Quantum Sheaves - An Outline of Results

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.