Unsharp Values, Domains and Topoi

TL;DR

This paper explores the topos approach to quantum theory, revealing how unsharp values and domain theory provide a new perspective on quantum states and observables, connecting to noncommutative algebras and potential models of quantum space-time.

Contribution

It introduces the analysis of the quantity-value object using domain theory and links the topos framework with noncommutative operator algebras and quantum space models.

Findings

The quantity-value object can be analyzed with domain theory.

The base category is a domain for matrix algebra-based systems.

Connections between topos theory, operator algebras, and domain theory are established.

Abstract

The so-called topos approach provides a radical reformulation of quantum theory. Structurally, quantum theory in the topos formulation is very similar to classical physics. There is a state object, analogous to the state space of a classical system, and a quantity-value object, generalising the real numbers. Physical quantities are maps from the state object to the quantity-value object -- hence the `values' of physical quantities are not just real numbers in this formalism. Rather, they are families of real intervals, interpreted as `unsharp values'. We will motivate and explain these aspects of the topos approach and show that the structure of the quantity-value object can be analysed using tools from domain theory, a branch of order theory that originated in theoretical computer science. Moreover, the base category of the topos associated with a quantum system turns out to be a domain if the underlying von Neumann algebra is a matrix algebra. For general algebras, the base category still is a highly structured poset. This gives a connection between the topos approach, noncommutative operator algebras and domain theory. In an outlook, we present some early ideas on how domains may become useful in the search for new models of (quantum) space and space-time.