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The Logos Categorical Approach to Quantum Mechanics: I. Kochen-Specker Contextuality and Global Intensive Valuations

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Abstract

In this paper we present a new categorical approach which attempts to provide an original understanding of QM. Our logos categorical approach attempts to consider the main features of the quantum formalism as the standpoint to develop a conceptual representation that explains what the theory is really talking about —rather than as problems that need to be bypassed in order to allow a restoration of a classical “common sense” understanding of what there is. In particular, we discuss a solution to Kochen-Specker contextuality through the generalization of the meaning of global valuation. This idea has been already addressed by the so called topos approach to QM —originally proposed by Isham, Butterfiled and Döring— in terms of sieve-valued valuations. The logos approach to QM presents a different solution in terms of the notion of intensive valuation. This new solution stresses an ontological (rather than epistemic) reading of the quantum formalism and the need to restore an objective (rather than classical) conceptual representation and understanding of quantum physical reality.

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Notes

  1. For a detailed analysis and discussion of the relation between conceptual representation, mathematical structures and measurement outcomes see [18].

  2. We use the term ‘ontic’ to make clear the distance with respect to the ‘epistemic’ views. See for a detailed discussion [17, 18]. As it has been remarked by van Fraassen, while the question of the ontological (or metaphysical) interpretation of a theory is something of main importance for the realist, for the empiricist this question is one of relative concern. He says [58, p. 242] that an ontological interpretation responds to the questions of “what would it be like for this theory to be true, and how could the world possibly be the way this theory says it is?”

  3. See also for a detailed analysis of contextuality in the topos approaach [23, 38]. For an analysis of conetxtuality in relation to modality see [32,33,34,35].

  4. For example Mermin states in [56, p. 3375] that: “The strengthening of Bell’s example by GHZ and the demonstration that GHZ also works as a KS theorem should liberate future generations of students of the foundations of quantum mechanics from having to cope with the geometrical intricacies of the original KS argument.”

  5. The idea of a contextual or partial truth valuation has been recently formalized by Karakostas and Zafiris [52, 53]. Our approach goes in line with the development of a potential truth as discussed in [22]. The analysis of this specific subject exceeds the scope of this paper and will be left for a future work.

  6. A similar definition is discussed in [51].

  7. This is what is called following the “minimal interpretation” a quantum state. For a detailed analysis of the notion of quantum state in the topos approach see [39].

  8. We avoid using the term “quantum state” for reasons that will become clear in the following.

  9. I am thankful to Bob Coecke for this linguistic insight. Cagliari, July 2014.

  10. The fact that even in classical physics we can find epistemically incompatible measurement situations has been discussed by Diederik Aerts in [3].

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Acknowledgments

This work was partially supported by the following grants: FWO project G.0405.08 and FWO-research community W0.030.06. CONICET RES. 4541-12 and the Project PIO-CONICET-UNAJ (15520150100008CO) “Quantum Superpositions in Quantum Information Processing”.

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Authors and Affiliations

  1. Philosophy Institute Dr. A. Korn, University of Buenos Aires - CONICET, Buenos Aires, Argentina

    C. de Ronde

  2. Center Leo Apostel for Interdisciplinary Studies, Foundations of the Exact Sciences - Vrije Universiteit Brussel, Ixelles, Belgium

    C. de Ronde

  3. Institute of Engineering, National University Arturo Jauretche, Buenos Aires, Argentina

    C. de Ronde

  4. Institute of Mathematical Investigations Luis A. Santaló, UBA - CONICET, Buenos Aires, Argentina

    C. Massri

  5. University CAECE, Buenos Aires, Argentina

    C. Massri

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Correspondence to C. de Ronde.

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de Ronde, C., Massri, C. The Logos Categorical Approach to Quantum Mechanics: I. Kochen-Specker Contextuality and Global Intensive Valuations. Int J Theor Phys 60, 429–456 (2021). https://doi.org/10.1007/s10773-018-3914-0

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