Abstract
The notion of atomic observable was introduced by S.Gudder for effect test spaces in 1997. In this paper an observable is a σ-homomorphism from the Borel algebra on a line to some logic. Roughly, an observable on a logic is atomic, if it is completely determined by its restriction to one-element subsets of its point spectrum. In particular, every discrete observable is atomic. We study some elementary properties of such observables, and discuss a possible notion of functional dependency between them. Algebraically, a dependency is a certain preorder relation on the set of all atomic observables, which induces an order relation on the set of all maximal orthogonal subsets of the logic. Several properties, as well as characteristics in terms of the underlying logic, of these relations are stated.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Beltrameti, E.G., Casinelli, G.: The Logic of Quantum Mechanics. Cambridge University Press, Cambridge (1984)
Cīrulis, J.: Knowledge representation in extended Pawlak’s information systems: algebraic aspects. In: Eiter, T. (ed.) e.a., FOIKS’02, Lect. Notes Comput. Sci. 2284, pp. 250–267. Springer, Heidelberg (2002)
Cīrulis, J.: Knowledge representation systems and skew nearlattices. In: Chajda, I. (ed.) e.a., Contrib. Gen. Algebra 14, pp. 43–51. Verl. J. Heyn, Klagenfurt (2004)
Cīrulis, J.: Algebraic models of question answering systems, Theory Days at Jõulumäe, October 3–5, 2008, http://www.ioc.ee/~tarmo/tday-joulumae/cirulis-slides.pdf
Gudder, S.: Systems of observables in axiomatic quantum mechanics. J. Math. Phys 8, 2109–2113 (1967)
Gudder, S.: Axiomatic quantum mechanics and generalized probability theory. Probab. Methods Appl. Math. 2, 53–129 (1970)
Gudder, S.: Spectral methods for a generalized probability theory. Trans. Amer. Math. Soc. 119, 428–442 (1965)
Gudder, S.: Effect test spaces. Int. J. Theoret. Phys. 36, 2681–2705 (1997)
Gudder, S.: A survey of e-test spaces. Int. J. Theoret. Phys. 37, 75–83 (1998)
Varadarajan, V.S.: Probability in physics and a theorem on simultaneous observability. Communs Pure Appl. Math. 15, 189–217 (1962). A correction: ibid: 18 (1965), 757
Varadarajan, V.S.: Geometry of Quantum Theory, 2nd. ed. Springer, New York (1985)
Acknowledgements
The author is thankful to the anonymous referee for comments that helped to improve the presentation of the paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was partially supported by ESF Project 2009/0216/1DP/1.1.1.2.0/09/ APIA/VIAA/044 and by Latvian Science Council, Grant No 271/2012
Rights and permissions
About this article
Cite this article
Cīrulis, J. Dependency Ordering of Atomic Observables. Int J Theor Phys 54, 4247–4259 (2015). https://doi.org/10.1007/s10773-014-2473-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10773-014-2473-2