Completely Positive Linear Mappings, Non-hamiltonian Evolution, And Quantum Stochastic Processes

129. 130. I. Ro Yukhnovsky and Yu. V. Kozitsky, "Phase transition in the generalized hierarchical model in the collective variable method," Inst. Teor. Fiz. Akad. Nauk Ukr. SSR, Preprint, ITP-80-60E (1980). E. Zalys, "On large deviations in the continuous spin hierarchical Dyson models, ~' in: Abstracts III Int. Conf. Prob. Theory and Math, Statist., Vilnius (1981), Vol. 3, pp. 367-368~ COMPLETELY POSITIVE LINEAR MAPPINGS, NON-HAMILTONIAN EVOLUTION, AND QUANTUM STOCHASTIC PROCESSES V. Io Oseledets L~C 519.218, 519.217~ 519o248o2 The present survey is mainly devoted to works published from 1969-!980 in Ref. Mat. Zh. in which completely positive linear mappings are studied that arise, in particular, in the quantum theory of open systems, the quantum theory of measurements, and in problems of the dynamics of a small system interacting with a large system. Here the probabilistic aspect is singled out, and analogies and connections with ordinary Markov processes are indicated. In this survey we consider a number of mathematical works of recent years in which completely positive linear mappings of algebras with an involution are studied. Interest in this topic was stimulated not only by the development of the theory of C*algebras but also by some mathematical problems of quantum physics. Such mappings arise in the quantum theory of open systems and the quantum theory of measurements. We note specially the problem on the dynamics of a small system interacting with a large system. The study of semigroups of completely positive linear mappings is connected with the rigorous derivation of the quantum kinetic equation in a series of works of Davies [59, 60, 61, 62, 64, 69, 75]. He introduced the concept of a quantum stochastic process which generalizes the concept of a Markov process~ Lindblad [137, 138] defined and studied non-Markov quantum sto