Non-orthogonal preferred projectors for modal interpretations of quantum mechanics

Abstract

Modal interpretations constitute a particular approach to associating dynamical variables with physical systems in quantum mechanics. Given the `quantum logical' constraints that are typically adopted by such interpretations, only certain sets of variables can be taken to be simultaneously definite-valued, and only certain sets of values can be ascribed to these variables at a given time. Moreover, each allowable set of variables and values can be uniquely specified by a single `preferred' projector in the Hilbert space associated with the system. In general, the preferred projector can be one of several possibilities at a given time. In previous modal interpretations, the different possible preferred projectors have formed an orthogonal set. This paper investigates the consequences of adopting a non-orthogonal set. We present three contributions on this issue: (1) we provide an argument for such non-orthogonality, based on the assumption that perfectly predictable measurements reveal pre-existing values of variables, an assumption which has traditionally constituted a strong motivation for the modal approach; (2) we generalize the existing framework for modal interpretations to accommodate non-orthogonal preferred projectors; (3) we present a novel type of modal interpretation wherein the set of preferred projectors is fixed by a principle of entropy minimization, and we discuss some of the successes and shortcomings of this proposal.