Parameter dependence and outcome dependence in dynamical models for state vector reduction

The authors apply the distinction between parameter independence and outcome independence to the linear and nonlinear models of a recent nonrelativistic theory of continuous state vector reduction. It is shown that in the nonlinear model there is a set of realizations of the stochastic process that drives the state vector reduction for which parameter independence is violated for parallel spin components in the EPR-Bohm setup. Such a set has an appreciable probability of occurrence (∼ 1/2). On the other hand, the linear model exhibits only extremely small parameter dependence effects. Some specific features of the models are investigated and it is recalled that, as has been pointed out recently, to be able to speak of definite outcomes (or equivalently of possessed objective elements of reality) at finite times, the criteria for their attribution to physical systems must be slightly changed. The concluding section is devoted to a detailed discussion of the difficulties met when attempting to take, as a starting point for the formulation of a relativistic theory, a nonrelativistic scheme which exhibits parameter dependence. Here the authors derive a theorem which identifies the precise sense in which the occurrence of parameter dependence forbids a genuinely relativistic generalization. Finally, the authors show how the appreciable parameter dependence of the nonlinear model gives rise to problems with relativity, while the extremely weak parameter dependence of the linear model does not give rise to any difficulty, provided the appropriate criteria for the attribution of definite outcomes are taken into account. 19 refs