Intuitionistic quantum logic of an n-level system

TL;DR

This paper develops an intuitionistic quantum logic framework for n-level systems using topos theory and C*-algebras, providing explicit structures and reformulating foundational theorems.

Contribution

It introduces an explicit topos-theoretic model for quantum n-level systems, combining C*-algebraic methods with internal language, and reformulates the Kochen--Specker Theorem.

Findings

Logical structure is intuitionistic, distributive but non-classical.

Provides explicit quantum phase space and Gelfand transform for n-level systems.

Reformulates the Kochen--Specker Theorem in a topos-theoretic context.

Abstract

A decade ago, Isham and Butterfield proposed a topos-theoretic approach to quantum mechanics, which meanwhile has been extended by Doering and Isham so as to provide a new mathematical foundation for all of physics. Last year, three of the present authors redeveloped and refined these ideas by combining the C*-algebraic approach to quantum theory with the so-called internal language of topos theory (see arXiv:0709.4364). The goal of the present paper is to illustrate our abstract setup through the concrete example of the C*-algebra of complex n by n matrices. This leads to an explicit expression for the pointfree quantum phase space and the associated logical structure and Gelfand transform of an n-level system. We also determine the pertinent non-probabilisitic state-proposition pairing (or valuation) and give a very natural topos-theoretic reformulation of the Kochen--Specker Theorem. The essential point is that the logical structure of a quantum n-level system turns out to be intuitionistic, which means that it is distributive but fails to satisfy the law of the excluded middle (both in opposition to the usual quantum logic).