Quasinormal Quantization in deSitter Spacetime

TL;DR

The paper develops a quasinormal-mode-based quantization for a scalar field in de Sitter space, using $SO(4,1)$ highest-weight representations and a finite, $R$-norm to regularize the inner product. This framework yields a complete, orthogonal basis formed by quasinormal modes and their descendants, while preserving de Sitter invariance; the Euclidean vacuum arises as the state annihilated by half the modes, and the Euclidean Green function is obtained via a mode-sum. An effective symmetry reduction to $SO(3,2)$ under the $R$-norm aligns with dS$_4$/CFT$_3$ expectations and underpins the CFT-like interpretation of the Hilbert space. The construction is extended to general light scalars ($m^2\ell^2<9/4$) and is argued to generalize to other dimensions and spins, with Southern/Rindler-type modes explored as potential probes of physics in a single causal diamond.

Abstract

A scalar field in four-dimensional deSitter spacetime (dS_4) has quasinormal modes which are singular on the past horizon of the south pole and decay exponentially towards the future. These are found to lie in two complex highest-weight representations of the dS_4 isometry group SO(4,1). The Klein-Gordon norm cannot be used for quantization of these modes because it diverges. However a modified `R-norm', which involves reflection across the equator of a spatial S^3 slice, is nonsingular. The quasinormal modes are shown to provide a complete orthogonal basis with respect to the R-norm. Adopting the associated R-adjoint effectively transforms SO(4,1) to the symmetry group SO(3,2) of a 2+1-dimensional CFT. It is further shown that the conventional Euclidean vacuum may be defined as the state annihilated by half of the quasinormal modes, and the Euclidean Green function obtained from a simple mode sum. Quasinormal quantization contrasts with some conventional approaches in that it maintains manifest dS-invariance throughout. The results are expected to generalize to other dimensions and spins.