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On joint distribution in quantum logics. II. Noncompatible observables
Applications of Mathematics, Tome 32 (1987) no. 6, pp. 436-450
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This paper i a continuation of the first part under the same title. The author studies a joint distribution in $\sigma$-finite measures for noncompatible observables of a quantum logic defined on some system of $\sigma$-independent Boolean sub-$\sigma$-algebras of a Boolean $\sigma$-algebra. We present some necessary and sufficient conditions fot the existence of a joint distribution. In particular, it is shown that an arbitrary system of obsevables has a joint distribution in a measure iff it may be embedded into a system of compatible observables of some quantum logic. The methods used are different from those developed for finite measures. Finally, the author deals with the connection between the existence of a joint distribution and the existence of a commutator of observables, and the quantum logic of a nonseparable Hilbert space is mentioned.
This paper i a continuation of the first part under the same title. The author studies a joint distribution in $\sigma$-finite measures for noncompatible observables of a quantum logic defined on some system of $\sigma$-independent Boolean sub-$\sigma$-algebras of a Boolean $\sigma$-algebra. We present some necessary and sufficient conditions fot the existence of a joint distribution. In particular, it is shown that an arbitrary system of obsevables has a joint distribution in a measure iff it may be embedded into a system of compatible observables of some quantum logic. The methods used are different from those developed for finite measures. Finally, the author deals with the connection between the existence of a joint distribution and the existence of a commutator of observables, and the quantum logic of a nonseparable Hilbert space is mentioned.
DOI : 10.21136/AM.1987.104275
Classification : 03G12, 28A60, 81B10, 81P10
Keywords: measure; noncompatible observables; joint distribution; commutators; quantum logic 
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     title = {On joint distribution in quantum logics. {II.} {Noncompatible} observables},
     journal = {Applications of Mathematics},
     pages = {436--450},
     year = {1987},
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     url = {http://geodesic.mathdoc.fr/articles/10.21136/AM.1987.104275/}
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Dvurečenskij, Anatolij. On joint distribution in quantum logics. II. Noncompatible observables. Applications of Mathematics, Tome 32 (1987) no. 6, pp. 436-450. doi: 10.21136/AM.1987.104275

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