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We consider the standard quantum logic ${\mathcal L}(H)$ associated to acomplex Hilbert space $H$, i.e. the lattice of closed subspaces of $H$ togetherwith the orthogonal complementation. The orthogonality and compatibilityrelations are defined for any logic. In the standard quantum logic, they have asimple interpretation in terms of operator theory. For example, two closedsubspaces (propositions in the logic ${\mathcal L}(H)$) are compatible if andonly if the projections on these subspaces commute. We present both classicaland more resent results on transformations of ${\mathcal L}(H)$ and theassociated Grassmannians which preserve the orthogonality or compatibilityrelation. The first result in this direction was classical Wigner's theorem.

Preserver problems for the logics associated to Hilbert spaces and related Grassmannians