UNIQUENESS AND CLUSTERING PROPERTIES OF GIBBS-STATES FOR CLASSICAL AND QUANTUM UNBOUNDED SPIN SYSTEMS

We consider quantum unbounded spin systems (lattice boson systems) in v-dimensional lattice space Z(v). Under appropriate conditions on the interactions we prove that in a region of high temperatures the Gibbs state is unique, is translationally invariant, and has clustering prop erties. The main methods we use are the Wiener integral representation , the cluster expansions for zero boundary conditions and for general Gibbs state, and explicitly P-dependent probability estimates. For one -dimensional systems we show the uniqueness of Gibbs states for any va lue of temperature by using the method of perturbed states. We also co nsider classical unbounded spin systems. We derive necessary estimates so that all of the results for the quantum systems hold for the class ical systems by straightforward applications of the methods used in th e quantum case.