The recent observational data are strongly in favor of a positive acceleration of the present universe. In a first approximation, the background SPACE-time that we live in, might be considered as a de SITTER SPACE-time. If this is so, it is important to understand how quantum gravity and should formulated in de SITTER SPACE. We discusses the some properties of linear quantum gravity necessitated to the indefinite metric quantization for preserving the de SITTER invariant and eliminating the infrared divergence in the graviton two points function. There appears two different types of negative norm states. The first one is similar to those non physical states dual to gauge states which appear in gauge quantum field theory in Minkowski SPACE. The other is due to the occurrence of negative frequency solutions in order to preserve de SITTER covariance. This is similar to the case of minimally coupled scalar field in dS SPACE. We discuss the possibility of generalization to the Minkowskian limit and their consequences in the renormalization theory.

The conformally invariant linearized de SITTER gravitational wave equation has recently received a lot of attention. In this article, using the ambient SPACE notations, we solve this field equation in five various cases. In each case, it has been shown that the solution can be written as the product of a generalized symmetric polarization tensor of rank 2 and a massless minimally or conformally coupled scalar field in de SITTER SPACE-time. The conformally covariant graviton two-point functions have been calculated in terms of the massless minimally or conformally coupled scalar two-point functions, using ambient SPACE notations. In the case of massless minimally coupled scalar field, the Krien SPACE quantization has been used to avoid the violation of de SITTER invariance. The two-point functions are expressed in terms of de SITTER intrinsic coordinates from their ambient SPACE counterparts, which are free of any theoretical problem.

The ”massless” vector field equation in de SITTER (dS) 4-dimensional SPACE is gauge invariant under some special gauge transformations. It is also gauge invariant in ambient SPACE notations in which the field equation is written in terms of the Casimir operators of dS group. In this paper the "massless" vector field equation has been solved in the physical case. It has been shown that the solution can be written as the multiplication of a generalized polarization vector and a ”massless” conformally coupled scalar field in the ambient SPACE notations. The physical vector twopoint function has been calculated using ambient SPACE formalism and its zero curvature limit has been considered. It is shown that the physical vector two-point can be written in terms of the conformally coupled massless scalar two-point function in the ambient SPACE notations. The two-point function is expressed in terms of dS intrinsic coordinates from its ambient SPACE counterpart, which is dS-invariant and is free from any divergences.

The harmonic oscillator in anti de SITTER SPACE(AdS) is discussed. We con-sider the harmonic oscillator potential and then time independent Schrodinger equation in AdS SPACE. Then, we apply the supersymmetric Quantum Mechanics approach to solve our di erential equation. In this paper, we have solved Schrodinger equation for harmonic oscillator in AdS SPACEtime by supersymmetry approach. The shape invari-ance, charge conjugation and other properties of supersymmetric Quantum Mechanics for our equation are discussed. We investigate the dynamical symmetry into de nite group. We also obtain the Casimir operator and thermodynamics properties of har-monic oscillator in Anti de SITTER SPACE. We have discussed the corresponding algebra for the N = 2 Supersymmetry. The energy spectrum of the harmonic oscillator is similar to the at SPACE but the constant terms of it are di erent, This constant terms depend on the geometrical parameter of the background. Finally, we have obtained the various thermodynamics quantity of the system.

Gravitational waves are the last unconfirmed prediction of the general relativity. These waves are tiny fluctuations in world frame that dessipate energy throghout SPACE. The gravitatinal waves spectra of fluctuations can be originated from the non-linear effects during different cosmic evolution periods, especially from initially non-linear and excited vacuum state in the very early universe. Based on this fact, in this paper introducing "excited-de SITTER vacuum" as a fundamental mode, the obtained power spectrum has been investigated. Corrected spectra obtained from Hilbert and Krein SPACEs are compared. The renormalization approach presented in this work, preserves the curved SPACE-time symmetry and stimulates us to use excited de SITTER mode. Also, the corrections obtained from the non-linear mode includes the second-order corrections and in the linear limit accords with the results from conventional methods.

In the previous paper (Behroozi et al., Phys Rev D 74:124014, 2006; Dehghani et al., Phys Rev D 77:064028, 2008), conformal invariance for massless tensor fields (scalar, vector and spin-2 fields) was studied and the solutions of their wave equations and two-point functions were obtained. In the present paper, conformally invariant wave equation for massless spinor field in de SITTER SPACE–time has been obtained. For this propose, we use Dirac’s six-cone formalism. The solutions of massless spin -1/2 and -3/2 equations, in the ambient SPACE notation, have been calculated.

The phase SPACE which is related to the motion of massive particle on 1+3- De SITTER SPACE is a 3-dimensional complex sphere. Our main aim in this study is discribing this movement in the frame quantum mechanics. Transfering from classical mechanic to quantum mechanics is possible by means of coherent states. Thus, after determination of this state, we quantize some of the classical observables.

We give in this paper an explicit construction of the covariant quantization of the rank-2 “massless” tensor field on de SITTER (dS) SPACE (linear covariant quantum gravity on a dS background). The main ingredient of the construction is an indecomposable representation of de SITTER group with different indecomposability channels. We here make the choice of a specific gauge fixing in order to get the simplest possible structure of the involved Gupta-Bleuler triplets. We describe the related Krein SPACE structure and covariant field operators kab(c). We show that our gauge fixing eliminates any infrared divergence in the two-point function for the traceless part of this field. But it is not possible to do the same for the pure trace part (conformal sector). This work is in the continuation of our previous ones concerning the “minimally coupled scalar fields and the “massive” tensor field on dS.

The field equation following the non-linear Lagrangian -2L + R + aR2 has been considered as an alternative to Einstein's theory. Based on the background field method, the linearized fourth order field equation has been obtained in both the 4-dimensional de SITTER SPACE-time and the flat 5-dimensional ambient SPACE notations. The field equation has been projected into the Minkowskian 4-dimensional SPACE by taking the zero curvature limit. It has been shown that the field equation transforms according to one of the unitary irreducible representations of de SITTER group denoted by P 2,2±, in discreet series.