## The boson and fermion Brownian motion as quantum central limits of the quantum Bernoulli processes.(English)Zbl 0858.60026

L. Accardi and A. Bach [in: Quantum probability and applications IV. Lect. Notes Math. 1396, 7-19 (1989; Zbl 0717.60034) and “The harmonic oscillators as quantum central limit of Bernoulli processes” (to appear)] have proved that the harmonic oscillator can be obtained as a quantum central limit of the Bernoulli processes and that the known boson Brownian motion arises from an invariance principle. In order to take over this shortcoming we now give a different proof of the results (loc.cit.) based on a technique which allows to obtain the analogous results in the Fermion case.

### MSC:

 60F05 Central limit and other weak theorems 60K40 Other physical applications of random processes 46N50 Applications of functional analysis in quantum physics

Zbl 0717.60034