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Quantum set theory extending the standard probabilistic interpretation of quantum theory. (English) Zbl 1396.81011

Summary: The notion of equality between two observables will play many important roles in foundations of quantum theory. However, the standard probabilistic interpretation based on the conventional Born formula does not give the probability of equality between two arbitrary observables, since the Born formula gives the probability distribution only for a commuting family of observables. In this paper, quantum set theory developed by Takeuti and the present author is used to systematically extend the standard probabilistic interpretation of quantum theory to define the probability of equality between two arbitrary observables in an arbitrary state. We apply this new interpretation to quantum measurement theory, and establish a logical basis for the difference between simultaneous measurability and simultaneous determinateness.

MSC:

81P05 General and philosophical questions in quantum theory
81P15 Quantum measurement theory, state operations, state preparations
46L10 General theory of von Neumann algebras
81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)
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