Quantum Sheaves - An Outline of Results

TL;DR

This paper develops a theory of presheaves on quantum lattices, revealing structural similarities between classical and quantum observables, with implications for operator algebras and quantum foundations.

Contribution

It introduces a presheaf framework on quantum lattices, connecting classical and quantum observables within a unified structural setting.

Findings

Classical and quantum observables share a common structural foundation.

Presheaf theory on quantum lattices has applications in operator algebras.

The approach offers insights into the foundations of quantum mechanics.

Abstract

In this paper we start with the development of a theory of presheaves on a lattice, in particular on the quantum lattice $\LL(\kH)$ of closed subspaces of a complex Hilbert space $\kH$, and their associated etale spaces. Even in this early state the theory has interesting applications to the theory of operator algebras and the foundations of quantum mechanics. Among other things we can show that classical observables (continuous functions on a topological space) and quantum observables (selfadjoint linear operators on a Hilbert space) are on the same structural footing.