On the Construction of Hartle–Hawking–Israel States Across a Static Bifurcate Killing Horizon

Abstract

We consider a linear scalar quantum field propagating in a spacetime of dimension d ≥ 2 with a static bifurcate Killing horizon and a wedge reflection. Under suitable conditions (e.g. positive mass), we prove the existence of a pure Hadamard state which is quasi-free, invariant under the Killing flow and restricts to a double β H -KMS state on the union of the exterior wedge regions, where β H is the inverse Hawking temperature. The existence of such a state was first conjectured by Hartle and Hawking (Phys Rev D 13:2188–2203, 1976) and by Israel (Phys Lett 57:107–110, 1976), in the more general case of a stationary black hole spacetime. Jacobson (Phys Rev D 50:R6031–R6032, 1994) has conjectured a similar state to exist even for interacting fields in spacetimes with a static bifurcate Killing horizon. The state can serve as a ground state on the entire spacetime and the resulting situation generalises that of the Unruh effect in Minkowski spacetime. Our result complements a well-known uniqueness result of Kay and Wald (Phys Rep 207:49–136, 1991) and Kay (J Math Phys 34:4519–4539, 1993), who considered a general bifurcate Killing horizon and proved that a certain (large) subalgebra of the free field admits at most one Hadamard state which is invariant under the Killing flow. This state is pure and quasi-free and in the presence of a wedge reflection it restricts to a β H -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, but only in the static case. Our proof follows the arguments of Sewell (Ann Phys 141: 201–224, 1982) and Jacobson (Phys Rev D 50:R6031–R6032, 1994), who exploited a Wick rotation in the Killing time coordinate to construct a corresponding Euclidean theory. In particular, we show that for the linear scalar field we can recover a Lorentzian theory by Wick rotating back. 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 analogues of analyticity and reflection positivity with respect to this coordinate.