Discussiones Mathematicae General Algebra and Applications 21(2) (2001) 239-253
doi: 10.7151/dmgaa.1041

[BIBTex] [PDF] [PS]

RING-LIKE STRUCTURES WITH UNIQUE SYMMETRIC DIFFERENCE RELATED TO QUANTUM LOGIC

Dietmar Dorninger, Helmut Länger

Technische Universität Wien
Institut für Algebra und Computermathematik
Wiedner Hauptstraß e 8-10, A-1040 Wien
e-mail: h.laenger@tuwien.ac.at   e-mail: d.dorninger@tuwien.ac.at

Maciej Maczyński

Warsaw University of Technology
Faculty of Mathematics and Information Science
Plac Politechniki 1, PL 00-661 Warsaw, Poland
e-mail: mamacz@alpha.mini.pw.edu.pl

Abstract

Ring-like quantum structures generalizing Boolean rings and having the property that the terms corresponding to the two normal forms of the symmetric difference in Boolean algebras coincide are investigated. Subclasses of these structures are algebraically characterized and related to quantum logic. In particular, a physical interpretation of the proposed model following Mackey's approach to axiomatic quantum mechanics is given.

Keywords: generalized Boolean quasiring, symmetric difference, quantum logic.

2000 AMS Mathematics Subject Classifications: 81P10, 03G12, 06C15, 06E99.

References

[1]D. Dorninger, H. Länger and M. Maczyński, The logic induced by a system of homomorphisms and its various algebraic characterizations, Demonstratio Math. 30 (1997), 215-232.
[2] D. Dorninger, H. Länger and M. Maczyński, On ring-like structures occurring in axiomatic quantum mechanics, Österreich.  Akad. Wiss. Math.-Natur. Kl. Sitzungsber. II 206 (1997), 279-289.
[3] D. Dorninger, H. Länger and M. Maczyński, On ring-like structures induced by Mackey's probability function, Rep. Math. Phys. 43 (1999), 499-515.
[4] D. Dorninger, H. Länger and M. Maczyński, Lattice properties of ring-like quantum logics, Intern. J. Theor. Phys. 39 (2000), 1015-1026.
[5] D. Dorninger, H. Länger and M. Maczyński, Concepts of measures on ring-like quantum logics, Rep. Math. Phys. 47 (2001), 167-176.
[6] G. W. Mackey, The mathematical foundations of quantum mechanics, Benjamin, Reading, MA, 1963.
[7] M. J. Maczyński, A remark on Mackey's axiom system for quantum mechanics, Bull. Acad. Polon. Sci. 15 (1967), 583- 587.
[8] V. S. Varadarajan, Geometry of quantum theory. Springer-Verlag, New York 1985.

Received 21 August 2001