Applications of geometric logic to topos approaches to quantum theory

量子物理学的一个深层次谜团是其内在的非决定论。对量子系统的测量结果具有随机性，不能解释为仅仅代表了我们对系统处于什么精确状态的不确定知识。从技术上讲(科兴-斯派克定理)，在大多数量子系统中，不可能在数学上描述一致且明确地说出每一种可能的测量将赋予什么值的经典状态。理解这一点的一种方法是帝国理工学院的伊沙姆的新现实主义，最近与杜林合作，也被兰德斯曼、斯皮特斯和海宁在奈梅亨研究。有一些方法可以经典地(用经典的状态)来看待系统，但没有一种方法描述了所有可能的测量，而且它们也不能连贯地结合在一起。伊沙姆的见解是，由此产生的系统逻辑可以总体上描述为一种非标准的内部逻辑，它根据采用的经典观点而有所不同，这种非标准的内部逻辑源于一种称为拓朴的数学结构--它包括基础空间上的波束，在这些量子应用中，基空间的点包括那些经典的观点。现在，逻辑问的不是某事是否属实，而是从哪里--从哪个角度看。在非标准逻辑中，量子系统呈现经典状态。然而，退回到标准逻辑，经典状态不能一直保留--尽管它们的概率分布可以--这就是我们在量子物理中看到的。拓扑态的内部逻辑--以及相应的数学--可能很难处理。有些标准原则行不通。此外，通常的拓扑空间点集思想(一组点和一些指定为开放的子集)必须被描述独立于点的开放的无点方法取代。这些点是随后构建的，尽管它们可能太少，以至于无法通过它们的点来唯一地区分开口处。它是在纯数学中发展起来的，已经被发现在一系列非标准逻辑中给出了很好的结果，也被应用到计算机科学中，与计算机程序上的观察理论有关的开放。直接在内部逻辑中使用无点拓扑是技术上的和困难的。然而(乔亚尔/蒂尔尼)，它们可以等价地被视为基础空间上的无点束-也就是说，从另一个空间到基础空间的地图。把一个映射称为一个丛，就是把它看作一个可变空间--对于基点的每个点，我们在它上面有一根纤维，在地图下面是那个点的逆像，随着这个点的变化，它的纤维也不同。理想情况下，我们关于内部无点空间的内在推理也应该适用于纤维，但这只适用于内部逻辑的某个几何片段。从技术上讲，束上的几何结构是那些通过束拉回而保留的几何结构，这覆盖了纤维。通过仔细的逻辑解释，几何推理也可以有效地通过无点空间的点，尽管它们可能是不足的。提出者开发了几何推理技术，特别是对幂区域(无点超空间或空间空间)的利用。该项目旨在利用Topos方法中的几何技术来研究量子物理，用更熟悉的拓扑概念-点、丛、纤维-而不是内部无点空间来重新表达它。其目标是使TOPO方法更容易为物理学家所接受，并有助于澄清其与其他物理形式主义的关系。它也是测试几何的一般数学范围的一个很好的案例研究。

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.