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Keywords:
measure; noncompatible observables; joint distribution; commutators; quantum logic
Summary:
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.
References:
[I] A. Dvurečenskij: Remark on joint distribution in quantum logics. I. Compatible observables. Apl. mat. 32, 427-435 (1987). MR 0916059
[19] T. Lutterová S. Pulmannová: An individual ergodic theorem on the Hilbert space logic. Math. Slovaca, 35, 361- 371 (1985). MR 0820633
[20] S. Pulmannová: Relative compatibility and joint distributions of observables. Found. Phys., 10, 614-653(1980). MR 0659345
[21] L. Beran: On finitely generated orthomodular lattices. Math. Nachrichten 88, 129-139 (1979). DOI 10.1002/mana.19790880111 | MR 0543398 | Zbl 0439.06005
[22] E. L. Marsden: The commutator and solvability in a generalized orthomodular lattice. Рас. J. Math., 33, 357-361 (1970). MR 0263712 | Zbl 0234.06004
[23] W. Puguntke: Finitely generated ortholattices. Colloq. Math. 33, 651-666 (1980).
[24] G. Grätzer: General Lattice Theory. Birkhauser - Verlag, Basel (1978). MR 0504338
[25] A. Dvurečenskij: On Gleason's theorem for unbounded measures. JINR, E 5-86-54, Dubna (1986).

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