Hartle-Hawking state in the real-time formalism

TL;DR

This paper demonstrates the equivalence between the Hartle-Hawking state and the double KMS state at Hawking temperature in a self-interacting scalar field theory on static spacetimes with a bifurcate Killing horizon, using the real-time formalism.

Contribution

It provides a perturbative proof of the equivalence between the Hartle-Hawking and double KMS states in the Schwinger-Keldysh formalism, extending previous path-integral results.

Findings

Equivalence between Hartle-Hawking and double KMS states established perturbatively.

Demonstrates the regularity and thermal properties of the states on the bifurcate Killing horizon.

Validates the real-time formalism approach for quantum fields in curved spacetimes.

Abstract

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 Kubo-Martin-Schwinger (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.