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Thermal equilibrium states of a linear scalar quantum field in stationary spacetimes

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 theory of 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. We also take the opportunity to weaken some of the technical assumptions of the earlier literature. In particular, we completely drop any assumptions on the behaviour of the norm and lapse of the Killing field. This is especially important for physical applications to the exterior region of a stationary black hole. Our review includes results on the existence of a unique ground state and of unique quasi-free KMS states, as well as an evaluation of the evidence that motivates the use of the KMS-condition to characterise thermal equilibrium. We especially draw attention to the poorly understood behaviour of the notion of temperature of a quantum field with respect to locality. If the spacetime is standard static (i.e. if it admits a Cauchy surface that is orthogonal to the Killing field) the analysis can be done even more explicitly. For compact Cauchy surfaces we consider Gibbs states and their properties. For general Cauchy surfaces we give a detailed and rigorous justification of the technique of Wick rotation, including the explicit determination of the Killing-time dependence of the quasi-free KMS states.