| Title:
|
On joint distribution in quantum logics. I. Compatible observables (English) |
| Author:
|
Dvurečenskij, Anatolij |
| Language:
|
English |
| Journal:
|
Aplikace matematiky |
| ISSN:
|
0373-6725 |
| Volume:
|
32 |
| Issue:
|
6 |
| Year:
|
1987 |
| Pages:
|
427-435 |
| Summary lang:
|
English |
| Summary lang:
|
Russian |
| Summary lang:
|
Slovak |
| . |
| Category:
|
math |
| . |
| Summary:
|
The notion of a joint distribution in $\sigma$-finite measures of observables of a quantum logic defined on some system of $\sigma$-independent Boolean sub-$\sigma$-algebras of a Boolean $\sigma$-algebra is studied. In the present first part of the paper the author studies a joint distribution of compatible observables. It is shown that it may exists, although a joint obsevable of compatible observables need not exist. (English) |
| Keyword:
|
compatibility |
| Keyword:
|
orthomodular poset |
| Keyword:
|
observables |
| Keyword:
|
joint distribution |
| Keyword:
|
measure |
| Keyword:
|
quantum logic |
| MSC:
|
03G12 |
| MSC:
|
28A60 |
| MSC:
|
81B10 |
| MSC:
|
81P10 |
| idZBL:
|
Zbl 0654.03050 |
| idMR:
|
MR0916059 |
| DOI:
|
10.21136/AM.1987.104274 |
| . |
| Date available:
|
2008-05-20T18:33:20Z |
| Last updated:
|
2020-07-28 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/104274 |
| . |
| Reference:
|
[1] F. S. Varadarajan: Geometry of Quantum Theory.Van Nostrand, Princeton (1968). Zbl 0155.56802 |
| Reference:
|
[2] R. Sikorski: Boolean Algebras.Springer-Verlag (1964). Zbl 0123.01303, MR 0126393 |
| Reference:
|
[3] G. Kalmbach: Orthomodular Lattices.Acad. Press, London (1983). Zbl 0528.06012, MR 0716496 |
| Reference:
|
[4] S. P. Gudder: Some unsolved problems in quantum logics.in: Mathematical foundations of quantum theory, A. R. Marlow ed. p. 87-103, Acad. Press, New York (1978). MR 0495813 |
| Reference:
|
[5] S. P. Gudder: Joint distributions of observables.J. Math. Mech. 18, 325-335 (1968). Zbl 0241.60092, MR 0232582 |
| Reference:
|
[6] A. Dvurečenskij S. Pulmannová: On joint distributions of observables.Math. Slovaca 32, 155-166 (1982). MR 0658249 |
| Reference:
|
[7] A. Dvurečenskij S. Pulmannová: Connection between joint distributions and compatibility of observables.Rep. Math. Phys. 19, 349-359 (1984). MR 0745430, 10.1016/0034-4877(84)90007-7 |
| Reference:
|
[8] A. Dvurečenskij: On two problems of quantum logics.Math. Slovaca 36, 253-265 (1986). MR 0866626 |
| Reference:
|
[9] S. Pulmannová: Compatibility and partial compatibility in quantum logics.Ann. Inst. Henri Poincaré, 34, 391-403. (1981). MR 0625170 |
| Reference:
|
[10] S. Pulmannová A. Dvurečenskij: Uncertainty principle and joint distribution of observables.Ann. Inst. Henri Poincaré 42, 253-265 (1985). MR 0797275 |
| Reference:
|
[11] S. Pulmannová: Commutators in orthomodular lattices.Demonstratio Math., 18, 187-208 (1985). MR 0816029 |
| Reference:
|
[12] A. Dvurečenskij: Joint distributions of observables and measures with infinite values.JINR, E 5-85-867, Dubna (1985). |
| Reference:
|
[13] P. Pták: Spaces of observables.Czech. Math. J. 34 (109), 552-561 (1984). MR 0764437 |
| Reference:
|
[14] L. H. Loomis: On the representation of $\sigma$-complete Boolean algebras.Bull. Amer. Math. Soc. 53, 757-760 (1947). Zbl 0033.01103, MR 0021084, 10.1090/S0002-9904-1947-08866-2 |
| Reference:
|
[15] M. Duchoň: A note on measures in Cartesian products.Acta F. R. N. Univ. Comen. Math., 23, 39-45 (1969). MR 0265546 |
| Reference:
|
[16] G. Birkhoff: Lattice Theory.Nauka, Moscow (1984). MR 0751233 |
| Reference:
|
[17] P. R. Halmos: Measure Theory.IIL Moscow (1953). |
| Reference:
|
[18] R. Sikorski: On an analogy between measures and homomorphisms.Ann. Soc. Pol. Math., 23, 1-20 (1950). Zbl 0041.17804, MR 0039697 |
| . |
