Contextual semantics in quantum mechanics from a categorical point of view.

*(English)*Zbl 1396.81014The technical core of this paper is a categorical adjunction between the sheaves of variable local Boolean frames and the category of quantum event algebras. The aspect which is analysed is the subobject classifier in the former category, which is a Grothendieck topos, that provides via a suitable adjunction a subobject classifier in the latter category, thus equipping it with a notion of non-classical truth allowing to interpret propositions in quantum mechanics, which is the ultimate purpose of the article.

The article appeared in Synthese, a leading journal in philosophy, and thus the focus of the work is to discuss the philosophical implications of the previous construction. Mathematically, the construction follows a standard path, and although it is by far non-elementary, the result is not surprising in a purely mathematical sense. In the physical sense, the result is, as far as the reviewer can understand, significant and of interest beyond the purely mathematical content, since it provides the conceptual framework to assemble observations in a coherent way, considering that the act of observing influences the outcome of an experiment. Of course, this is an over-simplification, adopted in this review to illustrate the content of the paper; however the same idea is discussed in depth in the first part of the work, in its full complexity, and with a particular emphasis on the physical and philosophical implications, e.g., its relation with the well-known Kochen-Specker’s theorem.

The paper starts providing the general framework which explains how the physical phenomena are linked to propositions, thus showing what a notion of truth means in the interpretation of a physical system in terms of its events. While in classical mechanics such an interpretation boils down to classical logic, as the semantics of propositions reduces to the lattice of subspaces of the phase space, which is Boolean (Section 1), the same interpretation in quantum Physics does not reduce to classical logic, as the structure of the phase space is not that of a Boolean lattice (Section 2). Studying the structure of the quantum algebra of events (Section 3), it becomes natural to interpret quantum event algebras in Category Theory (Section 4). This study is abstract, claiming for a concrete representation, which is pursued in Section 5 by means of pointed Boolean frames, and deepened in Section 6 by Boolean localization functors. The mathematical instruments developed in these sections allow to explicitly characterise the subobject classifier in the lattice of quantum event algebras (Section 7), and to provide an explicit interpretation of truth of physical propositions (Section 8) by means of the so-synthesised quantum truth-values algebras. Then, in Section 9, the contextual semantics valuation is analysed by its conceptual and philosophical implications. It turns out that elementary propositions associated with the behaviour of a quantum system receive a locally classical meaning with respect to an observational framework, while the proposition itself possesses a non-classical truth value which is linked to the classical truth-values it gets in all the possible observational framework by the corresponding Boolean localization functors. It is this link between the global quantum semantics and the local classical semantics that is the core of the paper and the discussed issue here.

The article is relevant to whoever is interested in the epistemology and ontology of quantum mechanics, as it provides not only a general discussion or an ideal framework, but rather it studies the fundamental mathematical tools to model the semantics of physical propositions along the proposed line of philosophical investigation. Mathematically, the paper provides a non-obvious use of the notion of subobject classifiers in a framework that requires a non-classical notion of truth to be synthesised, showing the power of topos-theoretic tools in this respect. Thus, the paper may have an independent interest for category theorists interested in the application of topos theory outside the strict domain of Mathematics.

The article appeared in Synthese, a leading journal in philosophy, and thus the focus of the work is to discuss the philosophical implications of the previous construction. Mathematically, the construction follows a standard path, and although it is by far non-elementary, the result is not surprising in a purely mathematical sense. In the physical sense, the result is, as far as the reviewer can understand, significant and of interest beyond the purely mathematical content, since it provides the conceptual framework to assemble observations in a coherent way, considering that the act of observing influences the outcome of an experiment. Of course, this is an over-simplification, adopted in this review to illustrate the content of the paper; however the same idea is discussed in depth in the first part of the work, in its full complexity, and with a particular emphasis on the physical and philosophical implications, e.g., its relation with the well-known Kochen-Specker’s theorem.

The paper starts providing the general framework which explains how the physical phenomena are linked to propositions, thus showing what a notion of truth means in the interpretation of a physical system in terms of its events. While in classical mechanics such an interpretation boils down to classical logic, as the semantics of propositions reduces to the lattice of subspaces of the phase space, which is Boolean (Section 1), the same interpretation in quantum Physics does not reduce to classical logic, as the structure of the phase space is not that of a Boolean lattice (Section 2). Studying the structure of the quantum algebra of events (Section 3), it becomes natural to interpret quantum event algebras in Category Theory (Section 4). This study is abstract, claiming for a concrete representation, which is pursued in Section 5 by means of pointed Boolean frames, and deepened in Section 6 by Boolean localization functors. The mathematical instruments developed in these sections allow to explicitly characterise the subobject classifier in the lattice of quantum event algebras (Section 7), and to provide an explicit interpretation of truth of physical propositions (Section 8) by means of the so-synthesised quantum truth-values algebras. Then, in Section 9, the contextual semantics valuation is analysed by its conceptual and philosophical implications. It turns out that elementary propositions associated with the behaviour of a quantum system receive a locally classical meaning with respect to an observational framework, while the proposition itself possesses a non-classical truth value which is linked to the classical truth-values it gets in all the possible observational framework by the corresponding Boolean localization functors. It is this link between the global quantum semantics and the local classical semantics that is the core of the paper and the discussed issue here.

The article is relevant to whoever is interested in the epistemology and ontology of quantum mechanics, as it provides not only a general discussion or an ideal framework, but rather it studies the fundamental mathematical tools to model the semantics of physical propositions along the proposed line of philosophical investigation. Mathematically, the paper provides a non-obvious use of the notion of subobject classifiers in a framework that requires a non-classical notion of truth to be synthesised, showing the power of topos-theoretic tools in this respect. Thus, the paper may have an independent interest for category theorists interested in the application of topos theory outside the strict domain of Mathematics.

Reviewer: Marco Benini (Buccinasco)

##### MSC:

81P13 | Contextuality in quantum theory |

81P05 | General and philosophical questions in quantum theory |

03G30 | Categorical logic, topoi |

18B25 | Topoi |

81P10 | Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects) |

##### Keywords:

Kochen-Specker theorem; category theory; categorical adjunction; subobject classifier; correspondence truth; contextual semantics
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\textit{V. Karakostas} and \textit{E. Zafiris}, Synthese 194, No. 3, 847--886 (2017; Zbl 1396.81014)

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