Applications of geometric logic to topos approaches to quantum theory

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A deep mystery of quantum physics is its inherent non-determinism. The outcome of a measurement on a quantum system has a randomness that cannot be explained away as representing just our uncertain knowledge of what precise state the system is in. Technically (the Kochen-Specker Theorem), there is mathematically no possibility in most quantum systems of describing classical states that consistently, and unequivocally, say what value every possible measurement would give.One approach to understanding this is the neo-realism of Isham at Imperial College, recently with Doering, and taken up also by Landsman, Spitters and Heunen at Nijmegen. There are ways of seeing the system classically (with classical states), but none describes all possible measurements and they cannot be fitted together coherently. Isham's insight is that the resulting logic of systems, which varies according to which classical viewpoint is adopted, can be described overall as a non-standard internal logic arising out of a mathematical structure known as a topos - comprising the sheaves over a base space whose points in these quantum applications include those classical viewpoints. Now logic asks not whether something is true, but where - from which points of view. In the non-standard logic, the quantum system appears classical and has classical states. Withdrawing to standard logic, however, the classical states cannot consistently be retained - although their probabilistic distributions can and these are what we see in quantum physics.The internal logic - and corresponding mathematics - of toposes can be difficult to work with. Some standard principles don't work. Also, the usual point-set idea of topological space (a set of points together with some subsets specified as open ) must be replaced by a point-free approach that describes the opens independently of points. The points are constructed subsequently, although there may be too few of them for the opens to be uniquely distinguished by their points. It was developed in pure mathematics, has been found to give excellent results with a range of non-standard logics, and has also been applied in computer science, with the opens related to theories of observations on computer programs.Working with the point-free topologies directly in the internal logic is technical and difficult. However (Joyal/Tierney), they can equivalently be viewed as point-free bundles over the base space - that is to say, maps from another space to the base. In referring to a map as a bundle, one is thinking of it as a variable space - for each point of the base, we have a fibre over it, the inverse image of that point under the map, and as the point varies so too does its fibre.Ideally, our internal reasoning about internal point-free spaces should also apply to the fibres, but this true only for a certain geometric fragment of the internal logic. Technically, the geometric constructions on the bundles are those that are preserved by bundle pullback, and this covers the fibres. By careful interpretation of logic, geometric reasoning also can work validly through the points of the point-free spaces, despite the possible shortage of them. Techniques of geometric reasoning have been developed by the proposer, with particular exploitation of powerlocales (point-free hyperspaces, or spaces of spaces).The project aims to exploit those geometricity techniques in the topos approach to quantum physics, reexpressing it in terms of more familiar topological concepts - points, bundles, fibres - instead of internal point-free spaces. The goal is to make the topos approach more accessible to physicists and help clarify its relationship with other physics formalisms. It is also an excellent case study for testing out the general mathematical scope of geometricity.