Observables I: Stone Spectra

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dc.creatorde Groote, Hans F.
dc.date2005-09-11
dc.date.accessioned2026-07-07T06:20:07Z
dc.date.available2026-07-07T06:20:07Z
dc.descriptionIn 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 $\mathcal{R}$ is a von Neumann algebra, a selfadjoint element $A \in \mathcal{R}$ induces a continuous function $f_{A} : \mathcal{Q}(\mathcal{P(R)}) \to \mathbb{R}$ defined on the \emph{Stone spectrum} $\mathcal{Q}(\mathcal{P(R)})$ of the lattice $\mathcal{P(R)}$ of projections in $\mathcal{R}$. The Stone spectrum $\mathcal{Q}(\mathbb{L})$ of a general lattice $\mathbb{L}$ is the set of maximal dual ideals in $\mathbb{L}$, equipped with a canonical topology. $\mathcal{Q}(\mathbb{L})$ coincides with Stone's construction if $\mathbb{L}$ is a Boolean algebra (thereby ``Stone'') and is homeomorphic to the Gelfand spectrum of an abelian von Neumann algebra $\mathcal{R}$ in case of $\mathbb{L} = \mathcal{P(R)}$ (thereby ``spectrum'').
dc.description77 pages, no figures
dc.identifierhttps://arxiv.org/abs/math-ph/0509020
dc.identifierhttp://arxiv.org/abs/math-ph/0509020
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/95246
dc.subjectMathematical Physics
dc.subjectOperator Algebras
dc.subjectQuantum Physics
dc.titleObservables I: Stone Spectra
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