Analytic Hadamard states, Calder\'on projectors and Wick rotation near analytic Cauchy surfaces

Understand

We consider the Klein-Gordon equation on analytic spacetimes with an analytic Cauchy surface.

• In this setting, we prove the existence of pure analytic Hadamard states.
• The proof is based on considering an elliptic operator obtained by Wick rotating the Klein-Gordon operator in a neighborhood of a Cauchy hypersurface.
• The Cauchy data of Hadamard two-point functions are constructed as Calder\'{o}n projectors (suitably generalized if the hypersurface is non-compact) for the elliptic operator.