Quantum logics with the Radon-Nikodym property
type: Item
instance of: scholarly article
title: Quantum logics with the Radon-Nikodym property
zbMATH Open document ID: 0643.03043
DOI: 10.1007/BF00714479
published in: Order
publication date: 1988-01-01T00:00:00Z
review text: A \(\sigma\)-orthomodular partially ordered set L is called a logic and a probability measure on L is called a state of L, as is usual in the so- called logico-algebraic approach to the foundations of quantum mechanics. A logic is said to have the Radon-Nikodym property and is called an RN- logic if it satisfies the following condition: If s, t are states on L and s is absolutely continuous with respect to t, then there is a central observable x such that \(s(a)=\int_{a}x dt\) for any \(a\in L\). The authors study the basic properties of RN-logics, including their closedness under the formation of epimorphisms and products. It is shown also that in many cases the concrete RN logics have to be Boolean \(\sigma\)-algebras, whereas there are concrete RN logics with an aribtrary degree of noncompatibility. This result extends their previous paper [J. Math. Phys. 24, 1450 (1983tion is reliable.
Mathematics Subject Classification ID: 03G12
Mathematics Subject Classification ID: 81P10
zbMATH DE Number: 4047716
zbMATH Keywords: Radon-Nikodym theorem
zbMATH Keywords: state
author: Mirko Navara
author: Pavel Pták
MaRDI profile type: Publication
cites work: QUANTUM LOGICS WITH JAUCH–PIRON STATES
cites work: Orthomodular lattices admitting no states
cites work: Q3881378
cites work: Generalized monotone convergence and Radon–Nikodym theorems
cites work: On the representation of 𝜎-complete Boolean algebras
cites work: On states on the product of logics
cites work: On the Radon–Nikodym property for σ-classes
cites work: Exotic logics
cites work: On the representation of Boolean algebras as fields of sets
cites work: On the inducing of homomorphisms by mappings
cites work: Q5537293
full work available at URL: https://doi.org/10.1007/bf00714479
OpenAlex ID: W2067977613
description: scientific article; zbMATH DE number 4047716