On the construction of Hartle-Hawking-Israel states across a static bifurcate Killing horizon

Understand

We consider a linear scalar quantum field propagating in a space-time with a static bifurcate Killing horizon and a wedge reflection.

• We prove the existence of a Hadamard state which is pure, quasi-free, invariant under the Killing flow and which restricts to a double KMS state at the inverse Hawking temperature on the union of the exterior wedge regions.
• The existence of such a state was first conjectured by Hartle and Hawking (1976) and Israel (1976) for stationary black hole space times.
• Our result complements a uniqueness result of Kay and Wald (1991), who considered a general bifurcate Killing horizon and proved that a certain (large) subalgebra of the free field algebra admits at most one Hadamard state which is invariant under the Killing flow.