Summary
In this paper we present a unified treatment of Wigner's unitarity-antiunitarity theorem simultaneously in the real and the complex case. Its elementary nature, emphasized by V. Bargmann in 1964, is underlined here by removing unnecessary hypotheses, the most important being the completeness of the inner product spaces involved. At the end, we shall obtain connections to some recent results in geometry.
Similar content being viewed by others
References
Almeida, D. F. andSharma, C. S.,The first mathematical proof of Wigner's theorem. J. Natural Geom.2 (1992), 113–123.
Alpers, B. andSchröder, E. M.,On mappings preserving orthogonality of non-singular vectors. J. Geom.41 (1991), 3–15.
Alsina, C. andGarcia-Roig, J. L.,On continuous preservation of norms and areas. Aequationes Math.38 (1989), 211–215.
Alsina, C. andGarcia-Roig, J. L.,On the functional equation |T(x) ⋅ T(y)| = |x ⋅ y|. InConstantin Carathéodory: An international tribute (Th. M. Rassias, ed.). World Sci. Publ., Singapore 1991, pp. 47–52.
Bargmann, V.,Note on Wigner's theorem on symmetry operations. J. Math. Phys.5 (1964), 862–868.
Darboux, G.,Sur le théorème fondamental de la géométrie projective. Math. Ann.17 (1880), 55–61.
Lomont, J. S. andMendelson, P.,The Wigner unitarity-antiunitarity theorem. Ann. Math.78 (1963), 548–559.
Rassias, Th. M. andSharma, C. S.,Properties of isometries. J. Natural Geom.3 (1993), 1–38.
Rätz, J.,Remarks on Wigner's theorem. Aequationes Math.47 (1994), 288–289.
Schröder, E. M.,Vorlesungen über Geometrie, Band 3. Wissenschaftsverlag, Mannheim—Leipzig—Wien—Zürich, 1992.
Sharma, C. S. andAlmeida, D. F.,A direct proof of Wigner's theorem on maps which preserve transition probabilities between pure states of quantum systems. Ann. Phys.197 (1990), 300–309.
Uhlhorn, U.,Representation of symmetry transformations in quantum mechanics. Ark. Fys.23 (1963), 307–340.
Wigner, E.,Gruppentheorie und ihre Anwendung auf die Quantenmechanik der Atomspektren. Vieweg, Braunschweig, 1931.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Rätz, J. On Wigner's theorem: Remarks, complements, comments, and corollaries. Aeq. Math. 52, 1–9 (1996). https://doi.org/10.1007/BF01818323
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF01818323