Abstract
We show that a test space consisting of nonzero vectors of a quadratic spaceE and of the set all maximal orthogonal systems inE is algebraic iffE is Dacey or, equivalently, iffE is orthomodular. In addition, we present another orthomodularity criteria of quadratic spaces, and using the result of Solèr, we show that they can imply thatE is a real, complex, or quaternionic Hilbert space.
Similar content being viewed by others
References
Amemiya, I., and Araki, H. (1966/67). A remark on Piron's paper,Publications.Research Institute for Mathematical Sciences Series A,2, 423–427.
Cattaneo, G., Franco, G., and Marino, G. (1987). Ordering on families of subspaces of pre-Hilbert spaces and Dacey pre-Hilbert spaces,Bolletino.Unione Matematica Italiana,1-B, 167–183.
Dvurečenskij, A. (1993).Gleason's Theorem and Its Applications, Kluwer, Dordrecht, and Ister Science Press, Bratislava.
Dvurečenskij, A., and Pulmannov'a, S. (1994). Test Spaces, Dacey Spaces, and Completeness of inner product spaces,Letters in Mathematical Physics,32, 299–306.
Foulis, D. J., and Bennett, M. K. (1993). Tensor products of orthoalgebras,Order,10, 271–282.
Foulis, D. J., and Randall, C. H. (1972). Operational statistics. I. Basic concepts,Journal of Mathematical Physics,13, 1667–1675.
Gross, H. (1990). Hilbert lattices: New results and unsolved problems,Foundations of Physics,20, 529–559.
Gudder, S. P. (1988).Quantum Probability, Academic Press, New York.
Holland, S. S., Jr. (1995). Orthomodularity in infinite dimensions; a theorem of M. Solèr,Bulletin of the American Mathematical Society (New Series),32, 205–234.
Keller, H. A. (1980). Ein nicht-klassischer Hilbertischen Raum,Mathematische Zeitschrift,172, 41–49.
Kolmogorov, A. N. (1933).Grundbegriffe der Wahrscheinlichkeitsrechnung, Berlin.
Maeda, F., and Maeda, S. (1970).Theory of Symmetric Lattices, Springer-Verlag, Berlin.
Morales, P., and Garcia-Mazaria, C. (n.d.). The support of a measure in ordered topological groups,Atti Seminario Matematico e Fisico Universita di Modena, to appear.
Morash, R. P. (1973). Angle bisection and orthoautomorphisms in Hilbert lattices,Canadian Journal of Physics,25, 261–271.
Piron, C. (1976).Foundations of Quantum Physics, Benjamin, Reading, Massachusetts.
Piziak, R. (1992). Orthostructures from sesquilinear forms. A prime,International Journal of Theoretical Physics,31, 871–879.
Solèr, M. P. (1995). Characterization of Hilbert spaces by orthomodular spaces,Communications in Algebra,23, 219–243.
Varadarajan, V. S. (1968).Geometry of Quantum Theory, Van Nostrand, Princeton.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Dvurečenskij, A. Test spaces and characterizations of quadratic spaces. Int J Theor Phys 35, 2093–2106 (1996). https://doi.org/10.1007/BF02302229
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02302229