The Krein–Gupta–Bleuler quantization in de Sitter space–time; Casimir energy–momentum tensor for a curved brane

Abstract

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]).

Massless spin-2 field in de Sitter space

“…Applying the unitary condition, it is proved that this quantization scheme in Minkowski space when interaction is taken into account truly yields the common results; the so-called radiative corrections are indeed the same as usual QFT (see the mathematical details in [53]). It also allows us to obtain the exact usual result for the black hole radiation, even regarding that the free field vacuum expectation value of the energy-momentum tensor is zero [54] (in this regard, see also [55][56][57]). On the other hand, following Wald, there exists a case where this quantization method seems in a very natural way.…”

Gupta-Bleuler quantization for linearized gravity in de Sitter spacetime

“…Here and before going any further, it would be interesting to characterize the gauge and the divergencelessness parts of the graviton bitensor two-point function (59). In a totally symmetric way, the gauge part D ′ 2 W Ξ g obeys…”

Casimir energy-momentum tensor for a quantized bulk scalar field in the geometry of two curved branes on Friedmann-Robertson-Walker background

“…In this regard, by considering the hyperspherical coordinates (r, θ 1 = θ, θ 2 , ..., θ D−1 ), the corresponding line element can be written as follows (1) in which γ = 1/ √ 1 + r 2 and dΩ 2…”

“…An intrinsic feature of these scenarios is the presence of boundaries and propagating fields in the bulk, which will naturally give Casimir-type contributions to the mean * h.pejhan@piau.ac.ir † s.rahbardehghan@iauctb.ac.ir 1 In the original brane-world scenarios such as the so-called RS models, standard model particles and fields are localized on a D-dimensional hyper-surface called the "brane" embedded in a higher-dimensional space-time called the "bulk", where gravity is the only field which has access to the extra dimensions.…”