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Quasi-set theory for a quantum ontology of properties

Federico Holik, Juan Pablo Jorge, Décio Krause, Olímpia Lombardi

Published Jan 1, 2021• 3 citations
Publisher

Abstract

In previous works, an ontology of properties for quantum mechanics has been proposed, according to which quantum systems are bundles of properties with no principle of individuality. The aim of the present article is to show that, since quasi-set theory is particularly suited for treating aggregates of items that do not belong to the traditional category of individual, it supplies an adequate meta-language to speak of the proposed ontology of properties and its structure.

DOI
10.13140/rg.2.2.33566.25925

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In previous works, an ontology of properties for quantum mechanics has been proposed, according to which quantum systems are bundles of properties with no principle of individuality. The aim of the present article is to show that, since quasi-set theory is particularly suited for treating aggregates of items that do not belong to the traditional category of individual, it supplies an adequate meta-language to speak of the proposed ontology of properties and its structure.
In this work, we discuss a formal way of dealing with the properties of contextual systems. Our approach is to assume that properties describing the same physical quantity, but belonging to different measurement contexts, are indistinguishable in a strong sense. To construct the formal theoretical structure, we develop a description using quasi-set theory, which is a set-theoretical framework built to describe collections of elements that violate Leibnitz's principle of identity of indiscernibles. This framework allows us to consider a new ontology in order to study the properties of quantum systems. This article is part of the theme issue 'Contextuality and probability in quantum mechanics and beyond'.
Abstract The issue about the nature of quantum systems usually hinges on the category of individual: the question is whether this category can be applied to understand the peculiarities of quantum behavior. However, some metaphysical subtleties are commonly overlooked in this discussion. What is the difference between the notions of object and of individual? What metaphysical principles are quantum systems supposed to fulfill? What ontological kinds of properties are involved in the quantum domain? These questions should be taken into account when the issue at stake is the ontological nature of quantum systems and, more generally, the structure of the quantum realm. In order to address these and other metaphysical questions, different ontological distinctions that are scarcely taken into account in the debate are brought to the fore in this chapter. In light of these distinctions, the ontological challenges that quantum mechanics poses to the ontological picture presupposed by traditional metaphysics will be discussed in the following chapter.
Abstract Standard quantum logics are typically defined over the orthomodular lattice associated with Hilbert space, yet there is no consensus on how this structure should be interpreted logically or ontologically. This paper offers a unified perspective on these issues by combining an epistemological three-layer model for scientific theories with quantum set theory. We argue that quasi-set theory is best understood not as an alternative semantics for quantum logics, but as an ontological framework operating at a different epistemological level: the ontological one. From this standpoint, the distinctive non-classical features of quantum logic are explained by representational assumptions about identity and individuality, rather than by positing any direct revisions of the logical principles themselves.
Abstract In Chapter 4, the MHI was introduced in terms of interpretative postulates that define the ontological reference of each physical term of quantum mechanics in its standard formulation. On the basis of that presentation, an ontology of properties, framed in the algebraic formalism of quantum mechanics, was proposed. The aim of this chapter is to set out in detail the structure of the ontology proposed by the MHI, now with the technical metaphysical resources developed in the previous chapter. In particular, the nature of the properties that constitute the fundamental ontological items is discussed, and it is argued that quantum systems are not objects at all, but mere collections of properties that do not retain their identity either synchronically or diachronically. More precisely, quantum systems turn out to be non-objectual bundles of properties.
We propose a new quantum ontology, in which properties are the fundamental building blocks. In this property ontology physical systems are defined as bundles of type-properties (specified by algebras of observables in a Hilbert space). Not all elements of such bundles are associated with definite case-properties, and this accommodates the Kochen and Specker theorem and contextuality. Moreover, we do not attribute an identity to the type-properties, which gives rise to a novel form of the bundle theory. There are no "particles" in the sense of classical individuals in this ontology, although the behavior of such individuals is mimicked in some circumstances. This picture leads in a natural way to the symmetrization postulates for systems of many "identical particles".
In this work, we discuss a formal way of dealing with the properties of contextual systems. Our approach is to assume that properties describing the same physical quantity, but belonging to different measurement contexts, are indistinguishable in a strong sense. To construct the formal theoretical structure, we develop a description using quasi-set theory, which is a settheoretical framework built to describe collections of elements that violate Leibnitz’s principle of identity of indiscernibles. This framework allows us to consider a new ontology in order to study the properties of quantum systems. \nThis article is part of the theme issue ‘Contextuality and probability in quantum mechanics and beyond’.