On joint distribution in quantum logics. II. Noncompatible observables

Anatolij Dvurečenskij

Aplikace matematiky (1987)

  • Volume: 32, Issue: 6, page 436-450
  • ISSN: 0862-7940

Abstract

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This paper i a continuation of the first part under the same title. The author studies a joint distribution in σ -finite measures for noncompatible observables of a quantum logic defined on some system of σ -independent Boolean sub- σ -algebras of a Boolean σ -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.

How to cite

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Dvurečenskij, Anatolij. "On joint distribution in quantum logics. II. Noncompatible observables." Aplikace matematiky 32.6 (1987): 436-450. <http://eudml.org/doc/15514>.

@article{Dvurečenskij1987,
abstract = {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.},
author = {Dvurečenskij, Anatolij},
journal = {Aplikace matematiky},
keywords = {measure; noncompatible observables; joint distribution; commutators; quantum logic; measure; noncompatible observables; joint distribution; commutators},
language = {eng},
number = {6},
pages = {436-450},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On joint distribution in quantum logics. II. Noncompatible observables},
url = {http://eudml.org/doc/15514},
volume = {32},
year = {1987},
}

TY - JOUR
AU - Dvurečenskij, Anatolij
TI - On joint distribution in quantum logics. II. Noncompatible observables
JO - Aplikace matematiky
PY - 1987
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 32
IS - 6
SP - 436
EP - 450
AB - 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.
LA - eng
KW - measure; noncompatible observables; joint distribution; commutators; quantum logic; measure; noncompatible observables; joint distribution; commutators
UR - http://eudml.org/doc/15514
ER -

References

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  1. A. Dvurečenskij, Remark on joint distribution in quantum logics. I. Compatible observables, Apl. mat. 32, 427-435 (1987). (1987) MR0916059
  2. T. Lutterová S. Pulmannová, An individual ergodic theorem on the Hilbert space logic, Math. Slovaca, 35, 361- 371 (1985). (1985) MR0820633
  3. S. Pulmannová, Relative compatibility and joint distributions of observables, Found. Phys., 10, 614-653(1980). (1980) MR0659345
  4. L. Beran, 10.1002/mana.19790880111, Math. Nachrichten 88, 129-139 (1979). (1979) Zbl0439.06005MR0543398DOI10.1002/mana.19790880111
  5. E. L. Marsden, The commutator and solvability in a generalized orthomodular lattice, Рас. J. Math., 33, 357-361 (1970). (1970) Zbl0234.06004MR0263712
  6. W. Puguntke, Finitely generated ortholattices, Colloq. Math. 33, 651-666 (1980). (1980) 
  7. G. Grätzer, General Lattice Theory, Birkhauser - Verlag, Basel (1978). (1978) MR0504338
  8. A. Dvurečenskij, On Gleason's theorem for unbounded measures, JINR, E 5-86-54, Dubna (1986). (1986) 

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