Ways to describe dynamical state-vector reduction

The continuous-spontaneous-localization (CSL) description of state-vector reduction allows one to describe reality by the state vector. The mathematical structure of the theory, based upon a c-number white-noise source interacting with the quantum system, is reviewed. Then this description is generalized in two ways. First, a generalization is made to a wide class of c-number stochastic processes: using the square root of the probability density functional of the process, it is shown how to construct an evolution equation for the state vector which results in state-vector reduction. Second, within this wider framework the c-number source is completely eliminated and replaced by quantum operators, dubbed ‘‘index’’ operators, representing new physical variables which interact with the quantum system. In this case the state vector cannot represent reality as in the previous case, but it is shown how to write it unambiguously as a sum of ‘‘preferred basis’’ states, any one of which can represent reality. Therefore this quantum theory is open to the ensemble interpretation, where ‘‘ensemble’’ refers to the collection of possible real states. Three requirements sufficient for a viable ensemble interpretation are specified. It is shown how to satisfy these requirements, obtaining a purely quantum model that is equivalent to CSL and capable of describing reality.