2005/09/11 by Hans F. de Groote, de Groote, Hans F. · 2 citations
Mathematics · Physics and Astronomy · #Advanced Operator Algebra Research #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Operator Algebras (math.OA) #Quantum Mechanics and Applications #Quantum Physics (quant-ph) #Random Matrices and Applications #math-ph #math.MP #math.OA #quant-ph
paper · pdf · doi:10.48550/arxiv.math-ph/0509020
77 pages, no figures
arxiv created 2005/09/11 · openalex publication_date 2005/09/11 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 part II 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 fA : Q(P(R)) → ℝ defined on the Stone spectrum Q(P(R)) of the lattice P(R) of projections in R. The Stone spectrum Q(\mathbbL) of a general lattice \mathbbL is the set of maximal dual ideals in \mathbbL, equipped with a canonical topology. Q(\mathbbL) coincides with Stone's construction if \mathbbL is a Boolean algebra (thereby ``Stone'') and is homeomorphic to the Gelfand spectrum of an abelian von Neumann algebra R in case of \mathbbL = P(R) (thereby ``spectrum'').