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Random planar curves and conformal field theory

随机平面曲线和共形场论

基本信息

批准号：
EP/D070643/1
负责人：
John Cardy
金额：
$ 40.07万
依托单位：
University of Oxford
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2007
资助国家：
英国
起止时间：
2007 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FD070643%2F1
关键词：
Random planar curves conformal field

项目摘要

Random fractal objects occur all around us. Examples are clouds, the shapes of coastlines, a head of cauliflower. They have the properties of being self-similar: if we take a picture of part of a cloud, choose just part of that picture and then blow it up to the size of the original, we can't say which one was in fact the original; and random: that is no two clouds look exactly the same although they are all obviously clouds. This example also illustrates a fundamental property of fractals in nature, as opposed to mathematical ones: we can in fact tell the pictures apart by the graininess of the photo: all physical fractals have some microscopic length scale. Unfortunately, although we know the equations which describe clouds, they are very difficult to analyse. Somewhat simpler examples occur in the physics of critical behaviour, for example magnets. If we look at a magnet closely enough, we see the individual atoms, each like a little magnet, or spin, regularly arranged on a regular lattice but pointing in random directions. However they form clusters, within which all the spins point the same way. In two dimensions the boundaries of these clusters are curves. At the critical temperature, just when the material becomes ferromagnetic, these curves are believed to be fractal, with a graininess given by the lattice. Physicists have, for the last 20 years, had a good theory of these systems, called conformal field theory (CFT). (It actually has applications in other branches of physics like string theory.) However it is not mathematically rigorous and in most cases no one has proved that it actually describes these lattice models. More recently, mathematicians have developed a theory called stochastic Loewner evolution (SLE) which describes the curves directly as mathematical fractals. It is the purpose of the proposed research to establish the full connection between CFT and SLE, as well as showing that some of the curves in these lattice models are in fact described by SLE if we ignore the graininess. Only then will we have a complete theory.

随机分形物体发生在我们周围。例如云、海岸线的形状、花椰菜的头。它们具有自相似的特性：如果我们拍一张云的一部分的照片，只选择该照片的一部分，然后将其放大到原图的大小，我们就不能说哪一张实际上是原图；随机：即没有两朵云看起来完全相同，尽管它们显然都是云。这个例子还说明了分形在自然界中的一个基本属性，而不是数学分形：事实上，我们可以通过照片的颗粒度来区分图片：所有物理分形都具有一定的微观长度尺度。不幸的是，尽管我们知道描述云的方程，但它们很难分析。一些更简单的例子出现在关键行为的物理学中，例如磁铁。如果我们足够仔细地观察磁铁，我们会看到单个原子，每个原子都像一个小磁铁或旋转，规则地排列在规则的晶格上，但指向随机的方向。然而，它们形成簇，其中所有自旋都指向相同的方向。在二维中，这些簇的边界是曲线。在临界温度下，当材料变成铁磁性时，这些曲线被认为是分形的，具有由晶格给出的颗粒度。在过去的 20 年里，物理学家对这些系统有了一个很好的理论，称为共形场论 (CFT)。（它实际上在弦理论等物理学的其他分支中也有应用。）然而，它在数学上并不严格，并且在大多数情况下，没有人证明它确实描述了这些晶格模型。最近，数学家发展了一种称为随机勒纳演化（SLE）的理论，它将曲线直接描述为数学分形。本研究的目的是建立 CFT 和 SLE 之间的完整联系，并表明如果我们忽略颗粒度，这些晶格模型中的一些曲线实际上是由 SLE 描述的。只有这样我们才有一个完整的理论。