Comparing intuitionistic quantum logics From orthomodular lattices to frames

The aim of this dissertation is to explore two different forms of intuitionistic quantum logic, one due to Isham, Butterfield and Döring, and the second due to Coecke. The two schemes have very different philosophical and physical motivations, and we explore how this leads to differences in the lattice extensions that they introduce. Isham, Butterfield and Döring suggested the so-called topos approach in which projections on a Hilbert space, or more generally in a von Neumann algebra, are mapped into a frame of subobjects of a particular presheaf by a map called daseinisation. Here, we present a factorization of the daseinisation embedding via an intermediate frame. The known properties of daseinisation are then recovered from this factorization. The relationship between our factorization of daseinisation and a free lattice construction is also considered. We also show that the frame generated by the codomain of daseinisation is in general strictly smaller than the frame of clopen subobjects considered to represent the quantum logic in this construction. Central to the Coecke approach is the injective hull of a meet semilattice, originally described by Bruns and Lakser, and key to this construction are sets within a meet semilattice with distributive joins. The properties of these sets are analyzed in detail, primarily from a geometric perspective. We give a geometric characterization of sets with distributive joins in the lattice of projections on an arbitrary Hilbert space. By abstracting to an order theoretic viewpoint, this result is then extended to a characterization of such sets in a large class of complete lattices, in terms of their completely join irreducible elements. Exploiting our factorization of daseinisation, we show the Coecke construction relates to the so called “inner daseinisation” of the topos approach, and symmetrically, daseinisation is related to the order theoretic dual of the Coecke construction. In the case of finite lattices, the topos construction is shown to be larger in size than the lattice of the Coecke approach. The question of universal properties for both schemes is also investigated, and a variety of adjunctions involving the two embeddings are described.