Natural Energy Bounds in Quantum Thermodynamics

We characterize KMS thermal equilibrium states, at Hawking temperature β, for the Killing evolution associated with a class of stationary quantum black holes in terms of a boundedness property, namely the localized state vectors should have energy density levels increasing β-subexponentially; a property which is similar in the form and weaker in the spirit than the modular compactness-nuclearity condition in Quantum Field Theory. In particular, for a Poincaré covariant net of C∗-algebras on the Minkowski spacetime, the boundedness property for the boosts (in the Rindler wedge) is shown to be equivalent to the Bisognano-Wichmann property. In general, the boundedness condition is equivalent to a holomorphic property closely related to the one recently considered by Ruelle and D’Antoni-Zsido and shared by a natural class of non-equilibrium steady states. Our holomorphic property is stronger than the Ruelle’s one and thus selects a restricted class of non-equilibrium steady states. The Hawking temperature is minimal for a thermodynamical system in the background of a black hole within this class of states. Given a stationary state for a noncommutative flow, we also introduce the complete boundedness condition and show this notion to be equivalent to the PuszWoronowicz complete passivity property, hence to the KMS equilibrium condition. Work partially supported by MURST and GNAFA-INDAM