Boolean properties of observables in axiomatic quantum mechanics

Frederic W. Shultz

Dietmar W. Dorninger

Jiří Souček

In this paper we describe a new (complete) axiomatization of quantum mechanics (QM) in which we add axioms describing the concept of an observation. We show that new axioms are clear and evident and based on common sense. Our approach is based on the idea of the observation of properties of an individual measuring system. We distinguish two concepts of a state: the properties of an individual measuring system and the state of an ensemble of systems. Then we prove that the ontic model of QM is inconsistent. This is our main result. This implies the necessity to consider “non-realistic” models for QM in which it is not true that each wave-function describes a possible state of an individual system. Our axiomatization is based on the postulate of the existence of definite outputs of an experiment. Thus the evident reality of the existence of definite outputs implies that our axiomatization is true and verifiable.

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.

Vladimir L. Vasyukov

Elias Zafiris

We propose a sheaf-theoretic framework for the representation of a quantum observable structure in terms of Boolean information sieves. The algebraic representation of a quantum observable structure in the relational local terms of sheaf theory effectuates a semantic transition from the axiomatic set-theoretic context of orthocomplemented partially ordered sets, la Birkhoff and Von Neumann, to the categorical topos-theoretic context of Boolean information sieves, la Grothendieck. The representation schema is based on the existence of a categorical adjunction, which is used as a theoretical platform for the development of a functorial formulation of information transfer, between quantum observables and Boolean localisation devices in typical quantum measurement situations. We also establish precise criteria of integrability and invariance of quantum information transfer by cohomological means.