The Hartle–Hawking–Israel state on spacetimes with stationary bifurcate Killing horizons.

We consider a free massive quantized Klein–Gordon field in a spacetime (M , g) containing a stationary bifurcate Killing horizon, i.e. a bifurcate Killing horizon whose Killing vector field is globally time-like in the right wedge ℳ + associated to the horizon. We prove the existence of the Hartle–Hawking–Israel (HHI) vacuum state, which is a pure state on the whole spacetime whose restriction to ℳ + is a thermal state ω T H for the time-like Killing field at Hawking temperature T H = κ (2 π) − 1 , where κ is the surface gravity of the horizon. We show that the HHI state is a Hadamard state and is the unique Hadamard state which is equal to the double T H − 1 -KMS state in the double wedge ℳ − ∪ ℳ + . We construct the HHI state by Wick rotation in Killing time coordinates, using the notion of the Calderón projector for elliptic boundary value problems. [ABSTRACT FROM AUTHOR]