Abstract
As Classical Propositional Logic finds its algebraic counterpart in Boolean algebras, the logic of Quantum Mechanics, as outlined within G. Birkhoff and J. von Neumann’s approach to Quantum Theory (Birkhoff and von Neumann in Ann Math 37:823–843, 1936) [see also (Husimi in I Proc Phys-Math Soc Japan 19:766–789, 1937)] finds its algebraic alter ego in orthomodular lattices. However, this logic does not incorporate time dimension although it is apparent that the propositions occurring in the logic of Quantum Mechanics are depending on time. The aim of the present paper is to show that tense operators can be introduced in every logic based on a complete lattice, in particular in the logic of quantum mechanics based on a complete orthomodular lattice. If the time set is given together with a preference relation, we introduce tense operators in a purely algebraic way. We derive several important properties of such operators, in particular we show that they form dynamic pairs and, altogether, a dynamic algebra. We investigate connections of these operators with logical connectives conjunction and implication derived from Sasaki projections in an orthomodular lattice. Then we solve the converse problem, namely to find for given time set and given tense operators a time preference relation in order that the resulting time frame induces the given operators. We show that the given operators can be obtained as restrictions of operators induced by a suitable extended time frame.
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Acknowledgements
The authors are grateful to the anonymous referees whose valuable remarks helped to increase the quality of the paper. This research was funded in whole or in part by the Austrian Science Fund (FWF) [10.55776/I4579], by the Czech Science Foundation (GACR), Project 20-09869 L, and, concerning the first author, by IGA, Project PrF 2023 010.
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Presented by Francesco Paoli; Received November 15, 2022.
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Chajda, I., Länger, H. Algebraic Structures Formalizing the Logic of Quantum Mechanics Incorporating Time Dimension. Stud Logica (2024). https://doi.org/10.1007/s11225-024-10103-7
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DOI: https://doi.org/10.1007/s11225-024-10103-7
Keywords
- Complete orthomodular lattice
- Event-state system
- Logic of Quantum Mechanics
- Tense operator
- Time frame
- Dynamic pair
- Dynamic algebra