Quantum Stochastic Calculus

Detereministic classical dynamics, reversible quantum dynamics in the Schrr odinger-Heinsenberg pictures and irreversible dynamics of diiusion processes driven by Brownian motion admitìnnnitesimal' or`diierential' descriptions which can be expressed in terms of derivations of * algebras. One of the central aims of quantum stochastic calculus is to unify all these and explore for similar algebraic features in the diierential description of irreversible quantum dynamical systems as well as classical Markov chains. Observables in quantum theory are, usually, selfadjoint operators in a Hil-bert space. A process of observables is described by a map t ! X(t) where t denotes time and X(t) is a selfadjoint operator. Our aim is to describe such processess diierentially as dX(t) = X i L i (t)d i (t) where d i are somèuniversal' diierentials including, of course, dt. In the case of irreversible diiusions d i 's diierent from dt are diierentials of independent Brownian motion processes. In the context of quantum theory it is natural to explore the possibility of using the creation, annihilation and number operator processes of free eld theory as candidates for the universal diierentials. The free eld operators in the boson Fock space ?(h L 2 (IR +)) can be expressed in terms of a chosen orthonormal (extended) canonical commutation relations (ECCR) : 0 0 (t) = tI; (t) y = (t);