Quantum Stochastic Processes

Abstract

A class of non-commutative stochastic processes is defined.These processes are defined up to equivalence by their multi-time correlation kernels.A reconstruction theorem, generalizing the Kolmogorov theorem for classical processes, is proved.Markov processes and their associated semigroups are studied, and some non-quasi free examples are constructed on the Clifford algebra, with the use of a perturbation theory of Markov processes.The connection with the Hepp-Lieb models is discussed.§ 0. IntroductionWe study a class of non-commutative stochastic processes which are determined up to equivalence by their multi-time correlations.They are analogues of classical processes in the sense of Doob [1], Meyer [2]; indeed, those processes are included as a special case.We define a stochastic process over a C*-algebra ^7, indexed by a set T, to consist of a C*-algebra j/, a family {j t : teT] of *-homomorphisms from 2$ into $0 and a state a> on 3$.This structure gives rise to a non-commutative stochastic process in the sense of Accardi [3], with local algebras defined by jtfj= v (j t (b): tel, b e &} for any subset / of T; observables which are "localized at different times" are not assumed to commute.We show (Proposition 1.1) that such a process is determined up to equivalence by its family of correlation kernels o)(j tl (a l )*---j t j[a n )*j tn (b n )---j tl (b l ))ithese are obtained by polarization from the expressions co(j t i(bi)*-~j tn (b n )*j tn (b n )'"j tl (b l )) 9 which are positive real numbers, and are the analogues of the finite-dimensional joint distributions of classical probability; they can, in principle, be determined by

Locations

• Publications of the Research Institute for Mathematical Sciences - View - PDF
• Cineca Institutional Research Information System (Tor Vergata University) - View - PDF