  The aim of this paper is to give a simple, geometric proof of Wigner's
theorem on the realization of symmetries in quantum mechanics that clarifies
its relation to projective geometry. Although several proofs exist already, it
seems that the relevance of Wigner's theorem is not fully appreciated in
general. It is Wigner's theorem which allows the use of linear realizations of
symmetries and therefore guarantees that, in the end, quantum theory stays a
linear theory. In the present paper, we take a strictly geometrical point of
view in order to prove this theorem. It becomes apparent that Wigner's theorem
is nothing else but a corollary of the fundamental theorem of projective
geometry. In this sense, the proof presented here is simple, transparent and
therefore accessible even to elementary treatments in quantum mechanics.
