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Physical Properties as Modal Operators in the Topos Approach to Quantum Mechanics

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Abstract

In the framework of the topos approach to quantum mechanics we give a representation of physical properties in terms of modal operators on Heyting algebras. It allows us to introduce a classical type study of the mentioned properties.

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Notes

  1. For a discussion about contradiction and superposition states see [11].

  2. For the construction of a lattice using convex sets instead of rays as states, see [16].

  3. In this line, we have built in a previous paper a QL that arises from considering a sheaf over a topological space associated to the Boolean sublattices of the ortholattice of closed subspaces of \({\mathcal H}\) [14]. To do so, we defined a valuation that respects contextuality (first translating the Kochen–Specker (KS) theorem to topological terms [14, Theorem 4.3]) and a frame for the Kripke model of the language. As frames are complete Heyting algebras, the resulting logic is an intuitionistic one—with restrictions on the allowed valuations arising from the KS theorem—, thus it has “good” properties as the distributive lattice structure and a nice definition of the implication as a residue of the conjunction.

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Acknowledgments

This work was partially supported by the following grants: VUB Project GOA67; FWO-research community W0.030.06; CONICET RES. 4541-12 (2013-2014) and PIP 112-201101-00636, CONICET.

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Authors and Affiliations

  1. Dipartimento di Filosofia, Università di Cagliari, Viale Merello 92, 09123, Cagliari, Italy

    H. Freytes

  2. Departamento de Matemática, UNR-CONICET, Av. Pellegrini 250, CP 2000, Rosario, Argentina

    H. Freytes

  3. Instituto de Filosofía “Dr. Alejandro Korn”, UBA-CONICET, Puan 480, Buenos Aires, Argentina

    C. de Ronde

  4. Center Leo Apostel (CLEA) - Brussels Free University, Brussels, Belgium

    G. Domenech & C. de Ronde

  5. Foundations of The Exact Sciences (FUND), Brussels Free University, Krijgskundestraat 33, 1160, Brussels, Belgium

    C. de Ronde

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  1. H. Freytes
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  2. G. Domenech
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  3. C. de Ronde
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Correspondence to G. Domenech.

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H. Freytes and C. de Ronde—Fellow of the Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET).

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Freytes, H., Domenech, G. & de Ronde, C. Physical Properties as Modal Operators in the Topos Approach to Quantum Mechanics. Found Phys 44, 1357–1368 (2014). https://doi.org/10.1007/s10701-014-9842-9

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