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We set up a general framework for Calder\'on projectors (and theirgeneralization to non-compact manifolds), associated with complex Laplacianse.g. obtained by Wick rotation of a Lorentzian metric. In the analytic case, weuse this to show that the Laplacian's Green's functions have analyticcontinuations whose boundary values are two-point functions of analyticHadamard states. The result does not require the metric to be stationary. As anaside, we describe how thermal states are obtained as a special case of thisconstruction if the coefficients are time-independent.

Wick rotation of the time variables for two-point functions on analytic backgrounds