Wigner-Type Theorems for Hilbert Grassmannians

"Wigner's theorem (67) provides a geometric characterization of unitary and anti-unitary operators as transformations of the set of rays of a complex Hilbert space, or equivalently, rank one projections. This statement plays an important role in mathematical foundations of quantum mechanics (11; 50; 63), since rays (rank one projections) can be identified with pure states of quantum mechanical systems. We present various types of extensions of Wigner's theorem onto Hilbert Grassmannians and their applications. Most of these results were obtained after 2000, but to completeness of the exposition we include some classical theorems closely connected to the main topic (for example, Kakutani-Mackey's result on the lattice of closed subspaces of a complex Banach space (31), Kadison's theorem on transformations preserving the convex structure of the set of states of quantum mechanical systems (30)). We use geometric methods related to the Fundamental Theorem of Projective Geometry and results in spirit of Chow's theorem (13)"--