On J. Czelakowski’s Contributions to Quantum Logic and the Foundation of Quantum Mechanics

In Jacek Malinowski & Rafał Palczewski (eds.), Janusz Czelakowski on Logical Consequence. Springer Verlag. pp. 233-264 (2024)
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Abstract

This paper provides an overview of Janusz Czelakowski’s contributions to the theory of partial Boolean (σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma $$\end{document}-)algebras, and, more in general, to the foundation of Quantum Mechanics. Particular attention is paid to the logic of partial Boolean σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma $$\end{document}-algebras, to characterizations of PBAs embeddable into Boolean (σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma $$\end{document}-)algebras, and their representation as self-adjoint idempotent elements of partial commutative algebras with involution. Also, applications to the theory of orthomodular posets as well as Czelakowski’s theory of partial Boolean algebras in a broader sense will be discussed. Finally, further representation theorems and their importance for quantum logic will be outlined.

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