Logical and mathematical structures of quantum mechanics

Rudolf Zahradník, R. Polák

Pitsri Atisoontornkul, Preedeeporn Limcharoeri

Several attempts have been made to establish the logical structure of quantum mechanics on an axiomatic foundation. It has been shown that, whereas the logical structure of the propositional calculus of classical mechanics is that of a Boolean lattice, the logical structure of the propositional calculus of quantum mechanics is non-Boolean. Moreover, a probability model of a quantum system, namely, the quantum probability model, has a mathematical structure completely different from that of the Kolmogorovian probability model, which is the standard probability model of a classical system. These results, which are still controversial, are analysed and put in critical perspective.

M. Esfahanian, Lodewijk Arntzen

We argue that limitations arise when using non-constructive mathematics (NCM) as a foundation for quantum mechanics, since the Boolean logical substrate of NCM fits a distributive propositional structure, but fits less naturally into the non-distributive lattice of quantum propositions. We distinguish three logical structures relevant to this discussion: Boolean algebras (classical, distributive, complemented), Heyting algebras (constructive, distributive, not necessarily complemented), and orthomodular lattices (quantum, non-distributive). The linear structure of quantum state space is compatible with NCM, but the propositional structure of quantum mechanics forms a non-distributive lattice rather than a Boolean algebra. Topos theory, one of the models of constructive mathematics, offers a way forward: the Döring–Isham framework provides an explicit bridge—the daseinization map—from quantum propositions to a topos whose internal logic is intuitionistic. Although some topoi carry Boolean logic, in general, topoi are not constrained to it. We argue that this bridged framework is more suitable for the mathematical foundation of quantum mechanics than one that retains the law of excluded middle (LEM) without such a bridge. Our discussion centers on the pivotal role of the LEM.

Philip L. Bowers

ANDREAS DÖRING, BENJAMIN EVA, MASANAO OZAWA

Abstract Quantum set theory (QST) and topos quantum theory (TQT) are two long running projects in the mathematical foundations of quantum mechanics (QM) that share a great deal of conceptual and technical affinity. Most pertinently, both approaches attempt to resolve some of the conceptual difficulties surrounding QM by reformulating parts of the theory inside of nonclassical mathematical universes, albeit with very different internal logics. We call such mathematical universes, together with those mathematical and logical structures within them that are pertinent to the physical interpretation, ‘Q-worlds’. Here, we provide a unifying framework that allows us to (i) better understand the relationship between different Q-worlds, and (ii) define a general method for transferring concepts and results between TQT and QST, thereby significantly increasing the expressive power of both approaches. Along the way, we develop a novel connection to paraconsistent logic and introduce a new class of structures that have significant implications for recent work on paraconsistent set theory.