Compatible quantum theory.

Formulations of quantum mechanics (QM) can be characterized as realistic, operationalist, or a combination of the two. In this paper a realistic theory is defined as describing a closed system entirely by means of entities and concepts pertaining to the system. An operationalist theory, on the other hand, requires in addition entities external to the system. A realistic formulation comprises an ontology, the set of (mathematical) entities that describe the system, and assertions, the set of correct statements (predictions) the theory makes about the objects in the ontology. Classical mechanics is the prime example of a realistic physical theory. A straightforward generalization of classical mechanics to QM is hampered by the inconsistency of quantum properties with classical logic, a circumstance that was noted many years ago by Birkhoff and von Neumann. The present realistic formulation of the histories approach originally introduced by Griffiths, which we call 'compatible quantum theory (CQT)', consists of a 'microscopic' part (MIQM), which applies to a closed quantum system of any size, and a 'macroscopic' part (MAQM), which requires the participation of a large (ideally, an infinite) system. The first (MIQM) can be fully formulated based solely on the assumption of a Hilbert space ontology and the noncontextuality of probability values, relying in an essential way on Gleason's theorem and on an application to dynamics due in large part to Nistico. Thus, the present formulation, in contrast to earlier ones, derives the Born probability formulas and the consistency (decoherence) conditions for frameworks. The microscopic theory does not, however, possess a unique corpus of assertions, but rather a multiplicity of contextual truths ('c-truths'), each one associated with a different framework. This circumstance leads us to consider the microscopic theory to be physically indeterminate and therefore incomplete, though logically coherent. The completion of the theory requires a macroscopic mechanism for selecting a physical framework, which is part of the macroscopic theory (MAQM). The selection of a physical framework involves the breaking of the microscopic 'framework symmetry', which can proceed either phenomenologically as in the standard quantum measurement theory, or more fundamentally by considering the quantum system under study to be a subsystem of a macroscopic quantum system. The decoherent histories formulation of Gell-Mann and Hartle, as well as that of Omnès, are theories of this fundamental type, where the physical framework is selected by a coarse-graining procedure in which the physical phenomenon of decoherence plays an essential role. Various well-known interpretations of QM are described from the perspective of CQT. Detailed definitions and proofs are presented in the appendices.