Topos Theory and Consistent Histories: The Internal Logic of the Set of all Consistent Sets

TL;DR

This paper applies topos theory to the consistent-histories approach in quantum mechanics, representing all consistent sets as varying sets and revealing a richer logical structure for quantum truth values.

Contribution

It introduces a topos-theoretic framework using presheaves on Boolean subalgebras to handle all consistent sets simultaneously in quantum theory.

Findings

Probabilistic predictions are context-dependent within consistent sets.

Truth values extend beyond binary to a Heyting algebra structure.

The approach unifies multiple quantum 'world-views' into a single logical framework.

Abstract

A major problem in the consistent-histories approach to quantum theory is contending with the potentially large number of consistent sets of history propositions. One possibility is to find a scheme in which a unique set is selected in some way. However, in this paper we consider the alternative approach in which all consistent sets are kept, leading to a type of `many world-views' picture of the quantum theory. It is shown that a natural way of handling this situation is to employ the theory of varying sets (presheafs) on the space $\B$ of all Boolean subalgebras of the orthoalgebra $\UP$ of history propositions. This approach automatically includes the feature whereby probabilistic predictions are meaningful only in the context of a consistent set of history propositions. More strikingly, it leads to a picture in which the `truth values', or `semantic values' of such contextual predictions are not just two-valued (\ie true and false) but instead lie in a larger logical algebra---a Heyting algebra---whose structure is determined by the space $\B$ of Boolean subalgebras of $\UP$.