Mathematical quantum theory. I: Random ultrafilters as hidden variables
type: Item
instance of: scholarly article
title: Mathematical quantum theory. I: Random ultrafilters as hidden variables
zbMATH Open document ID: 0935.81002
DOI: 10.1007/BF00413903
author: William Boos
published in: Synthese
publication date: 1999-10-05T00:00:00Z
review text: This paper aims to provide a natural setting for the stochasticity of von Neumann's classical formulation of non-relativistic quantum mechanics in a stochastic metatheory of random extensions \(V\)(A) of a set-theoretic universe \(V\) by Boolean algebras \(A\) of projections in a separable Hilbert space \(H\). These Boolean algebras are ranges of projection-valued selfadjoint operators on \(H\), i.e they are related to spectral values of the quantum observables. The ultimate aim of this approach is apparently to provide a framework of iterated measure-algebraic truth for mathematical assertions, a framework that is in which the truth-values for such assertions are elements of iterated Boolean measure-algebras. As the quoted aims perhaps suggest, this long paper is an unusual mix of very technical and detailed mathematical analysis, which is sourced to Kolmogorov's work, via Scott, Solovay, Krauss, and Takeuti, with excursions into the philosophical interpretive issues raised by this analysis. It is a strength of the author that his interpretive grand themes of this metatheoretic approach are not abandoned but revisited after the technical discussion, and it certainly makes for an unusual paper to combine very detailed mathematics with very broad brush flights of philosophy. Much of this discussion is of a highly formal nature. However ample references to sources and related papers are provided, and two sections of the paper summarise first von Neumann's analysis of quantum theory, and second provide a fairly thorough review of the basic theory of random extensions and Borel codes. Following these are a section on hidden states and the operator functional calculus, which reviews the nature of contextual and non-contextual hidden variable theories, with discussion of Gudder as well as Kochen and Specker's results, relating these to the preceding Boolean-valued analysis. Section 4, entitled ``Random hidden variables'', then uses the Boolean-valued tools developed earli…
Mathematics Subject Classification ID: 81P05
Mathematics Subject Classification ID: 81P10
Mathematics Subject Classification ID: 03C90
Mathematics Subject Classification ID: 03G12
zbMATH DE Number: 1322849
zbMATH Keywords: Boolean-valued models
zbMATH Keywords: hidden variables
zbMATH Keywords: quantum theory
zbMATH Keywords: stochasticity
zbMATH Keywords: non-relativistic quantum mechanics
zbMATH Keywords: random extensions of a set-theoretic universe
zbMATH Keywords: selfadjoint operators on Hilbert space
zbMATH Keywords: iterated Boolean measure algebras
zbMATH Keywords: operator functional calculus
zbMATH Keywords: interpretation of quantum statistics
zbMATH Keywords: random ultrafilters
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1322849