Quantum Stochastic Dynamics Ii

Abstract

We shortly review the progress in the domain of stochastic dynamics for quantum spin systems on a lattice. We also present some new results obtained in the framework of noncommutative [Formula: see text] spaces. In particular, using noncommutative Radon-Nikodym theorem of A. Connes we construct Markov generators of stochastic dynamics of spin flip type for systems at high temperatures or on one-dimensional lattice and with interactions of finite range at arbitrary temperatures.

QUANTUM Lp AND ORLICZ SPACES

“…Thus, this case will be treated in a separate section. (6) Furthermore, the case of the Hilbert space L 2 (M) gains in interest if we realize that the quantization of Markov-Feller processes can be done within the framework of noncommutative Hilbert spaces, see [49], [50], [51], [52]. (7) Hence, to get a simpler description of quantum maps arising both from Dirichlet forms as well as from bounded maps on L 2 (M), we combine some ideas given in [42] with the framework outlined in the previous sections.…”

On the structure of Markov flows

“…and, as a function of x, the right hand side of (20) is completely positive because j has this property. From (18) with n = 1, b = 1 B using j(1) = 1, we obtain (19).…”

“…Quantum analogues of these semigroups have also been considered by several authors (e.g. [20], [21], [19], [23], [24], . .…”

Remarks on the Structure of Dirichlet Forms on Standard Forms of Von Neumann Algebras

“…One of the reasons is that the general structure of Dirichlet forms for non-tracial states is not well-understood compared to the tracial case [2,3,6,12]. For constructions of Dirichlet forms for non-tracial states, we refer to [8,9,11,18,20,21,25,23] and the references there in. In [23], we gave a general construction method of Dirichlet forms on standard forms of von Neumann algebras.…”