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Illinois Journal of Mathematics

Orthogonality preserving transformations on the set of $n$-dimensional subspaces of a Hilbert space

Peter Šemrl

Published Apr 1, 2004• 40 citations
Publisher

Abstract

We characterize bijective transformations on the set of all $n$-dimensional subspaces of a Hilbert space that preserve orthogonality in both directions. This extends Uhlhorn's improvement of Wigner's classical theorem on symmetry transformations.

DOI
10.1215/ijm/1258138399

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Orthogonality preserving transformations on the set of nnn-dimensional subspaces of a Hilbert space
We characterize bijective transformations on the set of all nnn-dimensional subspaces of a Hilbert space that preserve orthogonality in both directions. This extends Uhlhorn's improvement of Wigner's classical theorem on symmetry transformations.
Transformations on the set of all nnn-dimensional subspaces of a Hilbert space preserving orthogonality
It is well known that transformations of $ \mathbb{C}^n $ preserving the standard inner product are unitary transformations. In this paper, all bijective transformations of isotropic sets of $ \mathbb{C}P^{n} $ preserving $ H $-orthogonality in both directions, called $ H $-orthogonal transformations, have been determined. This is a generalization of Uhlhorn's version of Wigner's unitary-antiunitary theorem. The group of $ H $-orthogonal transformations on some other sets of $ \mathbb{C}P^{n} $ were also determined.
Let H be a complex Hilbert space and let P ( H ) be the associated projective space (the set of rank-one projections). Suppose dim H ⩾ 3 . We prove the following Wigner-type theorem: if H is finite dimensional, then every orthogonality preserving transformation of P ( H ) is induced by a unitary or anti-unitary operator. This statement will be obtained as a consequence of the following result: every orthogonality preserving lineation of P ( H ) to itself is induced by a linear or conjugate-linear isometry ( H is not assumed to be finite-dimensional). As an application, we describe (not necessarily injective) transformations of Grassmannians preserving some types of principal angles.
Abstract Let H be either a complex inner product space of dimension at least two or a real inner product space of dimension at least three, and let us fix an α∈(0,π2)\alpha \in (0,\tfrac{\pi }{2} )α∈(0,2π​). The purpose of this paper is to characterise all bijective transformations on the projective space P(H) which preserve the quantum angle α\alphaα (or Fubini–Study distance α\alphaα) between lines in both directions. (Let us emphasise that we do not assume anything about the preservation of other quantum angles). For real inner product spaces and when H=C2H=\mathbb{C}^2H=C2 we do this for every α\alphaα, and when H is a complex inner product space of dimension at least three we describe the structure of such transformations for α≤π4\alpha \leq \tfrac{\pi }{4}α≤4π​. Our result immediately gives an Uhlhorn-type generalisation of Wigner’s theorem on quantum mechanical symmetry transformations, that is considered to be a cornerstone of the mathematical foundations of quantum mechanics. Namely, under the above assumptions, every bijective map on the set of pure states of a quantum mechanical system that preserves the transition probability cos⁡2α\cos ^2\alphacos2α in both directions is a Wigner symmetry (thus automatically preserves all transition probabilities), except for the case when H=C2H=\mathbb{C}^2H=C2 and α=π4\alpha = \tfrac{\pi }{4}α=4π​ where an additional possibility occurs. (Note that the classical theorem of Uhlhorn is the solution for the α=π2\alpha = \tfrac{\pi }{2}α=2π​ case). Usually in the literature, results which are connected to Wigner’s theorem are discussed under the assumption of completeness of H; however, here we shall remove this unnecessary hypothesis in our investigation. Our main tool is a characterisation of bijective maps on unit spheres of real inner product spaces which preserve one spherical angle in both directions.