×

Topologies on quantum logics induced by measures. (English) Zbl 0674.03021

The author discusses, from the topological point view, the compatibility of two or more elements of a quantum logic, i.e., a model for the collection of propositions on a general physical system. The physical importance of this has its origin in the uncertainty principle of Heisenberg. Starting with a logic \({\mathcal L}\), the author constructs a topological space (\({\mathcal L},{\mathcal T})\). The topological \({\mathcal T}\) is considered to be connected, in a sense, with the state of the corresponding physical system. The topology defined in the paper is linked with a finite measure \(\mu\) on \({\mathcal L}\) and the constructed topology is associated with this measure.
Reviewer: N.D.Sengupta

MSC:

03G12 Quantum logic
81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)
46N99 Miscellaneous applications of functional analysis
PDFBibTeX XMLCite
Full Text: EuDML

References:

[1] DVUREČENSKIJ A.: On convergences of signed states. Math. Slovaca 28, 1978, 289-295. · Zbl 0421.28003
[2] GLEASON A. M.: Measures on the closed subspaces of a Hilbert space. J. Math. Mech. 6, 1957, 885-893. · Zbl 0078.28803
[3] GUDDER S. P.: Spectral methods for a generalized probability theory. Trans. Amer. Math. Soc., 119, 1965, 428-442. · Zbl 0161.46105 · doi:10.2307/1994077
[4] HALMOS P. R.: Measure Theory. Van Nostrand, Princeton, 1968.
[5] HALMOS P. R.: A Hilbert Space Problem Book. Van Nostrand, Princeton, 1967. · Zbl 0144.38704
[6] KELLEY J. L.: General Topology. Van Nostrand, New York, 1955. · Zbl 0066.16604
[7] SARYMSAKOV T. A., AJUPOV Š. A., CHADŽIJEV, DŽ., ČILIN V. I.: Uporjadočennyje algebry. FAN, Taškent, 1983.
[8] SCHATTEN R.: Norm Ideals of Completely Continuous Operators. Springer, Berlin, 1970. · Zbl 0188.44103
[9] VARADARAJAN V. S.: Probability in physics and a theorem on simultaneous observability. Com. Pure Appl. Math., 15, 1962, 189-217. · Zbl 0109.44705 · doi:10.1002/cpa.3160150207
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.