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We revisit the construction by Sanders [S1] of the Hartle-Hawking-Israelstate for a free quantum Klein-Gordon field on a spacetime with a static,bifurcate Killing horizon and a wedge reflection. Using the notion of theCalder{\'o}n projector for elliptic boundary value problems andpseudodifferential calculus on manifolds, we give a short proof of its Hadamardproperty.

On the Hartle-Hawking-Israel states for spacetimes with static bifurcate Killing horizons