Quantum logic in algebraic approach
type: Item
instance of: scholarly article
title: Quantum logic in algebraic approach
zbMATH Open document ID: 0910.03038
author: Miklós Rédei
published in: Fundamental Theories of Physics
publication date: 1998-09-16T00:00:00Z
review text: The book explains quantum logic from the point of view of a mathematically oriented physicist. It is mathematically well founded, although some proofs are omitted when the original source is relatively easily available. The physical interpretations are presented in detail and will be helpful also to mathematicians working in related fields. The author's comments are extremely useful, because he always compares different possible approaches and their advantages and disadvantages. He presents a wide range of opinions providing a useful survey of the topic. The initial chapters describe the Hilbert space formalism of quantum mechanics, including basic facts from lattice theory, Hilbert spaces and von Neumann algebras. Also the logical aspects and the semantical approach to classical and quantum mechanics are described. A special chapter is devoted to the history of the Birkhoff and von Neumann concept of quantum logic. This seems to be a unique overview of the development of these ideas, accompanied with explanations why these classical authors modified the basic notions during their work and how their contribution differs from what is now considered the standard approach to quantum mechanics. Further, quantum conditionals and quantum conditional probability are studied. Special attention is paid to Stalnaker's possible worlds semantics. The problems of hidden variables are presented in detail, comparing several nonequivalent formulations. Bell's inequalities are derived and their physically verified consequences in the algebraic relativistic quantum field theory are described in detail. Logical independence and counterfactual probabilistic independence in quantum logic are explained. Finally, Reichenbach's common cause principle and superluminal correlations are discussed. The book is based on lectures on quantum logic given by the author since 1993/94. The reader is assumed to have only basic knowledge of functional analysis and probability theory. Selected parts of …
reviewed by: Mirko Navara
Mathematics Subject Classification ID: 03G12
Mathematics Subject Classification ID: 81P10
Mathematics Subject Classification ID: 03-02
Mathematics Subject Classification ID: 81-02
Mathematics Subject Classification ID: 47L90
Mathematics Subject Classification ID: 06C15
Mathematics Subject Classification ID: 81T05
Mathematics Subject Classification ID: 81Q10
Mathematics Subject Classification ID: 46L60
Mathematics Subject Classification ID: 46L10
Mathematics Subject Classification ID: 46N50
zbMATH DE Number: 1199704
zbMATH Keywords: quantum logic
zbMATH Keywords: Hilbert space
zbMATH Keywords: von Neumann algebra
zbMATH Keywords: projection
zbMATH Keywords: operator algebra
zbMATH Keywords: state
zbMATH Keywords: observable
zbMATH Keywords: orthomodular lattice
zbMATH Keywords: EPR paradox
zbMATH Keywords: Bell's inequalities
zbMATH Keywords: Gleason theorem
zbMATH Keywords: hidden variables
zbMATH Keywords: quantum conditional
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description: scientific article; zbMATH DE number 1199704