Environmental influence on the measurement process in stochastic reduction models

We consider the energy-driven stochastic state vector reduction equation for the density matrix, which for pure state density matrices can be written in two equivalent forms. We use these forms to discuss the decoupling of the noise terms for independent subsystems, and to construct 'environmental' stochastic density matrices whose time-independent expectations are the usual quantum statistical distributions. We then consider a measurement apparatus weakly coupled to an external environment, and show that in mean field (Hartree) approximation the stochastic equation separates into independent equations for the apparatus and environment, with the Hamiltonian for the apparatus augmented by the environmental expectation of the interaction Hamiltonian. We use the Hartree approximated equation to study a simple accretion model for the interaction of the apparatus with its environment, as part of a more general discussion of when the stochastic dynamics predicts state vector reduction, and when it predicts the maintenance of coherence. We also discuss the magnitude of decoherence effects acting during the reduction process. Our analysis supports the suggestion that a measurement takes place when the different outcomes are characterized by sufficiently distinct environmental interactions for the reduction process to be rapidly driven to completion.