## Symmetries of projective spaces and spheres
Geher, G.
(2018)
It is advisable to refer to the publisher's version if you intend to cite from this work. See Guidance on citing. To link to this item DOI: 10.1093/imrn/rny100 ## Abstract/SummaryLet $H$ 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.
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