# zbMATH — the first resource for mathematics

An elementary proof for the non-bijective version of Wigner’s theorem. (English) Zbl 1331.46066
Summary: The non-bijective version of Wigner’s theorem states that a map which is defined on the set of self-adjoint, rank-one projections (or pure states) of a complex Hilbert space and which preserves the transition probability between any two elements, is induced by a linear or antilinear isometry. We present a completely new, elementary and very short proof of this famous theorem which is very important in quantum mechanics. We do not assume bijectivity of the mapping or separability of the underlying space like in many other proofs.

##### MSC:
 46N50 Applications of functional analysis in quantum physics 46C05 Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product)
Full Text:
##### References:
 [1] Wigner, E. P., Gruppentheorie und ihre anwendung auf die quantenmechanik der atomspektrum, (1931), Fredrik Vieweg und Sohn · JFM 57.1578.03 [2] Uhlhorn, U., Representation of symmetry transformations in quantum mechanics, Ark. Fys., 23, 307-340, (1963) · Zbl 0108.21805 [3] Streater, R. F.; Wightman, A. S., PCT, spin and statistics, and all that, (1964), W.A. Benjamin, Inc. New York, Amsterdam · Zbl 0135.44305 [4] Lomont, J. S.; Mendelson, P., The Wigner unitary-antiunitary theorem, Ann. Math., 78, 548-559, (1963) · Zbl 0194.15204 [5] Bargmann, V., Note on Wigner’s theorem on symmetry operations, J. Math. Phys., 5, 862-868, (1964) · Zbl 0141.23205 [6] Casinelli, G.; de Vito, E.; Lahti, P.; Levrero, A., Symmetry groups in quantum mechanics and the theorem of Wigner on the symmetry transformations, Rev. Math. Phys., 8, 921-941, (1997) · Zbl 0907.46055 [7] Chevalier, G., Wigner’s theorem and its generalizations, (Handbook of Quantum Logic and Quantum Structures, (2007), Elsevier Sci. B.V. Amsterdam), 429-475 · Zbl 1126.81031 [8] Győry, M., A new proof of Wigner’s theorem, Rep. Math. Phys., 54, 159-167, (2004) · Zbl 1161.81381 [9] Molnár, L., Wigner’s unitary-antiunitary theorem via Herstein’s theorem on Jordan homeomorphisms, J. Nat. Geom., 10, 137-148, (1996) · Zbl 0858.46019 [10] Molnár, L., An algebraic approach to Wigner’s unitary-antiunitary theorem, J. Aust. Math. Soc., 65, 354-369, (1998) · Zbl 0943.46033 [11] Rätz, J., On Wigner’s theorem: remarks, complements, comments and corollaries, Aequ. Math., 52, 1-9, (1996) · Zbl 0860.39033 [12] Sharma, C. S.; Almeida, D. L., A direct proof of Wigner’s theorem on maps which preserve transition probabilities between pure states of quantum systems, Ann. Phys., 197, 300-309, (1990) · Zbl 0875.46008 [13] Simon, R.; Mukunda, N.; Chaturvedi, S.; Srinivasan, V., Two elementary proofs of the Wigner theorem on symmetry in quantum mechanics, Phys. Lett. A, 372, 6847-6852, (2008) · Zbl 1227.81189 [14] Mouchet, A., An alternative proof of Wigner theorem on quantum transformations based on elementary complex analysis, Phys. Lett. A, 377, 2709-2711, (2013) · Zbl 1301.81061 [15] Botelho, F.; Jamison, J.; Molnár, L., Surjective isometries on Grassmann spaces, J. Funct. Anal., 265, 10, 2226-2238, (2013) · Zbl 1301.47100 [16] Bau, S.; Beardon, A. F., The metric dimension of metric spaces, Comput. Methods Funct. Theory, 13, 2, 295-305, (2013) · Zbl 1282.51004 [17] Conway, J. B., A course in functional analysis, Grad. Texts Math., vol. 96, (1985), Springer-Verlag New York · Zbl 0558.46001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.