Billiard Field Theory

台球场论

• 批准号: EP/Y023005/1
• 负责人: Remy Dubertrand
• 金额: $ 52.85万
• 依托单位: Northumbria University
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2024
• 资助国家: 英国
• 起止时间: 2024 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FY023005%2F1
• 关键词: Billiard; Field; Theory

It is well known, from daily experience, that any physical system (e.g. a gas, a liquid) will reach equilibrium when left on its own for a long period of time. The process of reaching equilibrium is often called thermalisation. This can stand for the melting of an ice cube on a plate at room temperature, or the dissipation of small waves at the surface of a pond. Thermalisation is usually explained by assuming that there is a very complex dynamics at smaller scales. This is sometimes dubbed as 'microscopic chaos'. Another important ingredient for thermalisation is that the environment may act upon the considered system (the warmer air surrounding the ice cube, the still surface around the perturbation in a pond). Recently the role of both those fundamental ingredients to describe thermalisation has been challenged. The first ingredient, (microscopic) chaos, can be proved to be absent in the specfic case of an integrable system. Integrability is a very specific property, which claims e.g. that, apart from the total energy, there are infinitely many conserved quantities during the time evolution. The second ingredient can nowadays be made effectively absent in cold atom experiments. At very low temperature it is possible to observe a system where the interaction with its environment is negligible. Indeed it was observed that standard thermalisation fails!This project aims to tackle the question of thermalisation for a new class of models. Those models are especially relevant as they can be tuned to be integrable or fully chaotic at the microscopic level. Hence they sit in a unique position to enable one to fully understand the relevant and required assumptions for thermalisation to occur. For the sake of simplicity our models deal with isolated systems so our predictions will be of direct relevance for the experiments described above. A very powerful tool to describe the possible equilibria of a system is called statistical field theory. This has been successful to analyse the effects of the symmetry on the possible equilibria of a given system. Our models sits in a group of models called (non)linear sigma models. The main idea is to enforce the symmetry effects in a geometrical manner. It is remarkable that the standard sigma models have consisted only of geometries without edges (e.g. the surface of a torus or a sphere). One central aspect of this project is to study the effects of having a boundary (hard wall). Those effects connect sigma models to mathematical billiards. Those consist of tracing a ray of light trapped inside a table with an arbitrarily chosen shape. For a rectangular billiard table, the ray will have an integrable time evolution. If two half-disks are glued to the smallest sides, one gets a stadium-like shape for which the time evolution meets the strongest criterion for chaos. In particular two rays starting from neighbouring positions will depart quickly from each other.The second important aspect of the project is to focus on the subtle regime where quantum particles (or fields) start to show similarity with their non-quantum (classical) counterpart, typically at moderate or high energy. This regime is called semiclassical, for which a specific toolbox has been used to study the quantum version of mathematical billiards. Our aim is to transfer this accumulated expertise to fields in sigma models. We shall start with simpler billiard shapes, also to compare with numerous alternative approaches. Then we will implement the semiclassical tools for fields trapped in a billiard table of arbitrary shape. We believe that this can lead to field theories of new symmetry class and enable one to use non-perturbative techniques for non-integrable field theories.

从日常经验来看，众所周知，任何物理系统(例如气体、液体)在长时间保持其自身状态时都会达到平衡。达到平衡的过程通常被称为热敏化。这可以代表室温下盘子上的冰块融化，或者池塘表面的小浪消散。热化作用通常被解释为假设在较小的尺度上存在非常复杂的动力学。这有时被称为“微观混乱”。热化的另一个重要因素是环境可能作用于所考虑的系统(冰块周围较暖的空气，池塘中扰动周围的静止表面)。最近，描述热化的这两个基本要素的作用受到了挑战。第一个因素，(微观)混沌，可以证明在可积系统的特定情况下是不存在的。可积性是一个非常特殊的性质，它声称，在时间演化过程中，除了总能量之外，还有无限多的守恒量。第二种成分如今可以在冷原子实验中有效地消失。在很低的温度下，可以观察到一个与环境的相互作用可以忽略的系统。事实上，人们观察到标准的热敏化失败了！这个项目旨在解决新型号的热敏化问题。这些模型特别相关，因为它们可以在微观层面上调整为可积或完全混乱。因此，它们处于一个独特的位置，使人们能够充分理解发生热化所需的相关假设。为简单起见，我们的模型处理孤立的系统，因此我们的预测将与上述实验直接相关。描述系统可能均衡的一个非常强大的工具叫做统计场理论。这已经成功地分析了对称性对给定系统可能的平衡的影响。我们的模型位于称为(非线性)线性西格玛模型的一组模型中。其主要思想是以几何方式加强对称效果。值得注意的是，标准的西格玛模型只由没有边的几何图形组成(例如，环面或球面)。这个项目的一个核心方面是研究有边界(硬墙)的影响。这些效应将西格玛模型与数学台球联系起来。其中包括追踪被困在桌子内的一束光，形状是任意选择的。对于矩形台球桌，光线具有可积的时间演化。如果将两个半圆盘粘在最小的边上，就会得到一个类似体育场的形状，其时间演化满足最强的混沌标准。该项目的第二个重要方面是关注量子粒子(或场)开始显示出与其非量子(经典)对应物相似的微妙区域，通常是在中等或高能。这种机制被称为半经典，为此，一个特定的工具箱被用来研究量子版的数学台球。我们的目标是将积累的专业知识转移到西格玛模型领域。我们将从更简单的台球形状开始，也与许多替代方法进行比较。然后，我们将为困在任意形状的台球桌中的场实现半经典工具。我们相信，这可以导致新的对称类的场论，并使人们能够使用非微扰技术来处理不可积场论。