The Langevin equation from Markovian Quantum Central Limits

We establish the Langevin equation for the problem of Markovian approximations to the action of a Bose quantum field reservoir R on a quantum mechanical system S as a functional quantum central limit theory. In particular, we study the effect of the time-dependent interaction Υ (λ) t = E11⊗a + t (λ) a − t (λ) +E10⊗a + t (λ)+E01⊗a − t (λ) +E00⊗1 + R on the joint states space hS ⊗ hR where the a ± t (λ) are approximations to quantum white noises. Here the reservoir quanta may be emitted (E10-term), absorbed (E01-term) and also scattered (E11-term). The Heisenberg evolution is shown to converge to a quantum stochastic flow in the sense of matrix elements. The main technical difficulty is the proliferation of terms appearing in the Dyson series expansions now caused by the scattering, however, a uniform estimate is obtained by means of a generalization the Pulé inequalities. We re-sum the principal terms of the Heisenberg evolution series in the limit and determine the quantum stochastic differential equation for the limit process wherein the coefficients are determined by the Eαβ and the damping constants.