Natural energy bounds in quantum thermodynamics

Given a stationary state for a noncommutative flow, we study a boundedness condition. depending on a parameter beta > 0, which is weaker than the KMS equilibrium condition at inverse temperature beta. This 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 Ruelle's one and thus selects a restricted class of non-equilibrium steady states. We also introduce the complete boundedness condition and show this notion to be equivalent to the Pusz-Woronowicz complete passivity property, hence to the KMS condition. In Quantum Field Theory, the beta -boundedness condition can be interpreted as the property that localized state vectors have energy density levels increasing beta -subexponentially, a property which is similar in the form and weaker in the spirit than the modular compactness-nuclearity condition. In particular, for a Poincare covariant net of C*-algebras on Minkowski spacetime, the beta -boundedness property, beta greater than or equal to 2 pi, for the boosts is shown to be equivalent to the Bisognano-Wichmann property. The Hawking temperature is thus minimal for a thermodynamical system in the background of a Rindler black hole within the class of beta -holomorphic states. More generally, concerning the Killing evolution associated with a class of stationary quantum black holes, we characterize KMS thermal equilibrium states at Hawking temperature in terms of the boundedness property and the existence of a translation symmetry on the horizon.