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.