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Combinations of observables. (English) Zbl 1160.81306

Summary: This article begins with a review of the framework of fuzzy probability theory.The basic structure is given by the \(\sigma\)-effect algebra of effects (fuzzy events) \(E(\Omega,A)\) and the set of probability measures \(M^+_1 (\Omega, A)\) on a measurable space \((\Omega,A)\). An observable \(X: B \to E(\Omega, A)\) is defined, where \((\Lambda, B)\) is the value space of \(X\). It is noted that there exists a one-to-one correspondence between states on \(E(\Omega, A)\) and elements of \(M^+(\Omega, A)\) and between observables \(X: B \to E(\Omega,A)\) and \(\sigma\)-morphisms from \(E(\Lambda, B)\) to \(E(\Omega, A)\). Various combinations of observables are discussed. These include compositions, products, direct products,and mixtures. Fuzzy stochastic processes are introduced and an application toquantum dynamics is considered. Quantum effects are characterized from amonga more general class of effects. An alternative definition of a statistical map \(T:M^+_1 (\Omega, A)\to M^+_1 (\Lambda, B)\) is given.

MSC:

81P05 General and philosophical questions in quantum theory
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