A quantum-mechanical functional central limit theorem

Abstract

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 ( P n , Q n ) analogous to the random-walk approximating processes of the Donsker invariance principle are defined in terms of a sequence ( p i , q i ) of pairs of quantum mechanical canonical observables satisfying hypotheses analogous to those of the classical central limit theorem. It is shown that P n , Q n ) converges weakly to a canonical Wiener process.