On a Connection between Piron Lattices and Kripke Frames
Shengyang Zhong
Logical analysis of the (non-)orthogonality relation in quantum physics has a long history, and results in many well-known representation theorems for lattices arising from the study of quantum logic via Kripke frames, such as the result in [2] about ortholattices and orthoframes and the result in [3] about property lattices and state spaces. In my talk, I’m going to focus on Piron lattices, also called irreducible propositional systems, since they play an important role in the study of foundations of quantum theory. To represent Piron lattices, I will introduce the notion of quantum Kripke frame (QKF). A QKF is a Kripke frame in which elements of the set are interpreted as states of a quantum system and the binary relation as the non-orthogonality relation between states. With this interpretation, it is dened to satisfy four conditions which are mathematically simple and physically sensible. For example, one of the conditions is a direct translation of the superposition principle in quantum physics into our relational setting. Then, I will prove that the collection of all bi-orthogonally closed subsets of the set in a QKF forms a Piron lattice, and every Piron lattice is isomorphic to such a collection in some QKF. Therefore, we obtain a representation theorem for Piron lattices via QKFs. The technique used here is inspired by [1], in which Baltag and Smets give a representation theorem for Piron lattices (satisfying Mayet’s condition) via quantum dynamic frames. Moreover, I will show that, by replacing the condition corresponding to the superposition principle with a mathematically simple one, one can obtain Kripke frames which transparently represent the states and properties of classical systems. Connections between QKFs and other mathematical structures like Hilbert geometries and quantum dynamic frames will also be discussed. Finally, I will talk about some directions for future work, for example nding a complete axiomatization of QKFs in modal or hybrid logic and extending QKFs to probabilistic QKFs.
•Jan 1, 2005•0 citations