Wider Applications of Lattice Field Theory

量子场论是理论物理学中最成功、应用最广泛的方法之一，然而，在某些情况下，数学方程是无法用笔和纸来求解的。一种可能的第一性原理数值技术是点阵场论，在这种理论中，空间和时间的连续维度被限制在一个固定的网格中，这些网格之间有连接。通过这种方式，可以用计算机对系统进行建模，随着网格上的点数量的增加，精度也在增加。这种方法已被广泛用于研究量子色动力学，这是一种强核力理论，它控制着夸克之间的相互作用，由于相互作用的强度，它是不可解析解的。点阵场论也可用于研究许多物体系统的行为，例如石墨烯，这与凝聚态物理有关。越来越多的晶格场论被用于研究标准模型物理之外的行为，例如，包括超对称模型和弦理论的矩阵模型，如班克斯-费舍尔-申克-萨斯金德模型。其中一个特别有趣的领域是量子混沌。经典混沌被定义为初始封闭状态以指数速率彼此分离的路径。这种行为也可以在量子系统中观察到，并且在许多物体系统的热化中起着关键作用，例如重离子碰撞时产生的夸克-胶子等离子体。由于黑洞是最混沌的量子系统，对量子混沌的研究也可能为量子引力理论提供启示。在这个项目中，我们计划研究多体量子系统中近极大量子混沌的出现。这种处于最大混沌状态的系统将为我们提供类黑洞物理的微观模型。特别是，我们计划开发从欧几里得时间蒙特卡罗模拟结果中数值提取热化速率（最大李雅普诺夫指数）的方法。我们进一步计划将这些方法应用于强相互作用规范理论中Lyapunov指数的温度依赖性研究。除此之外，我们还将把晶格场理论应用到其他模型中，包括我之前合作的继续工作，用晶格场理论来证明手性对称性破缺的开始和限制和部分限制之间的Gross-Witten-Wadia跃迁，与Eguchi-Kawai模型相吻合。

Quantum field theory is one of the most successful, and widely used, approaches in theoretical physics however, in some cases the mathematical equations are not solvable using a pen and paper. One possible first-principle numerical technique is lattice field theory, in which the continuous dimensions of space and time are restricted to a fixed grid of points with links between them. In this way a system can be modelled using a computer, with increasing precision as the number of points on the grid is increased. This method has been widely used to investigate quantum chromodynamics, the theory of the strong nuclear force that governs the interactions between quarks, which is not analytically solvable due to the strength of the interactions. Lattice field theory may also be used to investigate the behaviour of many body systems, such as graphene, which is relevant in condensed matter physics. Increasingly lattice field theory is being used to investigate the behaviour of beyond the standard model physics, for example, models which include supersymmetry as well as matrix models of string theory such as the Banks-Fischler-Shenker-Susskind model. One area of particular interest is quantum chaos. Classically chaos is defined by the paths of initially close states separating from one another at an exponential rate. This behaviour can also be observed in quantum systems and plays a key role in the thermalization of many body systems, such as the quark-gluon plasma produced during collisions of heavy ions. With black holes being maximally chaotic quantum systems, studies of quantum chaos might also shed light on the theory of quantum gravity. In this project, we plan to study the emergence of nearly maximal quantum chaos in many-body quantum systems. Such systems in the regime of maximal chaos will provide us with microscopic models for black-hole-like physics. In particular, we plan to develop methods for the numerical extraction of thermalization rates (largest Lyapunov exponents) from the results of Monte-Carlo simulations in Euclidean time. We further plan to apply these methods to study the temperature dependence of Lyapunov exponents in gauge theories of strong interactions. In addition to this we will look to employ lattice field theory to a variety of other models, including continuing work that I collaborated on previously which used lattice field theory to demonstrate that the onset of chiral symmetry breaking and the Gross-Witten-Wadia transition, between confinement and partial confinement, coincide in the Eguchi-Kawai model.