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.
Highlights
A scalar field in four-dimensional deSitter spacetime has quasinormal modes which are singular on the past horizon of the south pole and decay exponentially towards the future
The quasinormal modes of the southern diamond, which have complex L0 eigenvalues and comprise four real or two complex highest-weight representations.1. They are singular on the past horizon and decay exponentially towards the future, as opposed to the conventional southern diamond modes which oscillate everywhere
We show that the Euclidean vacuum has the simple and manifestly dS-invariant definition as the state annihilated by two of the four sets of quasinormal modes
Summary
We wish to expand the scalar field operator in the (anti-)quasinormal modes. Towards this end it is useful to introduce an inner product. The Klein-Gordon norms of the Ω-modes are. Denote the two-point functions for a CFT3 operators with dimensions h±. The Klein-Gordon norm is not suitable for quantization of the quasinormal modes. SO(3, 2) is the symmetry group of a CFT in 2 + 1 dimensions This suggests that the quantum states on which these generators act could belong to a 2 + 1dimensional CFT, which fits in nicely with the dS4/CFT3 conjecture. The norm of the first descendant is (not summing over k). For the SO(3)-symmetric states, we provide the exact formula in appendix B
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