Quantum Markovian semigroups on quantum spin systems: Glauber dynamics

We study a class of KMS-symmetric quantum Markovian semigroups on a quantum spin system (A, tau, omega), where A is a quasi-local algebra, T is a strongly continuous one parameter group of *-automorphisms of A and to is a Gibbs state on A. The semigroups can be considered as the extension of semigroups on the nontrivial abelian subalgebra. Let H be a Hilbert space corresponding to the GNS representation constructed from omega. Using the general construction method of Dirichlet form developed in [8], we construct the symmetric Markovian semigroup {T-t}(t >= 0) on H. The sernigroup {T-t}(t >= 0) acts separately on two subspaces H-d and H-od of H, where H-d is the diagonal subspace and H-od is the off-diagonal subspace, H = Hd circle plus H-od. The restriction of the semigroup {T-t}(t >= 0) on H-d is Glauber dynamics, and for any eta is an element of H-od, T-t eta decays to zero exponentially fast as t approaches to the infinity.