Quantum logic in algebraic approach
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 the monograph may serve as a core text for an upper undergraduate course. With the advanced chapters, it may be used also in a graduate course. Many open problems are presented and this might become a base for further research. The book possesses both mathematical fundamentals understandable for physicists and physical interpretations explained for mathematicians. The author admits that the choice of topics was necessarily subjective, but he gives a meaningful and very interesting survey of the field. The comparison of many alternative approaches is of particular value. What is really exceptional for the book is its readability -- its lively style allows to study it with delight. The only objection (except for some clear small errors, e.g. in Definition 3.11) is that the author does not clearly distinguish between multiplication and composition of mappings. This requires some effort from the reader. The bibliography is also a valuable part of the work. It would be nice to add an index of notations, too. The author -- who is not a native English speaker -- need not be an expert in English and typesetting, but the role of the publisher should not be reduced to printing and advertising. Even a machine spell check was not made and the typesetting remains at the level of an average (not professional) user. As an extraordinarily readable monograph with a well-chosen topic, the book can be highly recommended to students and researchers in quantum physics as well as in mathematical fields related to quantum mechanics.
scientific article; zbMATH DE number 1199704