On joint distribution in quantum logics. II: Noncompatible observables. (English) Zbl 0654.03051
This is a continuation of the paper reviewed above (see Zbl 0654.03050). Here the author turns attention to noncompatible observables, and like others he mentions, relates the existence of joint distribution to the existence of commutators. He then distinguishes some necessary and sufficient conditions for the existence of joint distributions, generalizing some known results.
The major result of this section is to show that an arbitrary system of observables has a joint distribution in a measure iff it may be embedded into a system of compatible observables of some quantum logic. In this result measures may have infinite values, and furthermore the conditions do not depend on the existence of the commutator of a given system of observables. The relationship between commutators and joint distributions is further discussed in the last section of the paper.
The major result of this section is to show that an arbitrary system of observables has a joint distribution in a measure iff it may be embedded into a system of compatible observables of some quantum logic. In this result measures may have infinite values, and furthermore the conditions do not depend on the existence of the commutator of a given system of observables. The relationship between commutators and joint distributions is further discussed in the last section of the paper.
Reviewer: R.Wallace-Garden
MSC:
| 03G12 | Quantum logic |
| 81P10 | Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects) |
Citations:
Zbl 0654.03050References:
| [1] | A. Dvurečenskij: Remark on joint distribution in quantum logics. I. Compatible observables. Apl. mat. 32, 427-435 (1987). · Zbl 0654.03050 |
| [2] | T. Lutterová S. Pulmannová: An individual ergodic theorem on the Hilbert space logic. Math. Slovaca, 35, 361- 371 (1985). · Zbl 0597.46066 |
| [3] | S. Pulmannová: Relative compatibility and joint distributions of observables. Found. Phys., 10, 614-653(1980). |
| [4] | L. Beran: On finitely generated orthomodular lattices. Math. Nachrichten 88, 129-139 (1979). · Zbl 0439.06005 · doi:10.1002/mana.19790880111 |
| [5] | E. L. Marsden: The commutator and solvability in a generalized orthomodular lattice. Pac. J. Math., 33, 357-361 (1970). · Zbl 0234.06004 · doi:10.2140/pjm.1970.33.357 |
| [6] | W. Puguntke: Finitely generated ortholattices. Colloq. Math. 33, 651-666 (1980). |
| [7] | G. Grätzer: General Lattice Theory. Birkhauser - Verlag, Basel (1978). · Zbl 0385.06015 |
| [8] | A. Dvurečenskij: On Gleason’s theorem for unbounded measures. JINR, E 5-86-54, Dubna (1986). |
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