Orthogonality and quantum geometry: Towards a relational reconstruction of quantum theory

This thesis is an in-depth mathematical study of the non-orthogonality relation between the (pure) states of quantum systems. In Chapter 2, I define quantum Kripke frames, the protagonists of this thesis. A quantum Kripke frame is a Kripke frame in which the binary relation possesses some simple properties of the non-orthogonality relation in quantum theory. The structure of quantum Kripke frames is studied extensively from a geometric perspective. In the meantime, several kinds of projective geometries are discovered to be Kripke frames in disguise. In Chapter 3, maps between quantum Kripke frames are studied. I define continuous homomorphisms between quantum Kripke frames and study extensively their properties. Chapter 4 concerns the automated reasoning of quantum Kripke frames. I prove that the first-order theory of quantum Kripke frames is undecidable. Moreover, I characterize the first-order definable, bi-orthogonally closed subsets in a special kind of quantum Kripke frame. Chapter 5 is a pilot study of the transition probabilities between the states of quantum systems. They are the quantitative, more fine-grained version of the non-orthogonality relation. I define probabilistic quantum Kripke frames and quantum transition probability spaces, whose definitions capture some essential properties of the transition probabilities in quantum theory. Some elementary but useful results about these two kinds of structures are proved.