| Topos theory and noncommutative geometry are areas of modern mathematics that may both be seen as vast extensions of topology, each providing its own generalized notion of space. In topos theory one regards the so-called locales of lattice theory as spaces, whereas the C*-algebras of functional analysis define spaces in the noncommutative sense. The aim of this proposal is to relate these different notions of space to each other and to quantum theory. That this can be done in principle has recently been shown by the applicant in collaboration with C. Heunen and B. Spitters: given some C*-algebra A, we defined a locale L(A) in a certain topos T(A) with various interesting properties, including computability in principle. First, at least for certain classes of C*-algebras this would provide interesting new examples of locales, whose structure can be analyzed using the different perspectives and toolkits of topos theory and noncommutative geometry. For example, the K-theoretic invariants of the C*-algebra A should define corresponding invariants of the locale L(A), and more generally anything that can be said about A will have repercussions for L(A). Second, regarding the locale L(A) as a quantum phase space, the logical structure of the quantum system described by the C*-algebra A can be studied using the internal logic of topos theory. Since the latter happens to be intuitionistic, this proposal is also explicitly meant as a contribution to the emerging field of intuitionistic quantum logic (which is intended to clarify---if not replace---the bizarre nondistributive quantum logic proposed by Birkhoff and von Neumann in 1936). We request funding for one PhD student (4 years) at a total budget of 193.495 euro's. |
