On Quantum Stochastic Generators

From the notion of stochastic Hamiltonians and the flows that they generate, we present an account of the theory of stochastic derivations over both classical and quantum algebras and demonstrate the natural way to add stochastic derivations. Our discussion on quantum stochastic processes emphasizes the origin of the Itô correction to the Leibniz rule in terms of normal ordering of white noise operators. The key idea is to work in a Stratonovich calculus in which the Leibniz rule holds valid: in physical problems, this is closest to a Hamiltonian representation. This is approach is new for quantum stochastic evolutions. We outline a quantum white noise calculus which is justified by the fact that it formally reproduces the Hudson-Parthasarthy theory of quantum stochastic calculus and that it yields the appropriate Itô-Stratonovich conversion for physical models. (Dedicated to Prof. O.G. Smolianov on the occasion of his birthday.) 1 Stochastic Derivations Derivations play a fundamental role in the study of algebras, whether they be algebras of functions on a manifold or algebras of operators. Given the suitability of C*-algebras for modelling quantum mechanical variables it is natural to consider algebras of functions on Poisson manifolds as an intermediary between classical and quantum cases. This view is strengthened considerably by the deep analogies known to exist between Poisson and operator algebras [1] [2]. It is natural to return to these analogies when considering stochastic flows describing irreversible dynamical evolutions. We begin by extending the notion of a stochastic derivation.