Covariant and infrared-free graviton two-point function in de Sitter spacetime. II.

Abstract

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.

A small non-vanishing cosmological constant from the Krein–Gupta–Bleuler vacuum

“…Our aim in the present work, instead, will be to point out a potential relevance between the observable smallness of the cosmological constant and a choice of vacuum in the dS gravitational background of * bamba@sss.fukushima-u.ac.jp † Enayati@iauctb.ac.ir ‡ s.rahbardehghan@iauctb.ac.ir our expanding Universe, now known as the KGB vacuum. This vacuum, based on a new representation of the canonical commutation relations, was recently proposed as an alternative to the dS natural vacuum state (the Bunch-Davies state) that yields a fully covariant and coordinate-independent quantization (the KGB quantization) of linearized gravity in dS space [6][7][8][9][10][11][12]. [Due to the lack of the natural dS-invariant vacuum state for free gravitons, the fact that is now widely accepted in the physics community (see, for instance, [7,13,14]), the usual canonical quantization seems to break down for field theory of dS quantum gravity.]…”

Gupta-Bleuler quantization for linearized gravity in de Sitter spacetime

“…• It provides a remarkable automatic and covariant renormalization mechanism of the mean value of the stress tensor verifying the so-called Wald axioms (interestingly, the vacuum energy independent of the curvature is zero) [58][59][60].…”

Massive Rarita-Schwinger field in de Sitter space

“…Note that: (i) The above formulas have been found with the choices A · ξ = 0 = γ · A, for which, the calculation leads to the simplest form of the polarization vectors E α compatible with the Minkowski polarization vectors in the flat limit H → 0 (we will discuss this matter in more detail in the next subsection); (ii) Contrary to the flat space case, the polarization vectors E α are functions of spacetime; (iii) We have ξ ⊤ ·E = A·ξ = 0, since ξ ⊤ ·ξ ⊤ = (Hx·ξ) 2 ; (iv ) One can easily check that the spinor-vector waves (61) and (62) fulfill the auxiliary conditions (17). On the other hand, the adjoint spinor-vector waves ψ ( 3 2 ) satisfying the adjoint dS Rarita-Schwinger equation (24) are given by…”

“…All of these developments plead in favour of setting up a model of QFT in dS spacetime with the same level of completeness and rigor as its Minkowskian counterpart. In this regard, we refer in particular to a promising formulation of such a theory and its subsequent thermic interpretation that was originally put forward for the "massive" scalar fields in dS spacetime in the 1990's [7][8][9], and during recent two decades, it has been subject to scrutiny in a number of works to make explicit the extra algebraic structure inherent to other dS elementary systems (see, for instance, [10][11][12][13][14][15][16][17][18][19][20][21]). Technically, this model of dS QFT enjoys a robust group theoretical content.…”