Two projections commute if and only if their images are compatible. Using combinatorial methods wedescribe compatibility preserving bijective transformations of Grassmannians. In some cases, these transformations form a class greater than the class of transformations induced by unitary and anti-unitary operators.

Classical Wigner’s theorem characterizes unitary and anti-unitary operators as symmetries of pure states of quantum mechanical systems, i.e. rank one projections. We consider a non-injective version of Wigner’s theorem as well as Uhlhorn’s version concerning orthogonality preserving transformations and describe variousextensions of these results onto other Grassmannians.

This is a brief description of basic properties of the lattice formed by all subspaces of a vector space and the orthomodular lattice consisting of all closed subspaces of a complex Hilbert space. The first lattice is investigated in classical projective geometry, the second is related to the logical structure of quantum mechanical systems.

This final chapter provides some applications of Wigner’s theorem and its generalizations described in Chapter 4. The first is classical Kadison’s theorem concerning automorphisms of the convex set of all bounded positive operators of trace one. In the second section, we consider the real vector space formed by all self-adjoint operators of finite rank and investigate linear transformations sending projections of fixed rank $k$ to projections of rank $k$ as well as linear transformations which map projections of a fixed rank to projections of other fixed rank.

We present two closely connected classical results: a description of isomorphisms between the lattices of closed subspaces of infinite-dimensional normed spaces and Kakutani--Mackey’s theorem which characterizes the orthomodular lattices of infinite-dimensional complex Hilbert spaces.The final part of the chapter concerns extensions of order preserving transformations of Hilbert Grassmannians to isomorphisms of the corresponding lattices.

The chapter provides some geometrical characterizations of semilinear isomorphisms between vector spaces of an arbitrary (not necessarily finite) dimension: a modern version of the Fundamental Theorem of Projective Geometry, Chow’s theorem and its generalizations, apartment preserving transformations of Grassmannians. These results are useful tools for ourinvestigation of preserver problems related to quantum mechanics.