On the low density limit of Boson models

We study a nonrelativistic quantum system coupled, via a quadratic interaction (cf. formula (1.10) below), to a free Boson gas in the Fock state. We prove that, in the low density limit (z = fugacity → 0), the family of processes given by the collective Weyl operators and the collective coherent vectors, converge to the Fock quantum Brownian motion over L 2(R, dt; K), where K is an appropriate Hilbert space (cf. Section (1.) ). Moreover we prove that the matrix elements of the wave operator of the system at time t/z 2 in the collective coherent vectors converge to the matrix elements, in suitable coherent vectors of the quantum Brownian motion process, of a unitary Markovian cocycle satisfying a quantum stochastic differential equation ruled by some pure number process (i.e. no quantum diffusion part and only the quantum analogue of the purely discontinuous, or jump, processes).