Observables II : Quantum Observables

2005/09/30 by Hans F. de Groote, de Groote, Hans F. · 2 citations
Mathematics · Physics and Astronomy · #Advanced Operator Algebra Research #Quantum Mechanics and Applications #Spectral Theory in Mathematical Physics #math-ph #math.MP #math.OA #quant-ph

paper · pdf · doi:10.48550/arxiv.math-ph/0509075

51 pages, no figures

arxiv created 2005/09/30 · arxiv updated 2009/12/01

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 R is a von Neumann algebra, a selfadjoint element A ∈ R induces a continuous function f_A : Q(P(R)) → ℝ defined on the Stone spectrum Q(P(R)) (\citedeg3) of the lattice P(R) of projections in 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.

Cited by

• Observables as functions: Antonymous functions
• Complex Numbers, One-Parameter of Unitary Transformations and Stone's Theorem in Topos Quantum Theory
• On a canonical lattice structure on the effect algebra of a von Neumann algebra

Related

• Observables I: Stone Spectra
• Observables III: Classical Observables
• Self-adjoint Operators as Functions I: Lattices, Galois Connections, and the Spectral Order
• Self-adjoint Operators as Functions II: Quantum Probability
• Observables, Disassembled