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 βH-KMS state on the union of the exterior wedge regions, where βH 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 βH-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.
Mathematics Subject Classification
Quantum fields in curved spacetime thermal quantum field theory Hawking radiation
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