Abstract
A topology is introduced in a logic ℒ using the set of pure states of ℒ. It is shown that ℒ, equipped with this topology, under suitable conditions, determines the division ring ℝ, ℂ or 2e. With the continuity of the antiautomorphism of the division ring added, it is shown that these conditions are necessary and sufficient for the projective logic ℒ to be isomorphic with the projective logic of the projections in a Hilbert space over ℝ, ℂ or 2e.
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References
Artin, E. (1962).Algèbre Géométrique. Gauthier-Villars, Paris.
Dixmier, J. (1969).Les Algèbres d'Opérateurs dans l'Espace Hilbertien. Gauthier-Villars, Paris.
Jauch, J. M. (1968).Foundations of Quantum Mechanics. Addison Wesley. (1968)
Kato, T. (1966).Perturbation Theory of Linear Operators. Springer-Verlag, Berlin.
Kelley, J. L. (1965).General Topology. Van Nostrand Co., New York.
Piron, C. (1964). Axiomatique Quantique, Thése, Université de Lausanne, Faculté des Sciences, Bale, Imprimerie Birkhauser S.A.
Pontrjagin, L. (1946).Topological Groups. Princeton University Press, Princeton.
Varadarajan, V. S. (1968).Geometry of Quantum Theory, Vol. 1. Van Nostrand Co., Princeton, New Jersey.
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P. Cotta-Ramusino gratefully acknowledges a fellowship of the Consiglio Nazionale delle Ricerche.
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Cirelli, R., Cotta-Ramusino, P. On the isomorphism of a ‘quantum logic’ with the logic of the projections in a hilbert space. Int J Theor Phys 8, 11–29 (1973). https://doi.org/10.1007/BF00671575
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DOI: https://doi.org/10.1007/BF00671575