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 equiva lent to a holomorphic property closely related to the one recently consider ed by Ruelle and D'Antoni-Zsido and shared by a natural class of non-equili brium steady states. Our holomorphic property is stronger than Ruelle's one and thus selects a restricted class of non-equilibrium steady states. We a lso introduce the complete boundedness condition and show this notion to be equivalent to the Pusz-Woronowicz complete passivity property, hence to th e KMS condition. In Quantum Field Theory, the beta -boundedness condition can be interpreted as the property that localized state vectors have energy density levels in creasing beta -subexponentially, a property which is similar in the form an d weaker in the spirit than the modular compactness-nuclearity condition. I n particular, for a Poincare covariant net of C*-algebras on Minkowski spac etime, 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 t he 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 equil ibrium states at Hawking temperature in terms of the boundedness property a nd the existence of a translation symmetry on the horizon.