Thermal Equilibrium States of a Linear Scalar Quantum Field in Stationary Space–times

Abstract

The linear scalar quantum field, propagating in a globally hyperbolic spacetime, is a relatively simple physical model that allows us to study many aspects in explicit detail. In this review we focus on the thermal equilibrium (KMS) states of such a field in a stationary spacetime. Our presentation draws on several existing sources and aims to give a unified exposition, while weakening certain technical assumptions. In particular we drop all assumptions on the behaviour of the time-like Killing field, which is important for physical applications to the exterior region of a stationary black hole.Our review includes results on the existence and uniqueness of ground and KMS states, as well as an evaluation of the evidence supporting the KMS-condition as a characterization of thermal equilibrium. We draw attention to the poorly understood behaviour of the temperature of the quantum field with respect to locality.If the spacetime is standard static, the analysis can be done more explicitly. For compact Cauchy surfaces we consider Gibbs states and their properties. For general Cauchy surfaces we give a detailed justification of the Wick rotation, including the explicit determination of the Killing time dependence of the quasi-free KMS states. *

Hadamard state in Schwarzschild–de Sitter spacetime

“…Further properties of KMS states, also in curved spacetimes, can be found in the recent review [39].…”

Local incompatibility of the microlocal spectrum condition with the KMS property along spacelike directions in quantum field theory on curved spacetime

“…In this regard, it is worth noting that KMS states for linear quantum fields with respect to a timeevolution on a stationary spacetime, where the corresponding Killing vector field is everywhere timelike, have been shown to fulfill the microlocal spectrum condition [SV00]. The existence of KMS states for quantized linear fields on stationary spacetimes has also been established, see [Sa13] and references cited there.…”

Microlocal Analysis of Quantum Fields on Curved Spacetimes

“…It is known, see e.g. [S2,Prop. 3.1] that a stationary spacetime which is also globally hyperbolic (see Subsect.…”

“…The standard model of a static spacetime is the one above for w i dx i = 0. A static, globally hyperbolic spacetime is isometric to the standard model iff one can choose S to be a Cauchy surface, see [S2,Prop. 3.2].…”