GLAUBER DYNAMICS FOR QUANTUM LATTICE SYSTEMS

Abstract

Models of quantum mechanical anharmonic lattice systems ("anharmonic crystals") are described. Temperature quantum Gibbs states are represented by classical Gibbs measures for lattice systems of loop-valued spin variables. These Gibbs measures are also obtained as invariant (equilibrium) measures of a system of stochastic differential equations ("stochastic dynamics", "stochastic quantization"). Existence and uniqueness results for these equations are established and a construction of the solution via a finite volume approximation is given. The Markov property of this solution is also exhibited and properties of the Gibbs distributions (existence, a prioiri estimates, regularity of support) are characterized in terms of the stochastic dynamics. Ergodicity and uniqueness of the Gibbs distributions are also discussed.