Sergio Albeverio, Yuri Kondratiev, Yuri Kozitsky
Classical Limits of Euclidean Gibbs States for Quantum
Lattice Models
(48K, LaTeX)
ABSTRACT. Models of quantum and classical particles on the $d$--dimensional
lattice $\Zd$
with pair and many--particle interactions of a general type are
considered. The
classical model is obtained from the corresponding quantum one when
the
reduced physical mass of the
particle $m = \mu /\hbar^2 $ tends
to infinity. For these models, it is proved that in the case
of pair interactions every net of
quantum conditional Gibbs measures weakly converges to a
classical conditional Gibbs measure when $m$ tends to infinity.
In the case of general type interactions, it is proved that every
conditional Gibbs measure of the classical model may be obtained as a
weak limit
of corresponding conditional Gibbs measures of the quantum model.
The convergence of the probability kernels and periodic Gibbs
measures for
such models has also been proven. The latter convergence is then used
to show
the convergence of the order parameters which describe the phase
transitions
in the translation invariant models with pair interactions of the
type cosidered.