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De Sitter Invariant Vacuum States, Vertex Operators, and Conformal Field Theory Correlators

We show that there is only one physically acceptable vacuum state for quantum fields in de Sitter space-time which is left invariant under the action of the de Sitter-Lorentz group $SO(1,d)$ and supply its physical interpretation in terms of the Poincare invariant quantum field theory (QFT) on one dimension higher Minkowski spacetime. We compute correlation functions of the generalized vertex operator $:e^{i\hat{S}(x)}:$, where $\hat{S}(x)$ is a massless scalar field, on the $d$-dimensional de Sitter space and demonstrate that their limiting values at timelike infinities on de Sitter space reproduce correlation functions in $(d-1)$-dimensional Euclidean conformal field theory (CFT) on $S^{d-1}$ for scalar operators with arbitrary real conformal dimensions. We also compute correlation functions for a vertex operator $e^{i\hat{S}(u)}$ on the Łobaczewski space and find that they also reproduce correlation functions of the same CFT. The massless field $\hat{S}(u)$ is the nonlocal transform of the massless field $\hat{S}(x)$ on de Sitter space introduced by one of us.