An Approach to Quantum Statistical Mechanics

We start the discussion of the free Bose gas on Robertson-Walker space-times by introducing a new state on the Weyl algebra of the free massive Klein-Gordon eld on Robertson-Walker space-times. This state approximates a thermal equilibrium state and opens the possibility to do quantum statistical mechanics on expanding universes. We explain why this state is believed to be a physically relevant state in the framework of algebraic quantum eld theory. This remains true if we introduce a chemical potential but the value of the chemical potential is restricted. We also explain how to describe the time evolution of the state. It can be shown that the inverse temperature of the state changes proportional to the scale parameter in the Robertson-Walker metric. Beyond the standard model of cosmology the innationary scenario involving phase transitions and symmetry breaking has been intensively discussed during the last years. For an investigation of such phenomena it is necessary to describe the thermal behavior of quantum elds and states. In this work we start a discussion of quantum statistical mechanics on Robertson-Walker space-times in the framework of algebraic quantum eld theory. An analysis of the free Bose gas on Robertson-Walker space-times could serve as a model for more complicated quantum eld theories. This model may show a phase transition, namely Bose-Einstein condensation. In quantum eld theory on Minkowski space-time the Poincar e group as the symmetry group and the spectrum condition determine the Minkowski vacuum. On a generic space-time the symmetry group is trivial and there is no distinguished vacuum state. In general there is a whole class of physically relevant states, the Hadamard states, which are, roughly speaking, determined by their short-distance behaviour (see the book of Wald 7 and references therein). The Hadamard states allow the renormalization of the energy-momentum tensor (see Wald 7). Hadamard states also fullll the principle of local deeniteness (see Verch 6 for details). These are the main reasons to believe that Hadamard states are physically relevant. The class of adiabatic vacuum states, deened on the Weyl-algebra of the free massive Klein-Gordon eld on Robertson-Walker space-times, is known to be a subset of the class of Hadamard states 1. Adiabatic vacua can be understood as approximate vacuum states on Robertson-Walker space-times. On the other hand on ultrastatic space-times, such as the Einstein static universe it is possible to de-ne KMS states, i.e. thermal equilibrium states, of inverse temperature. Here, we …