Mathematical Problems in Statistical Mechanics and Quantum Theory

[9705779] King在这个提议中描述了三个项目，它们都与相互作用的多体系统的物理学有关。第一个是“光子的自发和受激发射”；这里的问题是在最简单的相互作用量子场论中，即双态原子与量子化辐射场耦合，找到一个有用的时间演化算符的近似，并在该模型中研究受激光子发射。第二个项目名为“相互避免自我避免随机漫步”；目标是确定晶格上非相交随机行走模型的精确相位边界，并确定每个相位的相关函数。第三个项目是“临界模型中相关性的尺度行为”，提出了在二维环面上的非相交随机环模型中寻找阶参数的尺度和大偏差特性。该项目的总体目标是通过从微观公式中得出数学结论，阐明由许多小部件组成的大系统的宏观行为。在微观层面上，物理学提供了易于理解的定律，允许一致的数学描述。挑战在于预测这些规律的宏观后果。本项目主要关注两个领域：第一，光子与原子的相互作用，其中宏观行为是光的发射和吸收；第二，超导体中涡旋线的空间波动，其中宏观行为是远程相关性的突然发生。在科学和技术的许多领域，地方法规可能合谋产生巨大的影响（例如高速通信网络的拥塞）。这个项目将使用数学分析来解决理论物理领域的这类问题。

9705779 King There are three projects described in this proposal, all concerned with the physics of interacting many-body systems. The first is "Spontaneous and stimulated emission of photons"; the problem here is to find a useful approximation for the time evolution operator in the simplest interacting quantum field theory, namely a two-state atom coupled to the quantized radiation field, and to investigate stimulated photon emission in this model. The second project is titled "Mutually avoiding self-avoiding random walks"; the goal is to locate the exact phase boundaries of a model of non-intersecting random walks on the lattice and to determine the correlation functions in each phase. The third project is "Scaling behavior of correlations in a critical model", where it is proposed to find the scaling and large deviation properties of an order parameter in a model of non-intersecting random loops on the two-dimensional torus. The broad goal of this project is to elucidate the macroscopic behavior of a large system which is composed of many small components by drawing mathematical conclusions from its microscopic formulation. At the microscopic level, physics provides well-understood laws which allow a consistent mathematical description. The challenge is to predict the macroscopic consequences of these laws. This project focuses on two areas: first, the interaction of photons with an atom, where the macroscopic behavior is the emission and absorption of light, and second, the spatial fluctuations of vortex lines in superconductors, where the macroscopic behavior is the sudden occurrence of long-range correlations. There are many areas in science and technology where local rules may conspire to produce huge effects (for example congestion in high-speed communications networks). This project will use mathematical analysis to address questions of this type in the realm of theoretical physics.