×

zbMATH — the first resource for mathematics

A variant of Wigner’s functional equation. (English) Zbl 1321.39025
The article deals with a new proof of the known Wigner’s theorem about mappings \(f:\;{\mathcal H} \to {\mathcal K}\) between inner product spaces \({\mathcal H}\) and \({\mathcal K}\) over the fields \({\mathbb R}\), \({\mathbb C}\), and \({\mathbb H}\). Two mappings \(f\) and \(g\) between \({\mathcal H}\) and \({\mathcal K}\) are phase equivalent if \(f(x) = \sigma(x)g(x)\) with \(|\sigma(x)| = 1\). Wigner’s theorem states that a mapping \(f\) satisfying the equation \[ |\langle f(x),f(y) \rangle| = |\langle x,y \rangle| \] is phase equivalent to a linear isometry in the case \({\mathbb F} = {\mathbb R}\) and \(\dim {\mathcal H} \geq 2\), is phase equivalent to either a linear isometry or to an anti-linear isometry in the case \({\mathbb F} = {\mathbb C}\) and \(\dim {\mathcal H} \geq 2\), and is phase equivalent to a linear isometry in the case \({\mathbb F} = {\mathbb H}\) and \(\dim {\mathcal H} \geq 3\). The proof presented in the article is simpler (the author’s opinion) than earlier known proofs.

MSC:
39B05 General theory of functional equations and inequalities
46C05 Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product)
46C50 Generalizations of inner products (semi-inner products, partial inner products, etc.)
47J05 Equations involving nonlinear operators (general)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bakić, D.; Guljaš, B., Wigner’s theorem in Hilbert \(C\)\^{*}-modules over \(C\)\^{*}-algebras of compact operators, Proc. Am. Math. Soc., 130, 2343-2349, (2002) · Zbl 1067.46052
[2] Bargmann, V., Note on wigner’s theorem on symmetry operations, J. Math. Phys., 5, 862-868, (1964) · Zbl 0141.23205
[3] Cohn P.M.: Algebra, vol. 3, 2nd edn. Wiley, Chichester (1991)
[4] Freed, D.S., On wigner’s theorem, Geom. Topol. Monogr., 18, 83-89, (2012) · Zbl 1259.81033
[5] Gehér, Gy.P., An elementary proof for the non-bijective version of wigner’s theorem, Phys. Lett. A, 378, 2054-2057, (2014) · Zbl 1331.46066
[6] Győry, M., A new proof of wigner’s theorem, Rep. Math. Phys., 54, 159-167, (2004) · Zbl 1161.81381
[7] Lomont, J.S.; Mendelson, P., The Wigner unitary-antiunitary theorem, Ann. Math., 78, 548-559, (1963) · Zbl 0194.15204
[8] Molnár, L., An algebraic approach to wigner’s unitary-antiunitary theorem, J. Austral. Math. Soc. Ser. A, 65, 354-369, (1998) · Zbl 0943.46033
[9] Rätz, J., On wigner’s theorem: remarks, complements, comments, and corollaries, Aequationes Math., 52, 1-9, (1996) · Zbl 0860.39033
[10] Šemrl, P., Generalized symmetry transformations on quaternionic indefinite inner product spaces: an extension of quaternionic version of wigner’s theorem, Commun. Math. Phys., 242, 579-584, (2003) · Zbl 1053.46012
[11] Sharma, C.S.; Almeida, D.F., 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
[12] Sharma, C.S.; Almeida, D.F., Additive isometries on a quaternionic Hilbert space, J. Math. Phys., 31, 1035-1041, (1990) · Zbl 0722.46009
[13] Sharma, C.S.; Almeida, D.F., The first mathematical proof of wigner’s theorem, J. Nat. Geom., 2, 113-123, (1992) · Zbl 0802.46033
[14] Uhlhorn, U., Representation of symmetry transformations in quantum mechanics, Ark. Fys., 23, 307-340, (1963) · Zbl 0108.21805
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.