Quantum Stochastic Integrals

Abstract

There have been many attempts to set up quantum analogues of the theory of stochastic processes and stochastic differential equations. I should mention the many papers of M. Lax [1] on “quantum noise”, and those of Senitzky [2]; these were inspired by the problem of describing a laser, and by quantum electronics. An account can be found in Haken [3]. Current-algebras led to my own research into continuous tensor products and infinitely divisible representations of algebraic structures [4], and to Araki’s theory of factorizabel representations [5]. Survey articles (as at 1972) are [6,7]. Then there is the work of Evans and Lewis [8] and the book by E. B. Davies [9]. Independently, Hudson was developing a “quantum” version of Brownian motion and the Feynman-Kac formula [10]. The subject was stimulated by the work of Lewis and Thomas [11], and lately by the axioms of Accardi, Frigerio and Lewis [12]. These authors attempt to specify by means of axioms what a quantum stochastic process is. The axioms are determined by what they consider to be the essential physics of an open quantum system. Because of this, any model obeying these axioms must be quite realistic and probably not very easy to construct (cf. Maasen, thesis, Groningen 1982). Unrealistic but explicitly known theories like Brownian motion do not fall into their axiomatic scheme. We take a more lenient set of axioms which includes Brownian motion. In this our point of view is similar to that of Hudson and Parthasarathy [15] and Emch [16].