Decoherent Histories Contextuality

Abstract

A consistent set of histories for a system is a set of histories where the probability assigned by a given quantum state of the system to a sum of histories (history operators) is equal to the sum of the probabilities assigned to the individual histories (each operator) by the given state. Abstract consistency alone admits multiple incompatible consistent sets of histories for the same initial state and Hamiltonian, where ‘incompatibility’ in this context means that the sets support conflicting probabilistic inferences. However, this predicament is also faced by any unitary quantum theory describing decoherence with respect to some system/environment split, and not just by the imposition of abstract consistency in isolation. While decoherence always occurs in an approximately unique basis, the final global state resulting from this decoherent evolution can always be rewritten *as if it had* branched along some other, incompatible, consistent set of histories. A quasi-classical realm is a set of decoherent histories where the initial state remains sharply peaked around one history in which dynamical variables approximately follow their classical equations of motion. The requirement of ‘quasi-classicality’ does not necessarily select a unique set or a set of decoherent histories recorded in the environment with the possibility of being measured. Why, then, do measurers of different subsystems of the universe in a final state, which could be expressed in terms of different sets of branching histories corresponding to incompatible quasi-classical realms, always agree on the realm that they are in? I will argue that they agree only because one of three scenarios always occurs: 1) they measure subregions of space on roughly the same length-scale and the set recorded at a specified length-scale is unique, 2) the records of measurement outcomes for different quasi-classical realms occur on different pure states in some global mixture of the universe, or 3) they measure subregions of an environment in a mixed state that fails to record one quasi-classical realm over another. The decoherent histories formalism seems to describe Bell-Kochen-Specker measurement contextuality persisting into the classical limit. By which I mean that the formalism allows that the probability of an outcome common to two different measurements, conditional on the measurement choice and an ontic state, can differ even for measurements of *pointer observables* (such as dial position). However, we never experience this contextuality, and I explain why by arguing that there will always be agreement on the quasi-classical realm between different observers sampling regions of the environment of roughly the same size.