Quantum Stochastic Processes

by C. Barnett, R. F. Streater and I. F. Wilde.

Lecture given at the workshop held at the Villa Mondragone, Rome, 6-11 Sept. 1982, on "Quantum Probability and Applications to the Quantum Theory of Irreversible Processes", pp 32-45 in Lecture Notes in Mathematics, 1055, Springer-Verlag, edited by L. Accardi, A. Frigerio and V. Gorini.

ABSTRACT

We present a view of what a quantum stochastic process is: it is a family of closed operators X(t), labelled by a parameter (or any index set), such that X(t) is affiliated to an A(t), an increasing family of W*-algebras, called the filtration. We introduce Gaussian processes, and illustrate the with the Ito-Clifford process, and the CCR and CAR processes, where the quantum field is split up into creators and annihilators, and the vacuum is a cyclic weight on each algebra of the filtration. We discuss stochastic differential equations, using the Ito-Clifford martingale, as well as some linear equations driven by positive-energy noise.