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