the generalized two-point function is calculated.


Introduction
In the previous work [1], the "massive" spin- 3 2 fields in the de Sitter space have been studied. The term "massive" refers to those the de Sitter fields which unambiguously contract to the Minkowskian massive fields at zero curvature limit. These fields are associated with the principal series of unitary irreducible representations of the de Sitter group SO 0 (1,4), with the Casimir operator eigenvalue < Q (1) ν >= ν 2 − 3 2 , ν > 3 2 (or equivalently "mass" m 2 p = H 2 (ν 2 − 3 2 )). The interpretation of m p as a mass becomes made possible by examining its null curvature limit. The quotation marks on "mass" are used because of the absence of the intrinsic notion of mass in the de Sitter relativity [2]. Indeed, the principal series of unitary irreducible representations admits a massive representation of the Poincaré group in the limit H = 0 [3,4,5].
In this paper, the "massless" spin-3 2 fields in the de Sitter space are considered. "Massless" is used by a reference to the conformal invariance and propagation on the light-cone. They are associated with the discrete series of unitary irreducible representations of the de Sitter group. There are two unitary irreducible representations of the de Sitter group in the discrete series with a Minkowskian meaning in the null curvature limit, which are denoted by Π ± 3 2 , 3 2 with the same the Casimir operator eigenvalue < Q (1) >= − 5 2 . The field equation of the "massless" spin- 3 2 fields is gauge invariant in the de Sitter space as well as the massless fields in the Minkowski space for s ≥ 1. It is well known that the quantization of gauge invariant theories usually requires theà la Gupta-Bleuler quantization [6,7]. It has been proved that the use of an indefinite metric is an unavoidable feature if one insists on the preserving of causality (locality) and covariance in gauge quantum field theories [8]. This means that one can not limit the state space of the massless fields (with s ≥ 1) to Hilbert space, the appearance of states with negative or null norms is necessary for the quantization of gauge fields. Therefore, an indecomposable structure appears inevitable, where the physical states belong to a subspace V of the solutions, but where the field operator must be defined on a larger gauge dependent space V λ (which contains negative norm states). The physical subspace V is invariant but not invariantly complemented in V λ . The same feature is repeated in V , where one can find the invariant (but again not invariantly complemented) subspace of gauge solutions V g . These gauge solutions have zero norms and are orthogonal to all the elements of V [9]. Consequently, one must eliminate them from the subspace V by considering the physical state space as the coset V /V g .
For simplicity, the following units are used: where c,h and H are light velocity, Planck constant and Hubble parameter respectively. The paper is organized as follows: In section 2, some of the useful notations of de Sitter space and unitary irreducible representations of the de Sitter group will be recalled. Section 3 is devoted to derivation the de Sitter-spin-3 2 "massless" field equation. In this section, we will show that the field equation is gauge invariant. We will adopt a very convenient value for the gauge fixing parameter λ. In this paper, we get the second order wave equation by using the Casimir operator, then it is converted to the first order equation. There are two different equations for this case; one of those is considered in the context and the other is studied in Appendix. In section 4, the solutions are calculated in terms of the coordinate-independent de Sitter plane waves. It will be shown that, for λ = 1 2 , the field solution has a simple form. In section 5, we will define the two-point function S(x, y) that satisfies the following conditions: a) indefinite sesquilinear form, b) locality, c) covariance and d) normal analyticity. Normal analyticity allows us to define the two-point function S(x, y) as the boundary value of the analytic function S(z 1 , z 2 ) from the tube domains. The normal analyticity is related to the Hadamard condition that selects a unique vacuum state. S(z 1 , z 2 ) is defined in terms of the spinor-vector de Sitter plane-waves in their tube domains. Section 6 contains a brief conclusion and the outlook.
On de Sitter space the tangential (or transverse) derivative defines as The unitary irreducible representations of 10-parameter group SO 0 (1, 4) (connected component of the identity) of the de Sitter space, which is one of the two possible deformations of the Poincaré group (the other one being SO 0 (2, 3)), are characterized by the eigenvalues of the two Casimir operators Q (1) and Q (2) . These operators that commute with the group generators are constant in each unitary irreducible representation. They read where ǫ αβγδη is the usual antisymmetrical tensor in IR 5 and L αβ = M αβ + S αβ is an infinitesimal generator. The orbital part M αβ is In order to precise the action of the spinorial part S αβ on a field tensor or spinor-tensor one must treat separately the integer and half-integer cases. Tensor fields of rank l, Ψ γ 1 ...γ l (x), show integer spin fields, and the spinorial action is [7] S (l) where (γ i → β) means γ i index replaced with β. Half-integer spin fields with spin s = l + and with the Dirac gamma matrices γ α [11,12,13] γ α γ β + γ β γ α = 2η αβ , γ α † = γ 0 γ α γ 0 , the useful representations, which are compatible with group as follows where σ i are Pauli matrices and I is a 2 × 2 unit matrix. The Casimir operators are simple to manipulate in ambient space notation. Since Q (1) is a second order derivative operator, it is convenint used for obtaining the field equation. In particular, it is easy to show that for a l-rank tensor field Ψ γ 1 ...γ l (x) one has Ψ ′ is the trace of the l-rank tensor Ψ(x) and Σ p is the non-normalized symmetrization operator: For half-integer spin fields with spin s = l + 1 2 , the S αβ acts only upon the index i, and we have [13,14] S ). In this case, the Casimir operator is Then we obtain As you will see in the next section, the spin-3 2 field equation can be written in terms of the Casimir operator Q (1) .

Field equation
As previously mentioned, the operator Q (1) 3 2 commutes with the group generators and consequently it is constant on each unitary irreducible representation. In fact, we can classify the spinor-vector unitary irreducible representations by using the eigenvalues of Q (1) . The field equation can be written as In Takahashi [15] and Dixmier [16], a general classification scheme for all the unitary irreducible representations of the de Sitter group, is expressed and may be labeled by a pair of parameters (p, q) with 2p ∈ N and q ∈ C as follows According to the possible values of p and q, two types of unitary irreducible representations of the spin-3 2 field are distinguished for de Sitter group SO(1, 4) namely, the principal and the discrete series. More particularity about mathematical details of the group contraction and the relationship between the de Sitter and the Poincaré groups are given in the [17,18]. The unitary irreducible representations of the spin- 3 2 field relevant to the present work are as follows: i) The unitary irreducible representations U 3 2 ,ν in the principal series where p = s = 3 2 and q = 1 2 + iν matching to the Casimir spectral values: Note that U ii) The unitary irreducible representations Π ± 3 2 ,q of the discrete series, where p = s = 3 2 , correspond to The physical content of the principal series and the discrete series representation from the point of view of a Minkowskian observer at the limit H = 0 have been expressed in [1].
The "massless" spin-3 2 field in the de Sitter space corresponds to the discrete series Π ± 3 2 , 3 2 and the field equation is If we compare this equation with the massive spin-3 2 field case (equation (3.7) in [1]) they are equivalent for the values ν = ±i but these values do not admissible for the unitary irreducible representations of the de Sitter group in the principal series. It is natural to use the solution of the equation (3.1), which are already given for the massive case [1]. The corresponding spin-3 2 field solution can be written under the form [1]: where E α (x, ξ), and ψ(x) also contain constant terms involving the parameters p and q, where do not diverge for the specific values p = q = 3 2 corresponding to the relation (3.3). One can simply see that the field solution for these values is singular (equation (4.12) in [1]). Clearly, the singularity appears only for the spin-3 2 massless field for representations Π ± . This singularity appears due to the gauge invariance, which will be discussed in the next subsection. Therefore the subspace ∂.Ψ = 0 alone can not be used for the construction of a quantum massless spin-3 2 field. One must solve the equation in a larger space, which includes the ∂ · Ψ = 0 types of solutions. As expected, one finds three main types of solutions: the general solutions which aren't divergenceless, the divergencelessness type and the gauge type solution.

Gauge transformation
The "massless" spin- 3 2 field, however with the subsidiary conditions ∂ ⊤ .Ψ = 0 = γ.Ψ(x), is singular. This type of singularity is actually due to the divergencelessness condition needed to associate this field with a specific unitary irreducible representation of the de Sitter group. To solve this problem, the subsidiary conditions must be dropped. Then the field equation (3.5) replaced with the following equation where ζ is an arbitrary spinor field. For providing gauge invariant the following identities are used: Q Let us introduce a gauge fixing parameter λ. The wave equation now reads as the role of λ is just to fix the gauge spinor field ζ.
It is convenient and usual to continue this work with the first order field equation for the spinor case. We can write the equation (3.5) such as whereD β α and D δ β are defined by There are two possibilities for the first order field equation as follows: where for physical states, means λ = 0 in equation ( This equation can be written as a simple form For prettiness it can be written as where (T rpr) means is the transverse projection. We should be recalled that the massless spin-3 2 is singular when ν = ±i. This singularity refereed to gauge invariant as we have this difficult for all fields with spin s ≥ 1. Now we rewrite the equation (3.16) as which is invariant under gauge transformation:

18)
ζ is an arbitrary spinor field. Let us introduce a gauge fixing parameter λ for this case. Now the wave equation reads as The above equation under gauge transformation (3.18) becomes With the choice of value λ = 1, the gauge field ∂ ⊤ α ζ is put into the field equation only if ζ obeys that means the gauge field ζ is fixed. The adjoint Ψ α (x) is defined as follows [11,13] which satisfy the equation As stated in [6], "the appearance of (the Gupta-Bleuler) triplet seems to be universal in gauge theories, and crucial for quantization". The ambient space formalism will allow to exhibit this triplet for the present field in exactly the same manner as it occurs for the electromagnetic field.
Let us now define the Gupta-Bleuler triplet V g ⊂ V ⊂ V λ carrying the indecomposable structure for the unitary irreducible representations of the de Sitter group appearing in our problem: -The space V λ is the space of all square integrable solutions of the field equation (3.19).
It is λ dependent so that one can actually adopt an optimal value of λ which has very simple form. In the next section, we will show that this value is λ = 1 2 , (more generally for a spin s field, we have λ = (2/(2s + 1)) [7]).
This invariant subspace V is not invariantly complemented in V λ . In view of equation (3.19), it is obviously λ independent.
-The subspace V g of V consists of the gauge solutions of the form ∂ ⊤ α ζ, and ∂ ⊤ ζ = 0 = Q 0 ζ. These are orthogonal to every element in V including themselves. They form an invariant subspace of V but admit no invariant complement in V .
The de Sitter group acts on the physical (or transverse) space V /V g through the massless, helicity ± 3 2 unitary representation Π + , which are called the central parts in this paper.

The de Sitter-Spin-3 2 Plane Waves
According to the de Sitter plane waves, which were presented by Bros et al. [19], the de Sitter-Dirac plane wave for spinor field was calculated [13]. In this section, the spinor-vector solution can be written by using de Sitter-Dirac plane wave as terms in the following form where Z is an arbitrary five-component constant vector field: By putting Ψ α in equation (3.19), we find that the spinor fields ψ 1 , ψ 2 , and ψ 3 must obey the following equations: For λ = 1 2 equation (4.4) becomes By multiplying equation (4.2) in ∂ ⊤ , we have Since the spinor fields ψ 1 ,ψ 2 , and ψ 3 are homogeneous function of variate x with the same degree of homogeneity, therefor by using equation (4.5), we obtain

6)
ψ 2 can be written in the following form: where ψ g is ψ g = φ g U and Q 0 φ g = 0. U is an arbitrary constant spinor field, and φ g is a massless minimally coupled scalar field. By using the following identities: and we obtain If equations (4.12) and (4.13) are substituted in equation (4.1), it can be seen that in this choice of gauge λ = 1 2 , the spinor field ψ g or equivalently ψ 3 is removed, and the solution can be written in the following simplest form Ψ α (x) = D α ψ 1 , (4.14) where 15) and ψ 1 is the solution of the de Sitter-Dirac field equation. In the previous paper, the spinor field ψ 1 was explicitly calculated, and the solutions are given by [11,13] ( where V(x, ξ) = x ξV(ξ) and The two spinors V(ξ) and U(ξ) are where 1, 0, ±1). Finally the two possible solutions for and Ψ a 2α (x) = By taking the derivation of the plane waves (x.ξ) σ , the explicit forms of U α and V α are obtained in term of ξ as follows and the spinor field ψ 1 satisfy the field equation It is corresponds to the massless conformally invariant field equation [13,20]. The arbitrariness introduced with the constant vector Z will be removed by comparison the solution with the Minkowskian limit. Unfortunately, our notations for the "massless" conformally coupled scalar field are not adapted to the computation of the limit H = 0 . It is due to the fact that contrary to the "massive" case the values σ = −1, −2 are constant [21]. In order to obtain the behavior of the field solutions in the limit H = 0 (at least for the scalar part), one can use the global conformal coordinate system where X 0 = ρ, X 1 = α, X 2 = θ, X 3 = φ [22]. The square-integrable solutions of the field equation are given by [23] φ(x) = φ(ρ, v) = cos ρ e ±i(L+1)ρ where Y Llm ( v) are the hyperspherical harmonics on S 3 = {v i ∈ R 4 |v 2 1 + v 2 2 + v 2 3 + v 2 4 = r 2 }. The unitary irreducible representations Π ± the functions (4.26) become, when suitably rescaled, the usual massless spherical waves (with k 2 = (k 0 ) 2 − ( k) 2 = 0) [24]. With these coordinates, the de Sitter spinor-vector field solutions read where V is a spinor transformation matrix from ambient space notation to intrinsic coordinate which is defined in [25]. The plane waves (x · ξ) σ are singular at x · ξ = 0 and they are not globally defined due to the ambiguity concerning to the phase factor. In contrast with the Minkowskian exponentials plane wave, these waves are singular on three-dimensional light-like manifolds and can at first be defined only on suitable halves of X H . We will need an appropriate iǫ-prescription (indicated below) to obtain global waves, for detail see [20]. For a complete determination, one may consider the solution in the complex de Sitter space-time X where T ± = IR 5 + iV ± and the V + (resp. V − ) stems from the causal structure on X H : (4.31) We introduce their respective intersections with X (c) (4.32) which will be called forward and backward tubes of the complex de Sitter space X (c) H . Then we define the "tuboid" above X (c) (4.33) More details are given in [20]. When z varies in T + (or T − ) and ξ lies in the positive cone C + : The sign of the imaginary part of (z.ξ) has fixed ,so the plane wave solutions are globally defined. The phase is chosen such as boundary value of (z.ξ) σ | x.ξ>0 > 0. (4.34) Finally we have Ψ 1α (z) = U (λ) α (z, ξ)(Hz · ξ) −1 , (4.35) where z ∈ X (c) H and ξ ∈ C + these solutions are globally defiend in the de Sitter hyperboloid and they are independant of the choise of the intrisic coordinate.

Two-point function
The two-point function of "massless" spin- 3 2 field is defined as where x, x ′ ǫX H . This function is a solution of the wave equation (3.19) with respect to x and the wave equation (3.22) with respect to x ′ . In Appendix A.3, the solution of equation (3.22) is obtained and here we consider the equation (3.19). The solution can be found in terms of the two-point function of spinor field, which was calculated in the previous paper [13]. By using the recurrence formula (4.1), we define By imposing the two-point function S αα ′ to obey equation (3.19) and by using the identities of equations (4.2)-(4.4), S 1 , S 2 , and S 3 must be satisfied by the following equations: (5.5) For λ = 1 2 , equation (5.5) becomes and we know that S 1 is the solution of equation (5.3) and it is given by [13] S 1 (x, x ′ ) = 1 4π 3P S 2 and S 3 are given by Finally the two-point function is obtained similar to equation(4.14) as where this function satisfies the following conditions. a) Indefinite sesquilinear form: For any spinor-vector test function f α ∈ D(X H ), we have an indefinite sesquilinear form that is defined by wheref is the adjoint of f , and dσ(x) denotes the de Sitter -invariant measure on X H [20]. D(X H ) is the space of C ∞ spinor-vector functions with compact support in X H and with values in l where Λ ∈ SO 0 (1, 4) , g ∈ Sp(2, 2) and gγ α g −1 = Λ α β γ β . i(g) is the group involution defined by i(g) = −γ 4 gγ 4 .
S αα ′ (z, z ′ ) is maximally analytic, i.e., can be analytically continued to the "cut domain" [13,20]: The two-point function S αα ′ (x, x ′ ) is the boundary value of S αα ′ (z, z ′ ) from T 12 and the "permuted two-point function" S α ′ α (x ′ , x) is the boundary value of S αα ′ (z, z ′ ) from the domain:

Conclusions
We have studied the "massless" spin-3 2 fields in the de Sitter space-time in the ambient space formalism. This formalism is independent from the chosen coordinate system. Gauge and conformal invariant are the properties of "massless" fields. Gauge invariant and the Gupta-Bleuler triplet, are discussed. We have shown that the field equation of "massless" spin- 3 2 fields is gauge invariant. The two-point function is calculated. We saw that the spinor field ψ 1 in equation (4.25) is conformal invariant. Conformal invariant of the spinor-vector field will be considered in the forthcoming paper [26]. Since the homogenized coefficient is the same for the functions ψ 1 ,ψ 2 , and ψ 3 therefor Q 0 ψ 2 = 0, Q 0 ψ 3 = 0.

A.3 Two-point function
Here, the two-point function is calculated with respect to x ′ , which satisfies equation (3.22). In this case, we obtain Finally the two-point function in this case (for λ = 1 2 ) is