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

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. In the presence of a wedge reflection this state reduces to a pure, quasi-free KMS state on the smaller subalgebra associated to one of the exterior wedge regions. Our result establishes the existence of such a state on the full algebra in the static case. Our proof follows the arguments of Sewell (1982) and Jacobson (1994), exploiting a Wick rotation in the Killing time coordinate to construct a corresponding Euclidean theory. Because the Killing time coordinate is ill-defined on the bifurcation surface we systematically replace it by a Gaussian normal coordinate. A crucial part of our proof is to establish that the Euclidean ground state satisfies the necessary analogs of analyticity and reflection positivity with respect to this coordinate.