Symmetry and Topology in Quantum Logic

The primordial quantum logic, the projection lattice of a Hilbert space, carries a rich topological and covariant structure. We might reasonably wish to equip abstract quantum logics – orthomodular lattices and posets, orthoalgebras, etc. – with similar structure. Topological OMLs (TOMLs) have already been studied in the literature, but the projection lattice of a Hilbert space is not a TOML: rather, it’s a topological orthoalgebra (TOA). After presenting some general results on TOAs, I’ll show how one can construct highly symmetric TOAs from compact groups. Time allowing, I’ll conclude with a selection of open problems.