Intuitionistic quantum logic of an \(n\)-level system (Q735205)

Continuing an earlier presentation of related material by \textit{C. Heunen, N. Landsman} and \textit{B. Spitters} [Commun. Math. Phys. 291, No. 1, 63--110 (2009; Zbl 1209.81147)] the authors of the paper under review combine the \(C^*\)-algebraic approach to quantum theory with internal language of topos theory and discuss the abstract setup of the \(C^*\)-algebra \(M_n(\mathbb C)\) of complex \(n \times n\) matrices. This interesting survey includes a review of the topos-theoretic approach to quantum mechanics by \textit{J. Butterfield} and \textit{C. J. Isham} [Int. J. Theor. Phys. 38, No. 3, 827--859 (1999; Zbl 1007.81009)], by \textit{A. Döring} and \textit{C. J. Isham} [J. Math. Phys. 49, No. 5, 053515 (2008; Zbl 1152.81408)] and others. In the authors' approach, ``the nondistributive lattice \({\mathcal P}( M_n(\mathbb C))\) of projections in \( M_n(\mathbb C)\) (which forms the basis of the traditional quantum logic of Birkhoff and von Neumann) is replaced by a specific distributive lattice \({\mathcal O}(\Sigma_n)\) of functions from the poset \({\mathcal C}(M_n(\mathbb C))\) of all unital commutative \(C^*\)-subalgebras \({\mathcal C}\) of \(M_n(\mathbb C)\) to \({\mathcal P}( M_n(\mathbb C))\). The lattice \({\mathcal O}(\Sigma_n)\) is essentially the (pointfree) topology of the quantum phase space \(\Sigma_n\), and as such defines a Heyting algebra. Each element of \({\mathcal O}(\Sigma_n)\) corresponds to a `Bohrified' proposition, in the sense that to each classical context \( C \in {\mathcal C}(M_n(\mathbb C))\) it associates a yes-no question (i.e., an element of the Boolean lattice \({\mathcal P}(C)\) of projections in \(C\)), rather than being a single projection as in standard quantum logic. Distributivity is recovered at the expense of the law of the excluded middle (Tertium Non Datur), whose demise is in the authors' opinion to be welcomed, not just in intuitionistic logic in the spirit of Brouwer, but also in quantum logic in spirit of von Neumann.'' The explicit description of the Gelfand spectrum through an associated frame \({\mathcal O}(\Sigma)\) in the category \textbf{Sets} is one of the central results of the paper. A comprehensive description and discussion of the history of evolution of concepts of quantum mechanics till 1967 is given in the book by \textit{M. Jammer} [Conceptual Development of Quantum Mechanics. New York: McGraw-Hill (1967)]. The paper has eight sections. The first one treats the topos-theoretic approach to quantum theory and indicates that the frame is the authors' novel quantum analogue of classical phase space. Some preliminary calculations, namely frames of the Gelfand spectrum of full \(n\)-dimensional complex linear algebras, are presented in Section 2. In Section 3 the authors explicitly compute the Gelfand spectrum of the algebras for any \(n\). Section 4 gives the Heyting algebra structure of the phase space. Section 5 presents results on explicit computation of the Gelfand transform. Section 6 contains the Kochen-Specker theorem in the authors's formulation and its new proof. In Section 7 the authors compute the non-probabilistic state-proposition pairing. Finally, the parametrization of the poset of unital commutative \(C^*\)-algebras of full complex matrix algebras of dimension \(n\) for any \(n\) is studied in Section 8. The appendix contains basic sheaf theory, basic topos theory, Heyting algebras and frames. The paper under review should be read by anyone who wishes to understand the intuitionistic quantum logic of finite dimensional complex algebras.