Mathematical Problems in Statistical Mechanics and Quantum Theory

9705779金在这项提案中描述了三个项目，都与相互作用的多体系统的物理学有关。第一个问题是“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为时间演化算符找到一个有用的近似，即一个耦合到量子化辐射场的两态原子，并在这个模型中研究受激光子发射。第二个项目名为“相互避免自避免随机游动”，其目标是定位格子上非相交随机游动模型的精确相边界，并确定每一相的关联函数。第三个项目是“临界模型中关联的标度行为”，其中提出了在二维环面上不相交的随机环的模型中找出序参数的标度和大偏差性质。这个项目的广泛目标是通过从微观公式中得出数学结论来阐明由许多小组件组成的大系统的宏观行为。在微观层面上，物理学提供了广为人知的定律，允许进行一致的数学描述。挑战在于预测这些法律的宏观后果。这个项目关注两个领域：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涨落，其中宏观行为是突然出现的远程关联。在科学和技术的许多领域，地方规则可能会合谋产生巨大的影响(例如，高速通信网络中的拥堵)。这个项目将使用数学分析来解决理论物理领域中的这类问题。

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.