Hartle-Hawking state in the real-time formalism

We study self-interacting massive scalar field theory in static spacetimes with a bifurcate Killing horizon and a wedge reflection. In this theory the Hartle-Hawking state is defined to have the $N$-point correlation functions obtained by analytically continuing those in the Euclidean theory, whereas the double KMS state is the pure state invariant under the Killing flow and the wedge reflection which is regular on the bifurcate Killing horizon and reduces to the thermal state at the Hawking temperature in each of the two static regions. We demonstrate in the Schwinger-Keldysh operator formalism of perturbation theory the equivalence between the Hartle-Hawking state and the double KMS state with the Hawking temperature, which was shown before by Jacobson in the path-integral framework.