Axiomatization of physical systems and ”quantum logic”. (Slovak) Zbl 0536.03042

In this review article, the Mackey’s approach to the mathematical description of a physical system is introduced. This approach is based on the triple (O,S,p), where O represents the set of observables, S represents the set of states and p(A,s,.) is interpreted as the probability distribution of the observable \(A\in O\) in the state \(s\in S\). If the triple (O,S,p) satisfies five Mackey’s axioms, then the physical system can be described by the couple (L,M), where L is a \(\sigma\)-orthomodular partially ordered set, which represents the set of all experimentally verifiable propositions about the physical system (it is called the ”logic” of the system), and M is an order-determining set of probability measures on L, which corresponds to the set of states S. This approach generalizes the description of a classical physical system as well as the traditional description of a quantum-mechanical system by means of a Hilbert space. Some recent problems and results arising in this approach are then introduced, for example, the problem of the existence of joint distributions of observables (in Gudder’s and Urbanik’s sense), the existence of sums of observables, laws of large numbers and central limit theorem for observables, ergodic properties and other problems of noncompatible probability theory. Another class of problems is connected with the superposition principle, the problem of representing the logic by a Hilbert space, the problem of ”hidden parameters”, of the coupling of logics, etc. References to the literature are given.


03G12 Quantum logic
81P05 General and philosophical questions in quantum theory
81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)
03-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mathematical logic and foundations
81-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to quantum theory
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