Observables II : Quantum Observables

Abstract

In this work we discuss the notion of observable - both quantum and classical - from a new point of view. In classical mechanics, an observable is represented as a function (measurable, continuous or smooth), whereas in (von Neumann's approach to) quantum physics, an observable is represented as a bonded selfadjoint operator on Hilbert space. We will show in the present part II and the forthcoming part III of this work that there is a common structure behind these two different concepts. If $\mathcal{R}$ is a von Neumann algebra, a selfadjoint element $A\in \mathcal{R}$ induces a continuous function $f_A:\mathcal{Q}(\mathcal{P}(\mathcal{R}))\to \mathbb{R}$ defined on the \emph{Stone spectrum} $\mathcal{Q}(\mathcal{P}(\mathcal{R}))$ (\cite{deg3}) of the lattice $\mathcal{P}(\mathcal{R})$ of projections in $\mathcal{R}$. $f_A$ is called the observable function corresponding to $A$. The aim of this part is to study observable functions and its various characterizations.

51 pages, no figures

References in corpus (2)

• On a canonical lattice structure on the effect algebra of a von Neumann algebra
• Some Reflections on the Status of Conventional Quantum Theory when Applied to Quantum Gravity

Cited by in corpus (5)

• A Topos Foundation for Theories of Physics: III. The Representation of Physical Quantities With Arrows
• On a canonical lattice structure on the effect algebra of a von Neumann algebra
• Observables as functions: Antonymous functions
• Self-adjoint Operators as Functions I: Lattices, Galois Connections, and the Spectral Order
• Complex Numbers, One-Parameter of Unitary Transformations and Stone's Theorem in Topos Quantum Theory