Billiard Field Theory

根据日常经验，我们都知道，任何物理系统（如气体、液体）在长时间独立运行后都会达到平衡。达到平衡的过程通常称为热化。这可以代表在室温下冰块在盘子上的融化，或者池塘表面小波浪的消散。热化通常是通过假设在较小的尺度上存在非常复杂的动力学来解释的。这有时被称为“微观混沌”。热化的另一个重要因素是环境可能对所考虑的系统起作用（冰块周围的温暖空气，池塘中扰动周围的静止表面）。最近，这两个基本成分在描述热化过程中的作用受到了挑战。在可积系统的特定情况下，可以证明第一要素（微观）混沌是不存在的。可积性是一个非常特殊的性质，它表明，在时间演化过程中，除了总能量之外，还有无限多个守恒量。第二种成分现在可以在冷原子实验中有效地制造出来。在极低的温度下，可以观察到一个系统与其环境的相互作用可以忽略不计。事实上，我们观察到标准的热化是失败的！该项目旨在解决一类新型模型的热化问题。这些模型是特别相关的，因为它们可以在微观水平上被调整为可积或完全混沌。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的相关和必要的假设。为简单起见，我们的模型处理的是孤立系统，因此我们的预测将与上述实验直接相关。描述系统可能的平衡的一个非常有力的工具被称为统计场论。这已经成功地分析了对称性对给定系统的可能平衡的影响。我们的模型属于一组称为（非线性）西格玛模型的模型。其主要思想是以几何的方式加强对称效果。值得注意的是，标准的西格玛模型只包括没有边缘的几何形状（例如环面或球体的表面）。这个项目的一个中心方面是研究有一个边界（硬墙）的影响。这些效应将sigma模型与数学台球联系起来。这些技术包括追踪一束被困在一张任意选择形状的桌子里的光线。对于矩形台球桌，射线将具有可积的时间演化。如果将两个半圆盘粘在最小的一面，其中一个就会得到一个类似体育场的形状，其时间演化符合混沌的最强标准。特别是两条从相邻位置出发的射线会很快远离对方。该项目的第二个重要方面是关注量子粒子（或场）开始显示与其非量子（经典）对应物相似的微妙状态，通常是在中等或高能量下。这种状态被称为半经典状态，为此已经使用了一个特定的工具箱来研究量子版的数学台球。我们的目标是将这些积累的专业知识转移到sigma模型的领域。我们将从更简单的台球形状开始，并与许多替代方法进行比较。然后，我们将实现半经典工具，用于捕获在任意形状的台球桌中的场。我们相信这将导致新的对称类场论，并使人们能够在非可积场论中使用非微扰技术。

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.