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arXiv:quant-ph/0003047

Indistinguishability and nonlocality in Einstein-Podolsky-Rosen experiment

Adonai S. Sant’Anna

Published Mar 14, 2000• 0 citations
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Abstract

Quasi-set theory allows us a non trivial relation between indistinguishability and nonlocality into the context of Einstein- Podolsky-Rosen experiment. Quasi-set theory is a set theory which provides a manner for dealing with collections of indistinguishable but not identical elementary particles.
DOI
10.48550/arxiv.quant-ph/0003047

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Quasi-set theory was developed to deal with collections of indistinguishable objects. In standard mathematics, there are no such kind of entities, for indistinguishability (agreement with respect to all properties) entails numerical identity. The main motivation underlying such a theory is of course quantum physics, for collections of indistinguishable (’identical’ in the physicists’ jargon) particles cannot be regarded as ’sets’ of standard set theories, which are collections of distinguishable objects. In this paper, a rationale for the development of such a theory is presented, motivated by Heinz Post’s claim that indistinguishability of quantum entities should be attributed ’right at the start’.
Quantum theory is the land of indiscernible things, of things that in some situations cannot be put apart. This contrasts in much with standard mathematics and classical logic, which were elaborated to deal with \textit{individuals}, discernible things. Many thinkers, among them the mathematician Yuri Manin, have proposed that a more general theory of `sets' than standard set theories (where sets are collections of discernible elements) should be elaborated to cope with quantum physics. Quasi-set theory was proposed with such an aim. In this paper, we consider some aspects of this theory and discuss the way we can use such a theory as a framework for constructing deformable (not rigid) structures where indiscernible things can be considered but with one additional and fundamental detail: these structures cannot be extended to rigid structures as it happens in the standard frameworks. Really, the way of dealing with indiscernible (or indistinguishable) elements within a `standard' framework such as a `classical' set theory (like the ZFC system) requires the confinement of the discussion to deformable (non-rigid) structures. But in ZFC every structure can be extended to a rigid structure (encompassing only the trivial automorphism -- the identity function), so that in the extended structure (or in the whole universe) one can realize that the supposed indiscernible objects are not indiscernible at all. Quasi-set theory is such that the universe of quasi-sets is deformable and in such a theory one can construct deformable structures which cannot be extended to rigid ones. The aim of this paper is to put these things in a clear way.
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'.
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’.
Quantum indistinguishability directly relates to the philosophical debate on the notions of identity and individuality. They are crucial for our understanding of multipartite quantum systems. Furthermore, the correct interpretation of this feature of quantum theory has implications that transcend fundamental science and philosophy, given that quantum indistinguishability is a resource in quantum information theory. Most of the conceptual analysis of quantum indistinguishability is restricted to studying the permutational invariance of quantum states, the concomitant quantum statistics and their entanglement. Here, we analyse the role of indistinguishability and non-individuality in other areas of quantum theory. We start by analysing how a very peculiar use of indistinguishability underlies Feynman's rules for summing amplitudes in interference phenomena. Next, we study how quantum indistinguishability is underestimated in several topics of debate in the quantum physics literature, such as the Einstein-Podolsky-Rosen argument, Bell's inequalities and the Bell-Kochen-Specker theorem. Finally, we argue that an ontology of truly indistinguishable entities can serve as a basis for a quantum ontology that can give interesting answers to the interpretational problems of quantum mechanics. We claim that, in addition to superposition, contextuality and entanglement, indistinguishability (understood in a robust ontological sense) is one of the central features of quantum physics. This article is part of the theme issue 'Identity, individuality and indistinguishability in physics and mathematics'.