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 Ito 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 Ito-Stratonovich conversion for physical models