  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.
