Symmetries of projective spaces and spheres

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. Let us fix an $\alpha\in \left(0,\tfrac{\pi}{2}\right)$. The purpose of this paper is to characterize all bijective transformations on the projective space $P(H)$ obtained from $H$ which preserves the angle $\alpha$ between lines in both directions. (We emphasize that we do not assume anything about other angles). For real inner product spaces and when $H=\mathbb{C}^2$ we do this for every $\alpha$, and when $H$ is a complex inner product space of dimension at least three we describe the structure of these transformations for $\alpha\leq\tfrac{\pi}{4}$. As an application, we give an Uhlhorn-type generalization of a famous theorem of Wigner which is considered to be a cornerstone of the mathematical foundations of quantum mechanics. Namely, we show that under the above assumptions, every bijective map on the set of pure states of a quantum mechanical system that preserves the transition probability $\cos^2\alpha$ in both directions is a Wigner symmetry (i.e. it automatically preserves all transition probability), except for the case when $H=\mathbb{C}^2$ and $\alpha = \tfrac{\pi}{4}$ where an additional possibility occurs. We note that the classical theorem of Uhlhorn is the solution for the $\alpha = \tfrac{\pi}{2}$ 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. Our main tools are a characterization of bijective maps on unit spheres of real inner product spaces which preserve an angle in both directions, and an extension of Uhlhorn's theorem for non-complete inner product spaces.