Article
Full entry |
PDF
(1.2 MB)
Feedback
PDF
(1.2 MB)
Feedback
Keywords:
compatibility; orthomodular poset; observables; joint distribution; measure; quantum logic
compatibility; orthomodular poset; observables; joint distribution; measure; quantum logic
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.
Related articles:
References:
[1] F. S. Varadarajan: Geometry of Quantum Theory. Van Nostrand, Princeton (1968). Zbl 0155.56802
[2] R. Sikorski: Boolean Algebras. Springer-Verlag (1964). MR 0126393 | Zbl 0123.01303
[3] G. Kalmbach: Orthomodular Lattices. Acad. Press, London (1983). MR 0716496 | Zbl 0528.06012
[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
[5] S. P. Gudder: Joint distributions of observables. J. Math. Mech. 18, 325-335 (1968). MR 0232582 | Zbl 0241.60092
[6] A. Dvurečenskij S. Pulmannová: On joint distributions of observables. Math. Slovaca 32, 155-166 (1982). MR 0658249
[7] A. Dvurečenskij S. Pulmannová: Connection between joint distributions and compatibility of observables. Rep. Math. Phys. 19, 349-359 (1984). DOI 10.1016/0034-4877(84)90007-7 | MR 0745430
[8] A. Dvurečenskij: On two problems of quantum logics. Math. Slovaca 36, 253-265 (1986). MR 0866626
[9] S. Pulmannová: Compatibility and partial compatibility in quantum logics. Ann. Inst. Henri Poincaré, 34, 391-403. (1981). MR 0625170
[10] S. Pulmannová A. Dvurečenskij: Uncertainty principle and joint distribution of observables. Ann. Inst. Henri Poincaré 42, 253-265 (1985). MR 0797275
[11] S. Pulmannová: Commutators in orthomodular lattices. Demonstratio Math., 18, 187-208 (1985). MR 0816029
[12] A. Dvurečenskij: Joint distributions of observables and measures with infinite values. JINR, E 5-85-867, Dubna (1985).
[13] P. Pták: Spaces of observables. Czech. Math. J. 34 (109), 552-561 (1984). MR 0764437
[14] L. H. Loomis: On the representation of $\sigma$-complete Boolean algebras. Bull. Amer. Math. Soc. 53, 757-760 (1947). DOI 10.1090/S0002-9904-1947-08866-2 | MR 0021084 | Zbl 0033.01103
[15] M. Duchoň: A note on measures in Cartesian products. Acta F. R. N. Univ. Comen. Math., 23, 39-45 (1969). MR 0265546
[16] G. Birkhoff: Lattice Theory. Nauka, Moscow (1984). MR 0751233
[17] P. R. Halmos: Measure Theory. IIL Moscow (1953).
[18] R. Sikorski: On an analogy between measures and homomorphisms. Ann. Soc. Pol. Math., 23, 1-20 (1950). MR 0039697 | Zbl 0041.17804
Search
Partner of
