John Harding: Orthomodular structures of decompositions
The key feature of a closed subspace of a Hilbert space is that it allows the vectors of the space to be represented as ordered pairs of vectors, one from the subspace and one from its orthogonal complement. So closed subspaces of a Hilbert space give direct product decompositions of the space, and conversely, every direct product decomposition of a Hilbert space gives a pair of complementary closed subspaces.
It turns out that the direct product decompositions of any set, vector space, group, ring, topological space, or uniform space naturally form an orthomodular poset, much as the closed subspaces of a Hilbert space form an orthomodular poset. Further, many common instances of orthomodular posets arise as such orthomodular posets of direct product decompositions.
In this talk we survey results of the past twenty years relating orthomodular posets of direct product decompositions to aspects of the quantum logic program. This includes the study of regularity, states, automorphisms, and their role in basic axiomatics. Also, as direct products have a standard categorical formulation, orthomodular posets of decompositions provide a means to link some of the recent categorical approaches to quantum mechanics to the quantum logic program.