A geometric approach to Wigner-type theorems

TL;DR

This paper proves a Wigner-type theorem for orthogonality-preserving transformations on projective spaces of complex Hilbert spaces, showing they are induced by unitary or anti-unitary operators, with applications to Grassmannian transformations.

Contribution

It introduces a geometric approach to Wigner-type theorems, extending results to infinite-dimensional spaces and characterizing transformations preserving principal angles.

Findings

Orthogonality-preserving transformations are induced by unitary or anti-unitary operators in finite dimensions.

Lineations preserving orthogonality are induced by linear or conjugate-linear isometries in general.

Applications include descriptions of transformations of Grassmannians preserving principal angles.

Abstract

Let $H$ be a complex Hilbert space and let ${\mathcal P}(H)$ be the associated projective space (the set of rank-one projections). Suppose that $\dim H\ge 3$. We prove the following Wigner-type theorem: if $H$ is finite-dimensional, then every orthogonality preserving transformation of ${\mathcal P}(H)$ 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 ${\mathcal P}(H)$ 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.