| Title:
|
On joint distribution in quantum logics. II. Noncompatible observables (English) |
| Author:
|
Dvurečenskij, Anatolij |
| Language:
|
English |
| Journal:
|
Aplikace matematiky |
| ISSN:
|
0373-6725 |
| Volume:
|
32 |
| Issue:
|
6 |
| Year:
|
1987 |
| Pages:
|
436-450 |
| Summary lang:
|
English |
| Summary lang:
|
Russian |
| Summary lang:
|
Slovak |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
measure |
| Keyword:
|
noncompatible observables |
| Keyword:
|
joint distribution |
| Keyword:
|
commutators |
| Keyword:
|
quantum logic |
| MSC:
|
03G12 |
| MSC:
|
28A60 |
| MSC:
|
81B10 |
| MSC:
|
81P10 |
| idZBL:
|
Zbl 0654.03051 |
| idMR:
|
MR0916060 |
| DOI:
|
10.21136/AM.1987.104275 |
| . |
| Date available:
|
2008-05-20T18:33:23Z |
| Last updated:
|
2020-07-28 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/104275 |
| . |
| Reference:
|
[I] A. Dvurečenskij: Remark on joint distribution in quantum logics. I. Compatible observables.Apl. mat. 32, 427-435 (1987). MR 0916059 |
| Reference:
|
[19] T. Lutterová S. Pulmannová: An individual ergodic theorem on the Hilbert space logic.Math. Slovaca, 35, 361- 371 (1985). MR 0820633 |
| Reference:
|
[20] S. Pulmannová: Relative compatibility and joint distributions of observables.Found. Phys., 10, 614-653(1980). MR 0659345 |
| Reference:
|
[21] L. Beran: On finitely generated orthomodular lattices.Math. Nachrichten 88, 129-139 (1979). Zbl 0439.06005, MR 0543398, 10.1002/mana.19790880111 |
| Reference:
|
[22] E. L. Marsden: The commutator and solvability in a generalized orthomodular lattice.Рас. J. Math., 33, 357-361 (1970). Zbl 0234.06004, MR 0263712 |
| Reference:
|
[23] W. Puguntke: Finitely generated ortholattices.Colloq. Math. 33, 651-666 (1980). |
| Reference:
|
[24] G. Grätzer: General Lattice Theory.Birkhauser - Verlag, Basel (1978). MR 0504338 |
| Reference:
|
[25] A. Dvurečenskij: On Gleason's theorem for unbounded measures.JINR, E 5-86-54, Dubna (1986). |
| . |
