Quantum Markovian Approximation as a Quantum Central Limit

We formulate the problem of Markovian approximations for 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). We show that the unitary family governing the evolution will converge to a unitary, adapted, quantum stochastic process: the convergence is weakly in the sense of matrix elements. The main technical difficulty is the proliferation of terms appearing in the Dyson series expansion 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 series in the limit and the quantum stochastic differential equation for the limit unitary process wherein the coefficients are determined by the Eαβ and the damping constants.