Propagators and Gaussian effective actions in various patches of de Sitter space

We consider time-ordered (or Feynman) propagators between two different $\alpha-$states of a linear de Sitter Quantum Field in the global de Sitter manifold and in the Poincar\'e patch. We separately examine $\alpha-\beta$, In-In and In-Out propagators and find the imaginary contribution to the effective actions. The In-In propagators are real in both the Poincar\'e patch and in the global de Sitter manifold. On the other side the In-Out propagators at coincident points contain finite imaginary contributions in both patches in even dimensions, but they are not equivalent. In odd dimensions in both patches the imaginary contributions are zero. For completeness, we also consider the Static patch and identify in our construction the state that is equivalent to the Bunch-Davies one in the Poincar\'e patch.