  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 \L obaczewski 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.
