  We consider scalar field theory in de Sitter space with a general vacuum
invariant under the continuously connected symmetries of the de Sitter group.
We begin by reviewing approaches to define this as a perturbative quantum field
theory. One approach leads to Feynman diagrams with pinch singularities in the
general case, which renders the theory perturbatively ill-defined. Another
approach leads to well-defined perturbative correlation functions on the
imaginary time continuation of de Sitter space. When continued to real-time, a
path integral with a non-local action generates the time-ordered correlators.
Curiously, observables built out of local products of the fields show no sign
of this non-locality. However once one couples to gravity, we show acausal
effects are unavoidable and presumably make the theory ill-defined. The
Bunch-Davies vacuum state is the unique de Sitter invariant state that avoids
these problems.
