Scopes and limits of modality in quantum mechanics

Abstract

We develop an algebraic frame for the simultaneous treatment of actual and possible properties of quantum\ud systems.We show that, in spite of the fact that the language is enriched with the addition of a modal operator\ud to the orthomodular structure, contextuality remains a central feature of quantum system

A Topological Study of Contextuality and Modality in Quantum Mechanics

“…In view of (Maeda and Maeda, 1970; Lemma 29.16), complete orthomodular lattices with an operator e(a) = {z ∈ Z(L) : z ≤ a}, are examples of Boolean saturated orthomodular lattices. They form a variety of algebras A, ∧, ∨, ¬, 2, 0, 1 of type 2, 2, 1, 1, 0, 0 , noted as OML 2 (Domenech et al, 2006). OML 2 are axiomatized as follows:…”

“…We refer to A as a modal extension of L. In this case we can see the lattice L as a subset of A. (Domenech et al, 2006;Theorem 14). The possibility space represents the modal content added to the discourse about properties of the system.…”

“…At first sight, it may be thought that the enrichment of the set of (actual) propositions with modal ones could allow to circumvent the contextual character of quantum mechanics. We have faced the study of this issue and given a Kochen-Specker type theorem for the enriched lattice (Domenech et al, 2006). In this paper, we give a topological version of that theorem, i.e.…”

Modal‐type orthomodular logic

“…In [9] and [10], we have introduced an orthomodular structure enriched with a modal operator called Boolean saturated orthomodular lattice. This structure has a rigorous physical motivation and allows to establish algebraic-type versions of the Born rule and the well known Kochen-Specker (KS) theorem [18].…”

Semilattices global valuations in the topos approach to quantum mechanics

“…The problem of contextuality has been studied from different approaches. One of them is the modal algebraic approach version related to partial valuations of the orthomodular lattice of closed subspaces of Hilbert space developed in (Domenech et al 2006Freytes et al 2009). This proposal allows us to identify the constraints imposed by the structure to the relation between actuality and possibility and the discourse that includes both type of propositions.…”