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.
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References
Artin, E.: Geometric Algebra. Interscience Publ., New York (1957)
Bargmann, V.: Note on Wigner’s theorem on symmetry operations. J. Math. Phys. 5(7), 862–868 (1964)
Bracci, L., Morchio, G., Strocchi, F.: Wigner’s theorem on symmetries in indefinite metric spaces. Commun. Math. Phys. 41, 289–299 (1975)
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)
Emch, G., Piron, C.: Symmetry in quantum theory. J. Math. Phys. 4(4), 469–473 (1963)
Hagedorn, R.: Note on symmetry operations in quantum mechanics. Nuovo Cimento XII(X), 553–566 (1959)
Keller, K.J.: Über die Rolle der projektiven Geometrie in der Quantenmechanik. Master’s thesis, Johannes Gutenberg-Universität Mainz, http://wwwthep.physik.uni-mainz.de/Publications/theses/dip-keller.pdf (2006)
Klein, F.: Elementarmathematik vom höheren Standpunkte aus II. In: Die Grundlagen der mathematischen Wissenschaften in Einzeldarstellungen, vol. XV, pp. 96–98. Springer, Berlin (1925)
Lomont, J.S., Mendelson, P.: The wigner unitary-antiunitary theorem. Ann. Math. 78(3), 548–559 (1963)
Uhlhorn, U.: Representation of symmetry transformations in quantum mechanics. Ark. Fysik 23(30), 307–340 (1962)
Varadarajan, V.S.: Geometry of Quantum Theory. The University Series in Higher Mathematics, vol. 1. D. Van Nostrand Company, Inc., Princeton, NJ (1968)
Weinberg, S.: The Quantum Theory Of Fields, vol. 1. University Of Cambridge, Cambridge (1995)
Wigner, E.P.: Gruppentheorie und ihre Anwendung auf die Quantenmechanik der Atomspektren. Vieweg, Braunschweig (1931)
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Keller, K.J., Papadopoulos, N.A. & Reyes-Lega, A.F. On the realization of symmetries in quantum mechanics . Math. Semesterber. 55, 149–160 (2008). https://doi.org/10.1007/s00591-008-0035-5
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DOI: https://doi.org/10.1007/s00591-008-0035-5