Abstract
Some algebraic structures of the set of all effects are investigated and summarized in the notion of a(weak) orthoalgebra. It is shown that these structures can be embedded in a natural way in lattices, via the so-calledMacNeille completion. These structures serve as a model ofparaconsistent quantum logic, orthologic, andorthomodular quantum logic.
Similar content being viewed by others
References
S. Bugajski, “The Inner Language of Operational Quantum Mechanics,” in B. C. van Fraassen and E. G. Beltrametti (eds.),Current Issues in Quantum Logic (Plenum, New York, 1981), pp. 283–299.
S. Bugajski and P. Lahti, “Fundamental Principles of Quantum Theory,”Int. J. Theor. Phys. 24, 1075–1104 (1985).
P. Busch, “Elements of Unsharp Reality in the EPR Experiment,” in P. Lahti and P. Mittelstaedt (eds.),Symposium on the Foundations of Modern Physics (World Scientific, Singapore, 1985), 342–357.
P. Busch, “Unsharp Reality and the Question of Quantum Systems,” inSymposium on the Foundations of Modern Physics, June 8, 1987, Joensuu, Finland.
M. L. Dalla Chiara, “Quantum Logic,” in D. M. Gabbay and F. Guenther (eds.),Handbook of Philosophical Logic, Vol. III (Reidel, Dordrecht, 1986), pp. 427–469.
M. L. Dalla Chiara and R. Giuntini, “Paraconsistent Quantum Logics,”Found. Phys. 19, 891–904 (1989).
E. B. Davies, “Quantum Stochastic Processes,”Commun. Math. Phys. 15, 277–304 (1969).
E. B. Davies and J. T. Lewis, “An Operational Approach to Quantum Probability,”Commun. Math. Phys. 17, 239–260 (1970).
P. D. Finch, “Orthogonality relations and Orthomodularity,”Bull. Aust. Math. Soc. 2, 125–128 (1970).
D. J. Foulis and C. H. Randall, “Empirical Logic and Tensor Products,” in H. Neumann (ed.),Interpretations and Foundations of Quantum Theory (Grundlagen der exakten Naturwissenschaften, Bd. 5) (Bibliographisches Institut, Mannheim, 1981).
C. Garola, “Embedding of Posets into Lattices of Quantum Logic,”Int. J. Theor. Phys. 24, 423–433 (1985).
G. Kalmbach,Orthomodular Lattices (Academic Press, New York, 1983).
R. Haag and D. Kastler, “An Algebraic Approach to Quantum Field Theory,”J. Math. Phys. 5, 848–861 (1964).
G. Ludwig,Foundations of Quantum Mechanics, Vols. I and II (Springer, Berlin, 1983).
M. D. MacLaren, “Atomic Orthocomplemented Lattices,”Pac. J. Math. 14, 597–612 (1964).
P. Mittelstaedt,Quantum Logic (Reidel, Dordrecht, Holland).
G. Lüders, “Über der Zustandsänderung durch den Meßprozeß,”Ann. Phys. 6, Ser.8, 322–328 (1951).
J. C. Pool, “Baer-*-Semigroups and the Logic of Quantum Mechanics,”Commun. Math. Phys.,9, 118–141 (1968).
E. W. Stachow, “Logical Foundation of Quantum Mechanics,”Int. J. Theor. Phys. 19, 251–304 (1980).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Giuntini, R., Greuling, H. Toward a formal language for unsharp properties. Found Phys 19, 931–945 (1989). https://doi.org/10.1007/BF01889307
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01889307