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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

Abstract

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.

Keywords

Wigner theorem  projective geometry 

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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

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