International Journal of Theoretical Physics

, Volume 35, Issue 10, pp 2093–2106 | Cite as

Test spaces and characterizations of quadratic spaces

  • Anatolij Dvurečenskij


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.


Hilbert Space Field Theory Elementary Particle Quantum Field Theory Nonzero Vector 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Amemiya, I., and Araki, H. (1966/67). A remark on Piron's paper,Publications.Research Institute for Mathematical Sciences Series A,2, 423–427.MathSciNetGoogle Scholar
  2. 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.MathSciNetGoogle Scholar
  3. Dvurečenskij, A. (1993).Gleason's Theorem and Its Applications, Kluwer, Dordrecht, and Ister Science Press, Bratislava.Google Scholar
  4. Dvurečenskij, A., and Pulmannov'a, S. (1994). Test Spaces, Dacey Spaces, and Completeness of inner product spaces,Letters in Mathematical Physics,32, 299–306.Google Scholar
  5. Foulis, D. J., and Bennett, M. K. (1993). Tensor products of orthoalgebras,Order,10, 271–282.CrossRefMathSciNetGoogle Scholar
  6. Foulis, D. J., and Randall, C. H. (1972). Operational statistics. I. Basic concepts,Journal of Mathematical Physics,13, 1667–1675.CrossRefMathSciNetGoogle Scholar
  7. Gross, H. (1990). Hilbert lattices: New results and unsolved problems,Foundations of Physics,20, 529–559.CrossRefMathSciNetGoogle Scholar
  8. Gudder, S. P. (1988).Quantum Probability, Academic Press, New York.Google Scholar
  9. 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.zbMATHMathSciNetGoogle Scholar
  10. Keller, H. A. (1980). Ein nicht-klassischer Hilbertischen Raum,Mathematische Zeitschrift,172, 41–49.zbMATHMathSciNetGoogle Scholar
  11. Kolmogorov, A. N. (1933).Grundbegriffe der Wahrscheinlichkeitsrechnung, Berlin.Google Scholar
  12. Maeda, F., and Maeda, S. (1970).Theory of Symmetric Lattices, Springer-Verlag, Berlin.Google Scholar
  13. 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.Google Scholar
  14. Morash, R. P. (1973). Angle bisection and orthoautomorphisms in Hilbert lattices,Canadian Journal of Physics,25, 261–271.zbMATHMathSciNetGoogle Scholar
  15. Piron, C. (1976).Foundations of Quantum Physics, Benjamin, Reading, Massachusetts.Google Scholar
  16. Piziak, R. (1992). Orthostructures from sesquilinear forms. A prime,International Journal of Theoretical Physics,31, 871–879.CrossRefzbMATHMathSciNetGoogle Scholar
  17. Solèr, M. P. (1995). Characterization of Hilbert spaces by orthomodular spaces,Communications in Algebra,23, 219–243.zbMATHMathSciNetGoogle Scholar
  18. Varadarajan, V. S. (1968).Geometry of Quantum Theory, Van Nostrand, Princeton.Google Scholar

Copyright information

© Plenum Publishing Corporation 1996

Authors and Affiliations

  • Anatolij Dvurečenskij
    • 1
  1. 1.Mathematical InstituteSlovak Academy of SciencesBratislavaSlovakia

Personalised recommendations