ar X iv : m at h - ph / 0 50 90 20 v 1 1 1 Se p 20 05 Observables I : Stone Spectra
نویسنده
چکیده
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(L) of a general lattice L is the set of maximal dual ideals in L, equipped with a canonical topology. Q(L) coincides with Stone’s construction if L is a Boolean algebra (thereby “Stone”) and is homeomorphic to the Gelfand spectrum of an abelian von Neumann algebra R in case of L = P(R) (thereby “spectrum”). Moreover, Q(L) appears quite naturally in the construction of the sheafification of presheaves on a lattice L. On the other hand, measurable or continuous functions can be described by spectral families and, therefore, as functions on appropriate Stone spectra. In this first part of our work, we investigate general properties of Stone spectra and, in more detail, Stone spectra of two specific classes of lattices: σ-algebras and projection lattices P(R) of von Neumann algebras R.
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تاریخ انتشار 2008