  We analyze the renormalization properties of quantum field theories in de
Sitter space and show that only two of the maximally invariant vacuum states of
free fields lead to consistent perturbation expansions. One is the Euclidean
vacuum, and the other can be viewed as an analytic continuation of Euclidean
functional integrals on $RP^d$. The corresponding Lorentzian manifold is the
future half of global de Sitter space with boundary conditions on fields at the
origin of time. We argue that the perturbation series in this case has
divergences at the origin, which render the future evolution of the system
indeterminate without a better understanding of high energy physics.
