Generalized symmetry transformations on quaternionic indefinite inner product spaces: An extension of quaternionic version of Wigner's theorem
Recall Wigner's famous unitary-antiunitary theorem, which plays a fundamental role in quantum physics and reads in the mathematical language as follows: If \(\Phi\) is a bijective transformation on the set of all one-dimensional linear subspaces of a complex Hilbert space \({\mathcal H}\) preserving the angle between every pair of such subspaces (transition probability), then there exists a unitary or an antiunitary operator \(U:{\mathcal H}\to{\mathcal H}\) such that \(\Phi (L)=\{Ux\,| \,x\in L\}\) for every one-dimensional linear subspace \(L\subset{\mathcal H}\). If \(\text{ dim}({\mathcal H})\geq 3\), Uhlhorn proved the above under the weaker assumption that \(\Phi\) only preserves the orthogonality between the one-dimensional linear subspaces of \({\mathcal H}\). Since the indefinite inner product spaces represent a more useful mathematical setting for describing several physical problems than definite ones, the results of Wigner and Uhlhorn were generalized to the more general setting of indefinite inner product spaces in [\textit{L. Molnár}, J. Funct. Anal. 194, 248--262 (2002; Zbl 1010.46023)]. The paper under review is concerned with quaternionic Hilbert spaces (left vector spaces over the skew-field of quaternions together with a positive definite inner product \(\langle.,.\rangle\) which makes the resulting normed linear space complete). The author proves Molnár's result for the case of quaternionic Hilbert spaces, i.e., the generalization of Uhlhorn's result to the case of quaternionic Hilbert spaces with indefinite metric. The proof of that result is based on different methods than those used by Molnár, and it leads also to a shorter proof of Molnár's result.
scientific article; zbMATH DE number 2051481
scientific article