Sorkin's coevent interpretation shifts the focus of quantum logic from the structure of a propositional lattice to the nature of truth valuations thereon. We apply this shift in emphasis to a simple formulation of the consistent histories approach, expressing it in terms of truth valuations which are brought together in a logical framework. We see that these consistent histories valuations are related to Sorkin's multiplicative coevents, and that they can be naturally described by an adaptation of Isham's early topos-theoretic approach.
A formulation of the consistent histories approach to quantum mechanics in terms of generalized observables (POV measures) and effect operators is provided. The usual notion of ‘‘history’’ is generalized to the notion of ‘‘effect history.’’ The space of effect histories carries the structure of a D-poset. Recent results of J. D. Maitland Wright imply that every decoherence functional defined for ordinary histories can be uniquely extended to a bi-additive decoherence functional on the space of effect histories. Omnès’ logical interpretation is generalized to the present context. The result of this work considerably generalizes and simplifies the earlier formulation of the consistent effect histories approach to quantum mechanics communicated in a previous work of this author.
An introduction is given to an algebraic formulation and generalisation of the consistent histories approach to quantum theory. The main technical tool in this theory is an orthoalgebra of history propositions that serves as a generalised temporal analogue of the lattice of propositions of standard quantum logic. Particular emphasis is placed on those cases in which the history propositions can be represented by projection operators in a Hilbert space, and on the associated concept of a `history group'.