Mathematische Semesterberichte

, Volume 55, Issue 2, pp 149–160 | Cite as

On the realization of symmetries in quantum mechanics

  • Kai Johannes KellerEmail author
  • Nikolaos A. Papadopoulos
  • Andrés F. Reyes-Lega
Forschung, Lehre und Anwendung


The aim of this paper is to give a simple, geometric proof of Wigner’s theorem on the realization of symmetries in quantum mechanics that clarifies its relation to projective geometry. Although several proofs exist already, it seems that the relevance of Wigner’s theorem is not fully appreciated in general. It is Wigner’s theorem which allows the use of linear realizations of symmetries and therefore guarantees that, in the end, quantum theory stays a linear theory. In the present paper, we take a strictly geometrical point of view in order to prove this theorem. It becomes apparent that Wigner’s theorem is nothing else but a corollary of the fundamental theorem of projective geometry. In this sense, the proof presented here is simple, transparent and therefore accessible even to elementary treatments in quantum mechanics.


Wigner theorem  projective geometry 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Artin, E.: Geometric Algebra. Interscience Publ., New York (1957)zbMATHGoogle Scholar
  2. 2.
    Bargmann, V.: Note on Wigner’s theorem on symmetry operations. J. Math. Phys. 5(7), 862–868 (1964)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bracci, L., Morchio, G., Strocchi, F.: Wigner’s theorem on symmetries in indefinite metric spaces. Commun. Math. Phys. 41, 289–299 (1975)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Cassinelli, G., DeVito, E., Lahti, P.J., Levrero, A.: Symmetry groups in quantum mechanics and the theorem of Wigner on the symmetry transformations. Rev. Math. Phys. 9(8), 921–941 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Emch, G., Piron, C.: Symmetry in quantum theory. J. Math. Phys. 4(4), 469–473 (1963)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Hagedorn, R.: Note on symmetry operations in quantum mechanics. Nuovo Cimento XII(X), 553–566 (1959)Google Scholar
  7. 7.
    Keller, K.J.: Über die Rolle der projektiven Geometrie in der Quantenmechanik. Master’s thesis, Johannes Gutenberg-Universität Mainz, (2006)Google Scholar
  8. 8.
    Klein, F.: Elementarmathematik vom höheren Standpunkte aus II. In: Die Grundlagen der mathematischen Wissenschaften in Einzeldarstellungen, vol. XV, pp. 96–98. Springer, Berlin (1925)Google Scholar
  9. 9.
    Lomont, J.S., Mendelson, P.: The wigner unitary-antiunitary theorem. Ann. Math. 78(3), 548–559 (1963)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Uhlhorn, U.: Representation of symmetry transformations in quantum mechanics. Ark. Fysik 23(30), 307–340 (1962)Google Scholar
  11. 11.
    Varadarajan, V.S.: Geometry of Quantum Theory. The University Series in Higher Mathematics, vol. 1. D. Van Nostrand Company, Inc., Princeton, NJ (1968)zbMATHGoogle Scholar
  12. 12.
    Weinberg, S.: The Quantum Theory Of Fields, vol. 1. University Of Cambridge, Cambridge (1995)Google Scholar
  13. 13.
    Wigner, E.P.: Gruppentheorie und ihre Anwendung auf die Quantenmechanik der Atomspektren. Vieweg, Braunschweig (1931)Google Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  • Kai Johannes Keller
    • 1
    Email author
  • Nikolaos A. Papadopoulos
    • 1
  • Andrés F. Reyes-Lega
    • 2
  1. 1.Inst. f. Physik (WA THEP)Johannes Gutenberg-Universität MainzMainzGermany
  2. 2.Departamento de FísicaUniversidad de los AndesBogotáColombia

Personalised recommendations