Quantum Stochastic Generators

From the notion of a stochastic Hamiltonian 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 new for quantum stochastic evolutions. (Dedicated to Prof. O.G. Smolyanov 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. 1.1 Notation Let A be an associative unital topological algebra with involution †. We denote the set of linear maps L : A × A 7→ A satisfying L ( A ) = L (A), ∀A ∈ A, by L (A).