Quantum White Noises and The Master Equation for Gaussian Reference States

We show that a basic quantum white noise process formally reproduces quantum stochastic calculus when the appropriate normal / chronological orderings are prescribed. By normal ordering techniques for integral equations and a generalization of the Araki-Woods representation, we derive the master and random Heisenberg equations for an arbitrary Gaussian state: this includes thermal and squeezed states. 1 Quantum White Noises It is possible to develop a formal theory of quantum white noises which nevertheless provides a powerful insight into quantum stochastic processes. We present the “bare bones” of the theory: the hope is that the structure will be more apparent without the mathematical gore and viscera. Essentially, we need a Hilbert space H0 to describe a system of interest. We postulate a family of pseudo-operators { at : t ≥ 0 } called creation noises; a formally adjoint family { a−t : t ≥ 0 } called annihilation noises and a vector Ψ called the vacuum vector. The noises describe the environment and are assumed to act trivially on the observables of H0. The first structural relations we need is a−t Ψ = 0 (QWN1) which implies that the annihilator noises annihilate the vacuum. The second