THERMAL EQUILIBRIUM STATES OF A LINEAR SCALAR QUANTUM FIELD IN STATIONARY SPACE–TIMES

Abstract

The linear scalar quantum field, propagating in a globally hyperbolic space–time, 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 space–time. 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 behavior 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 behavior of the temperature of the quantum field with respect to locality. If the space–time 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.