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Applications of geometric logic to topos approaches to quantum theory

几何逻辑在量子理论拓扑方法中的应用

基本信息

批准号：
EP/G046298/1
负责人：
Steven Vickers
金额：
$ 52.33万
依托单位：
University of Birmingham
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2009
资助国家：
英国
起止时间：
2009 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FG046298%2F1
关键词：
Applications geometric logic topos approaches

项目摘要

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.

量子物理学的一个深层奥秘是它固有的非决定论。量子系统的测量结果具有随机性，不能解释为仅仅代表我们对系统精确状态的不确定知识。从技术上讲（柯钦-斯佩克定理），在数学上，大多数量子系统都不可能描述经典状态，从而一致地、毫不含糊地说明每种可能的测量结果会给出什么值。理解这一点的一种方法是帝国理工学院的Isham的新现实主义，最近与Doering合作，奈梅亨大学的Landsman、Spitters和Heunen也采用了这种方法。有一些方法可以经典地观察系统（具有经典状态），但没有一种方法可以描述所有可能的测量，而且它们不能连贯地组合在一起。Isham的见解是，系统的最终逻辑根据采用的经典视点而变化，总体上可以描述为由称为拓扑的数学结构产生的非标准内部逻辑——拓扑由基空间上的束组成，这些量子应用中的点包括那些经典视点。现在逻辑问的不是某物是否为真，而是在哪里——从哪个观点出发。在非标准逻辑中，量子系统表现为经典并具有经典状态。然而，撤回到标准逻辑，经典状态不能始终保持不变——尽管它们的概率分布可以，这就是我们在量子物理学中看到的。题目的内部逻辑和相应的数学是很难处理的。一些标准原则不起作用。此外，拓扑空间中通常的点集思想（一组点和一些被指定为开放的子集）必须被一种描述独立于点的开放的无点方法所取代。这些点随后被构造，尽管它们可能太少，无法通过它们的点来唯一地区分开孔。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出色的结果，并且也被应用于计算机科学，与计算机程序的观察理论有关。直接在内部逻辑中处理无点拓扑在技术上是困难的。然而（Joyal/Tierney），它们同样可以被视为基地空间上的无点束——也就是说，从另一个空间到基地的映射。当我们把地图看作一个束的时候，我们可以把它看作一个可变的空间——对于基底的每一点，我们都在它上面有一个纤维，在地图下面是那个点的逆像，随着点的变化，它的纤维也会变化。理想情况下，我们关于内部无点空间的内部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，束上的几何结构是通过束回拉保存下来的，这覆盖了纤维。通过对逻辑的仔细解释，几何推理也可以通过无点空间的点有效地工作，尽管它们可能缺乏。提出者开发了几何推理技术，特别是利用了powerlocales（无点超空间，或空间的空间）。该项目旨在利用量子物理拓扑方法中的几何性技术，用更熟悉的拓扑概念（点、束、纤维）来重新表达它，而不是内部无点空间。目标是使topos方法对物理学家来说更容易理解，并帮助澄清它与其他物理形式化的关系。它也是测试几何性的一般数学范围的一个极好的案例研究。