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Quantum Mechanics: Ontology Without Individuals

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Abstract

The purpose of the present paper is to consider the traditional interpretive problems of quantum mechanics from the viewpoint of a modal ontology of properties. In particular, we will try to delineate a quantum ontology that (i) is modal, because describes the structure of the realm of possibility, and (ii) lacks the ontological category of individual. The final goal is to supply an adequate account of quantum non-individuality on the basis of this ontology.

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Notes

  1. Although van Fraassen does not endorse scientific realism (from his “constructive empiricism”, the aim of science is only empirical adequacy), he admits that a meaningful account of reality is necessary for a scientific theory to be intelligible.

  2. According to the algebraic formalism of quantum mechanics, given a *-algebra \(\mathcal {A}\) of operators, (i) the set of the self-adjoint elements of \( \mathcal {A}\) is the space \(\mathcal {O}\), whose elements represent observables, \(O\in \mathcal {O}\) , and (ii) states are represented by functionals on \(\mathcal {O}\), that is, by elements of the dual space \( \mathcal {O}^{\prime }\), \(\rho \in \mathcal {O}^{\prime }\). In the case of a C*-algebra of operators, it can be represented by a Hilbert space \(\mathcal {H }\) (GNS theorem) and \(\mathcal {O}=\mathcal {O}^{\prime }\); therefore, both \( \mathcal {O}\) and \(\mathcal {O}^{\prime }\) are represented by \(\mathcal { H\otimes H}\).

  3. Mathematically, a quantum system \(\mathcal {S}^{1}\) is represented by the set of operators \(\mathcal {O}^{1}\), or by the Hilbert space \(\mathcal {H}^{1}\) if \(\mathcal {O}^{1}=\mathcal {H}^{1}\otimes \mathcal {H}^{1}\).

  4. In the case of working with C*-algebras of operators, \(\mathcal {O}^{1}\) and \( \mathcal {O}^{2}\) are represented by \(\mathcal {H}^{1}\otimes \) \(\mathcal {H} ^{1}\) and \(\mathcal {H}^{2}\otimes \) \(\mathcal {H}^{2}\) respectively. Therefore, \(\mathcal {O}^{c}=\mathcal {O}^{1}\otimes \mathcal {O}^{2}=\mathcal {H }^{1}\otimes \) \(\mathcal {H}^{1}\otimes \) \(\mathcal {H}^{2}\otimes \) \(\mathcal { H}^{2}\).

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Acknowledgments

The authors want to thank Décio Krause for his organization of the Workshop “Logical Quantum Structures” in the context of the 4th World Congress and School on Universal Logic, and also the attendants to the workshop for their constructive comments to the oral version of this work. This paper was partially supported by grants of the Buenos Aires University (UBA), the National Research Council (CONICET) and the National Research Agency (FONCYT) of Argentina.

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  1. CONICET - Facultad de Ciencias Exactas y Naturales, University of Buenos Aires, Buenos Aires, Argentina

    Newton da Costa & Olimpia Lombardi

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Correspondence to Olimpia Lombardi.

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da Costa, N., Lombardi, O. Quantum Mechanics: Ontology Without Individuals. Found Phys 44, 1246–1257 (2014). https://doi.org/10.1007/s10701-014-9793-1

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