Glauber dynamics for quantum lattice systems

Models of quantum mechanical anharmonic lattice systems ("anharmonic crysta ls") are described. Temperature quantum Gibbs states are represented by cla ssical Gibbs measures for lattice systems of loop-valued spin variables. Th ese Gibbs measures are also obtained as invariant (equilibrium) measures of a system of stochastic differential equations ("stochastic dynamics", "sto chastic quantization"). Existence and uniqueness results for these equation s are established and a construction of the solution via a finite volume ap proximation is given. The Markov property of this solution is also exhibite d and properties of the Gibbs distributions (existence, a prioiri estimates , regularity of support) are characterized in terms of the stochastic dynam ics. Ergodicity and uniqueness of the Gibbs distributions are also discusse d.