2017/02/10 by Mark Pankov, Pankov, Mark
Computer Science · Mathematics · #Advanced Algebra and Logic #Advanced Topics in Algebra #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Mathematical Physics (math-ph) #Matrix Theory and Algorithms
paper · pdf · doi:10.48550/arxiv.1702.03157
openalex publication_date 2017/02/10 · openalex created_date 2017/03/03 · openalex updated_date 2026/07/28
We consider the standard quantum logic \mathcal L(H) associated to a complex Hilbert space H, i.e. the lattice of closed subspaces of H together with the orthogonal complementation. The orthogonality and compatibility relations are defined for any logic. In the standard quantum logic, they have a simple interpretation in terms of operator theory. For example, two closed subspaces (propositions in the logic \mathcal L(H)) are compatible if and only if the projections on these subspaces commute. We present both classical and more resent results on transformations of \mathcal L(H) and the associated Grassmannians which preserve the orthogonality or compatibility relation. The first result in this direction was classical Wigner's theorem.