Quantum Stochastic Processes

Abstract

1. You might well ask: Why on earth should we concern ourselves with quantum stochastic processes? My answer would be: Because that is what the quantum theory of open systems is really about; whenever a system is coupled to a heat bath or reservoir its evolution has a stochastic element which is absent from the Hamiltonian evolution of a closed quantum mechanical system. However, others might give you very different answers because they have other examples in mind. I do not intent to make a proprietary claim to the name “Quantum Stochastic Process”; my talk might be entitled more accurately (though more awkwardly) “A Class of Non-Commutative Stochastic Processes analogous to Classical Stochastic Processes in the sense of Doob”. What I hope to do is to convince you that there is a very strong analogy between the evolution of an open quantum system and the evolution of a classical stochastic process in Doob’s sense, and that it is possible to construct a mathematical theory which embraces them both. Moreover, I hope to persuade you that the exercise is worthwhile by showing you how it clarifies the dilation problem for Quantum Dynamical Semi-groups.