Abstract
This paper provides an overview of Janusz Czelakowski’s contributions to the theory of partial Boolean (\(\sigma \)-)algebras, and, more in general, to the foundation of Quantum Mechanics. Particular attention is paid to the logic of partial Boolean \(\sigma \)-algebras, to characterizations of PBAs embeddable into Boolean (\(\sigma \)-)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.
Second reader: Sonja Smets
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Notes
- 1.
- 2.
See e.g. Hooker (1975) for extensive discussions on these topics.
- 3.
Cf. e.g. D.J. Foulis’ work (Foulis, 1960) on orthomodular lattices.
- 4.
Here \(\varphi \leftrightarrow \psi \) is short for \((\varphi \rightarrow \psi )\wedge (\psi \rightarrow \varphi )\), where \(\varphi \rightarrow \psi :=\lnot \varphi \vee \psi \).
- 5.
Here \(h(\Gamma )=1\) means \(h(\gamma )=1\), for any \(\gamma \in \Gamma \).
- 6.
Note that there is a one to one correspondence between closed subspaces and projection operators (see e.g. Beran 1984, Theorem 4.1).
- 7.
See also Czelakowski (1973b).
- 8.
- 9.
Here \(\textrm{C}_{ n}( {a_{1},\dots ,a_{n}})\) stands for \(({a_{1},\dots ,a_{n}})\in \textrm{C}_{n}\).
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Acknowledgements
This work has been partly produced during a research stay at the Nicolaus Copernicus University of Toruń (PL) under the Excellence-Initiative Research Programme. The author thanks R. Gruszczyński, T. Jarmużek, M. Klonowski, and all the members of the Department of Logic for the valuable support received. Finally, the author expresses his gratitude to R. Giuntini, A. Ledda, and F. Paoli, for the insightful and exciting discussions we had over the last years on topics of the present work.
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Fazio, D. (2024). On J. Czelakowski’s Contributions to Quantum Logic and the Foundation of Quantum Mechanics. In: Malinowski, J., Palczewski, R. (eds) Janusz Czelakowski on Logical Consequence. Outstanding Contributions to Logic, vol 27. Springer, Cham. https://doi.org/10.1007/978-3-031-44490-6_7
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