On the low density limit of Boson models

Abstract

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).

On leave of absence from Beijing Normal Uniersity