We consider time-ordered (or Feynman) propagators between two different α−states of a linear de Sitter Quantum Field in the global de Sitter manifold and in the Poincaré patch. We separately examine α − β, In-In and In-Out propagators and find the imaginary contribution to the effective actions. The In-In propagators are real in both the Poincaré 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 Poincaré patch.
We construct explicit mode expansions of various tree-level propagators in the Rindler-de Sitter universe, also known as the static (or compact) patch of the de Sitter spacetime. We construct in particular the Wightman functions for thermal states having a generic temperature T. We give a fresh simple proof that the only thermal Wightman propagator that respects the de Sitter isometry is the restriction to the Rindler-de Sitter wedge of the propagator for the Bunch-Davies state. It is the thermal state with T ¼ ð2πÞ −1 in the units of de Sitter curvature. We show that propagators with T ≠ ð2πÞ −1 are only time translation invariant and have extra singularities on the boundary of the static patch. We also construct the expansions for the so-called alpha-vacua in the static patch and discuss the flat limit.
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