Group-valued measures on coarse-grained quantum logics. (English) Zbl 1174.03350
Summary: In [S. Gudder and J.-P. Marchand [Bull. Acad. Pol. Sci., Sér. Sci. Math. 28, 557–564 (1980; Zbl 0499.28002)] it was shown that a (real) signed measure on a cyclic coarse-grained quantum logic can be extended, as a signed measure, over the entire power algebra. Later [P. G. Ovchinnikov, Konstr. Teor. Funkts. Funkts. Anal. 8, 95–98 (1992; Zbl 0825.03058)] this result was re-proved (and further improved) and, moreover, the non-negative measures were shown to allow for extensions as non-negative measures. In both cases the proof techniques used were those of linear algebra. In this paper we further generalize the results cited by extending group-valued measures on cyclic coarse-grained quantum logics (or non-negative group-valued measures for lattice-ordered groups). Obviously, the proof techniques are entirely different from those of the preceding papers. In addition, we provide a new combinatorial argument for describing all atoms of cyclic coarse-grained quantum logics.
MSC:
| 03G12 | Quantum logic |
| 06C15 | Complemented lattices, orthocomplemented lattices and posets |
| 28B10 | Group- or semigroup-valued set functions, measures and integrals |
| 81P10 | Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects) |
References:
| [1] | M. R. Darnel: Theory of Lattice-Ordered Groups. Dekker, New York, 1995. |
| [2] | A. De Simone, M. Navara and P. Pták: Extensions of states on concrete finite logics. To appear in Intern. J. Theoret. Phys. · Zbl 1118.81007 |
| [3] | S. Gudder and J. P. Marchand: A coarse-grained measure theory. Bull. Polish Acad. Sci. Math. 28 (1980), 557–564. · Zbl 0499.28002 |
| [4] | S. Gudder: Stochastic Methods in Quantum Mechanics. North Holland, 1979. · Zbl 0439.46047 |
| [5] | S. Gudder: Quantum probability spaces. Proc. Amer. Math. Soc. 21 (1969), 286–302. · Zbl 0183.28703 · doi:10.1090/S0002-9939-1969-0243793-1 |
| [6] | S. Gudder: An extension of classical measure theory. SIAM 26 (1984), 71–89. · Zbl 0559.28003 · doi:10.1137/1026002 |
| [7] | P. de Lucia and P. Pták: Quantum logics with classically determined states. Colloq. Math. 80 (1999), 147–154. · Zbl 0930.03097 |
| [8] | M. Navara and P. Pták: Almost Boolean orthomodular posets. J. Pure Appl. Algebra 60 (1989), 105–111. · Zbl 0691.03045 · doi:10.1016/0022-4049(89)90108-4 |
| [9] | P. Ovtchinikoff: Measures on the Gudder-Marchand logics. Constructive Theory of Functions and Functional Analysis 8 (1992), 95–98. (In Russian.) zbl |
| [10] | P. Pták: Some nearly Boolean orthomodular posets. Proc. Amer. Math. Soc. 126 (1998), 2039–2046. · Zbl 0894.06003 · doi:10.1090/S0002-9939-98-04403-7 |
| [11] | P. Pták: Concrete quantum logics. Internat. J. Theoret. Phys. 39 (2000), 827–837. · Zbl 0965.03076 · doi:10.1023/A:1003626929648 |
| [12] | P. Pták and S. Pulmannová: Orthomodular Structures as Quantum Logics. Kluwer, Dordrecht/Boston/London, 1991. · Zbl 0743.03039 |
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.
