Abstract/Summary

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 $\alpha\in\left(0,\tfrac{\pi}{2}\right)$. The purpose of this paper is to characterise all bijective transformations on the projective space $P(H)$ which preserve the quantum angle $\alpha$ (or Fubini-Study distance $\alpha$) 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 $H=\mathbb{C}^2$ 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 such transformations for $\alpha\leq\tfrac{\pi}{4}$. 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 $\cos^2\alpha$ in both directions is a Wigner symmetry (thus automatically preserves all transition probabilities), except for the case when $H=\mathbb{C}^2$ and $\alpha = \tfrac{\pi}{4}$ where an additional possibility occurs. (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 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.