In this paper, the two-point function of linearized gravitons on de Sitter (dS) space is presented. Technically, respecting the dS ambient space notation, the field equation is given by the coordinateindependent Casimir operators of the de Sitter group. Analogous to the quantization of the electromagnetic field in Minkowski space, the field equation admits gauge solutions. The notation allows us to exhibit the formalism of Gupta-Bleuler triplets for the present field in exactly the same manner as it occurs for the electromagnetic field. In this regard, centering on the spin-two part (the traceless part, K t ), the field solution is written as a product of a generalized polarization tensor and a minimally coupled massless scalar field. Then, admitting a de Sitter-invariant vacuum through the so-called "Krein space quantization", the de Sitter fully covariant two-point function is calculated. This function is interestingly free of pathological large distance behavior (infrared divergence). Moreover, the spin-zero part (the pure-trace part; K pt ) of the field is discussed in this paper. It is shown that the implications of the dS group unitary irreducible representations restrict the gauge-fixing parameter to the optimal value, which remarkably results in the pure-trace part be written in terms of a conformally coupled massless scalar field.

The solution to the linearized Einstein equation in de Sitter (dS) spacetime and the corresponding two-point function are explicitly written down in a gauge with two parameters 'a' and 'b'. The quantization procedure, independent of the choice of the coordinate system, is based on a rigorous group theoretical approach. Our result takes the form of a universal spin-two (transverse-traceless) sector and a gauge-dependent spin-zero (pure-trace) sector. Scalar equations are derived for the structure functions of each part. We show that the spin-two sector can be written as the resulting action of a second-order differential operator (the spin-two projector) on a massless minimally coupled scalar field (the spin-two structure function). The operator plays the role of a symmetric rank-2 polarization tensor and has a spacetime dependence. The calculated spin-two projector grows logarithmically with distance and also no dS-invariant solution for either structure functions exist. We show that the logarithmically growing part and the dS-breaking contribution to the spin-zero part can be dropped out, respectively, for suitable choices of parameters 'a' and 'b'. Considering the transverse-traceless graviton two-point function, however, shows that dS breaking is universal (cannot be gauged away). More exactly, if one wants to respect the covariance and positiveness conditions, the quantization of the dS graviton field (as for any gauge field) cannot be carried out directly in a Hilbert space and involves unphysical negative norm states. However, a suitable adaptation (Krein spaces) of the Gupta-Bleuler scheme for massless fields, based on the group theoretical approach, enables us to obtain the corresponding two-point function satisfying the conditions of locality, covariance, transversality, index symmetrizer, and tracelessness.

From the group theoretical point of view, it is proved that the theory of linear conformal gravity should be written in terms of a tensor field of rank-3 and mixed symmetry [Binegar et al., Phys. Rev. D 27, (1983) 2249. We obtained such a field equation in de Sitter space [Takook et al, J. Math. Phys. 51, (2010) 032503]. In this paper, a proper solution to this equation is obtained as a product of a generalized polarization tensor and a massless scalar field and then the conformally invariant two-point function is calculated. This two-point function is de Sitter invariant and free of any pathological large-distance behavior. * Electronic address: takook@razi.ac.ir † Electronic address: m_tanhayi@iauctb.ac.ir

In this paper, considering the linearized Einstein equation with a two-parameter family of linear covariant gauges in de Sitter spacetime, we examine possible vacuum states for the gravitons field with respect to invariance under the de Sitter group SO0(1, 4). Our calculations explicitly reveal that there exists no natural de Sitter-invariant vacuum state (the Euclidean state) for the gravitons field. Indeed, on the foundation of a rigorous group theoretical reasoning, we prove that if one insists on full covariance as well as causality for the theory, has to give up the positivity requirement of the inner product. However, one may still look for states with as much symmetry as possible, more precisely, a restrictive version of covariance by considering the gravitons field and the associated vacuum state which are, respectively, covariant and invariant with respect to some maximal subgroup of the full de Sitter group. In this regard, we treat SO(4) case, and find a family of SO(4)-invariant states. The associated SO(4)-covariant quantum field is given, as well.

In this paper, vacuum expectation value (VEV) of the energy-momentum tensor for a conformally coupled scalar field in de Sitter space-time is investigated through the Krein-Gupta-Bleuler construction. This construction has already been successfully applied to the de Sitter minimally coupled massless scalar field and massless spin-2 field to obtain a causal and fully covariant quantum field on the de Sitter background. We also consider the effects of boundary conditions. In this respect, Casimir energy-momentum tensor induced by Dirichlet boundary condition on a curved brane is evaluated. * e-mail: h.pejhan@piau.ac.ir 1 In this formalism, Vg stands for the space of gauge states (longitudinal photon states), while the space of positive frequency solutions of the field equation which satisfy the Lorentz condition is defined by V . Meanwhile, V ′ is allotted to the all positive frequency solutions space which includes un-physical states. These spaces verify Vg ⊂ V ⊂ V ′ . The Fock space is constructed over V ′ , which is not a Hilbert space, but an indefinite inner product space; the Klein-Gordon inner product determines the Poincaré and locally and conformally invariant indefinite inner product on V ′ . It should be noted that, all three spaces carry representations of the Poincaré group but Vg and V are not covariantly complemented. The quotient space V /Vg of states up to a gauge transformation is the space of physical one-photon states (for more mathematical details, one can refer to [7,8]).