  Let $H$ be a complex Hilbert space and let ${\mathcal F}_{s}(H)$ be the real
vector space of all self-adjoint finite rank operators on $H$. We prove the
following non-injective version of Wigner's theorem: every linear operator on
${\mathcal F}_{s}(H)$ sending rank one projections to rank one projections
(without any additional assumption) is either induced by a linear or
conjugate-linear isometry or constant on the set of rank one projections.
