A quantum-mechanical functional central limit theorem

Continuing an earlier work [4], properties of canonical Wiener processes are investigated. An analog of the sample path continuity property is obtained. A noncommutative counterpart of weak convergence is formulated. Operator processes (Pn, Qn) analogous to the random-walk approximating processes of the Donsker invariance principle are defined in terms of a sequence (pi, qi) of pairs of quantum mechanical canonical observables satisfying hypotheses analogous to those of the classical central limit theorem. It is shown that Pn, Qn) converges weakly to a canonical Wiener process.