Classical Limits of Euclidean Gibbs States for Quantum Lattice Models

Models of quantum and classical particles on the d–dimensional lattice ZZ with pair interparticle interactions are considered. The classical model is obtained from the corresponding quantum one when the reduced physical mass of the particle m = μ/h̄ tends to infinity. For these models, it is proposed to define the convergence of the Euclidean Gibbs states, when m → +∞, by the weak convergence of the 1 corresponding local Gibbs specifications, determined by conditional Gibbs measures. In fact it is proved that all conditional Gibbs measures of the quantum model weakly converge to the conditional Gibbs measures of the classical model. A similar convergence of the periodic Gibbs measures and, as a result, of the order parameters, for such models with pair interactions possessing the translation invariance, has also been proven.