Stochastic Processes and Their Representations in Hilbert Space

Abstract

Beginning with an intuitive consideration of sequences of measurements, we define a time-ordered event space representing the collection of all imaginable outcomes for measurement sequences. We then postulate the generalized distributive relation on the event space and examine the class of measurements for which this relation can be experimentally validated. The generalized distributive relation is shown to lead to a σ-additive conditional probability on the event space and to a predictive and retrodictive formalism for stochastic processes. We then show that this formalism has a predictive and a retrodictive representation in a separable Hilbert space H, which has no counterpart in unitary quantum dynamics.