Causal structure of quantum stochastic integrators

A class of concrete representations of a non-commutative Stratonovich calculus is defined and its relationship with the quantum Ito calculus of Hudson and Parthasarathy is made explicit. Given a quantum field interacting with a quantum mechanical system, it is possible to extract a quantum noise description for the field using a suitable scaling limit (here the weak coupling limit). The motivation for our construction is to discuss the relationship between the micro-causality of a quantum field and the notion of macro-causality of the quantum noise which replaces it. We derive the Stratonovich quantum stochastic differential equation for the limit evolution operator and show that it agrees with the quantum stochastic limit theory of Accardi, Frigerio and Lu when we convert to the Ito form. The Stratonovich approach, being inherently closer to the physical microscopic equations, leads to an overwhelmingly simplified derivation of the quantum stochastic limit equations of motion. The unification of the two quantum stochastic calculi is given and their physical origins explained.