×

zbMATH — the first resource for mathematics

Effect algebras and unsharp quantum logics. (English) Zbl 1213.06004
Summary: The effects in a quantum-mechanical system form a partial algebra and a partially ordered set which is the prototypical example of the effect algebras discussed in this paper. The relationships among effect algebras and such structures as orthoalgebras and orthomodular posets are investigated, as are morphisms and group- valued measures (or charges) on effect algebras. It is proved that there is a universal group for every effect algebra, as well as a universal vector space over an arbitrary field.

MSC:
06C15 Complemented lattices, orthocomplemented lattices and posets
03G12 Quantum logic
81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)
PDF BibTeX Cite
Full Text: DOI
References:
[1] Aerts, D., ”Description of many physical entities without the paradoxes encountered in quantum mechanics,”Found. Phys. 12, 1131–1170 (1982).
[2] Ali, S. T., ”Stochastic localization, quantum mechanics on phase space and quantum space-time,”Nuovo Cimento 8, No. 11, 1–128 (1985).
[3] Beltrametti, E., and Cassinelli, G.,The Logic of Quantum Mechanics (Encyclopaedia of Mathematics and Its Applications, Gian-Carlo Rota, ed., Vol. 15) (Addison-Wesley, Reading, Massachusetts, 1981). · Zbl 0491.03023
[4] Bennett, M. K., and Foulis, D., ”Tensor products of orthoalgebras,”Order 10, No. 3, 271–282 (1993). · Zbl 0798.06015
[5] Birkhoff, G.,Lattice Theory, 3rd edn. (American Mathematical Society Colloquium Publications, XXV, Providence, Rhode Island, 1967). · Zbl 0153.02501
[6] Boole, G.,An Investigation of the Laws of Thought (Macmillan, London, 1854; reprinted by Dover, New York, 1967).
[7] Bunce, L., and Maitland Wright, J., ”The Mackey-Gleason problem,”Bull. Am. Math. Soc. 26, No. 2, 288–293 (1992). · Zbl 0759.46054
[8] Busch, P., Lahti, P., and Mittelstaedt, P.,The Quantum Theory of Measurement (Lecture Notes in Physics, New Series m2) (Springer, Berlin, 1991). · Zbl 0868.46051
[9] Cattaneo, G., and Nistico, G., ”Brouwer-Zadeh posets and three-valued Lukasiewicz posets,”Int. J. Fuzzy Sets Syst. 33, 165–190 (1989). · Zbl 0682.03036
[10] Della Chiara, M. L., and Giuntini, R., ”Paraconsistent quantum logics”,Found. Phys. 19, No. 7, 891–904 (1989).
[11] Dvurečenskij, A., ”Tensor product of difference posets,” to appear inProc. Am. Math. Soc.
[12] Foulis, D., Greechie, R., and Rüttimann, G., ”Filters and supports in orthoalgebras,”Int. J. Theor. Phys. 31, No. 5, 789–802 (1992). · Zbl 0764.03026
[13] Foulis, D., Piron, C., and Randall, C., ”Realism, operationalism, and quantum mechanics,”Found. Phys. 13, No. 8, 813–842 (1983).
[14] Fuchs, L.,Partially Ordered Algebraic Systems (International Series of Monographs on Pure and Applied Mathematics, Vol. 28) (Pergamon, Oxford, 1963). · Zbl 0137.02001
[15] Giuntini, R., and Greuling, H., ”Toward a formal language for unsharp properties,”Found. Phys. 19, No. 7, 931–945 (1989).
[16] Hardegree, G., and Frazer, P., ”Charting the labyrinth of quantum logics,” inCurrent Issues in Quantum Logic, E. Beltrametti and B. van Fraassen, eds. (Ettore Majorana International Science Series, 8) (Plenum, New York, 1981).
[17] Kalmbach, G.,Orthomodular Lattices (Academic, New York, 1983). · Zbl 0512.06011
[18] Kläy, M., Randall, C., and Foulis, D., ”Tensor products and probability weights,”Int. J. Theor. Phys. 26, No. 3, 199–219 (1987). · Zbl 0641.46049
[19] Kôpka, F., and Chovanec, F., ”D posets,”Math. Slovaca 44, No. 1, 21–34 (1994). · Zbl 0789.03048
[20] Lock, P., and Hardegree, G., ”Connections among quantum logics, Parts 1 and 2, Quantum prepositional logica,”Int. J. Theor. Phys. 24, No. 1, 43–61 (1984). · Zbl 0592.03051
[21] Ludwig, G.,Foundations of Quantum Mechanics, Vols. I and II (Springer, New York, 1983/1985).
[22] Ludwig, G.,An Axiomatic Basis for Quantum Mechanics, Vol. 2 (Springer, New York, 1986/87).
[23] Maeda, F.,Kontinuierliche Geometrien (Springer, Berlin, 1958).
[24] Navara, M., An orthomodular lattice admitting no group-valued measures,Proc. Am. Math. Soc. 122, No. 1, 7–12 (1994). · Zbl 0809.06008
[25] Navara, M., and Pták, P., ”Difference posets and orthoalgebras,” Department of Mathematics Report Series, Czech Technical University in Prague, Faculty of Electrical Engineering, No. 93–8 (1993), pp. 1–5.
[26] Neumann, J. von,Continuous Geometry (Princeton University Press, Princeton, New Jersey, 1960).
[27] Obeid, M., ”Pastings and Centeria of Orthoalgebras,” Ph.D. Thesis, Kansas State University, Manhattan, Kansas, 1990.
[28] Piron, C.,Foundations of Quantum Physics (Mathematical Physics Monograph Series, A. Wightman, ed.) (Benjamin, Reading, Massachusetts, 1976). · Zbl 0333.46050
[29] Prugovecki, E.,Stochastic Quantum Mechanics and Quantum Space Time, 2nd edn. (Reidel, Dordrecht, 1986).
[30] Schroeck, F., and Foulis, D., ”Stochastic quantum mechanics viewed from the language of manuals,”Found. Phys. 20, No. 7, 823–858 (1990).
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.