Quantum Stochastic Calculus on Full Fock Modules

Abstract

AbstractWe develop a quantum stochastic calculus on full Fock modules over arbitrary Hilbert B–B-modules. We find a calculus of bounded operators where all quantum stochastic integrals are limits of Riemann–Stieltjes sums. After having estalished existence and uniqueness of solutions of a large class of quantum stochastic differential equations, we find necessary and sufficient conditions for unitarity of a subclass of solutions. As an application we find dilations of a conservative CP-semigroup (quantum dynamical semigroup) on B with arbitrary bounded (Christensen–Evans) generator. We point out that in the case B=B(G) the calculus may be interpreted as a calculus on the full Fock space tensor initial space G with arbitrary degree of freedom dilating CP-semigroups with arbitrary Lindblad generator. Finally, we show how a calculus on the boolean Fock module reduces to our calculus. As a special case this includes a calculus on the boolean Fock space