Mathematical Problems in Statistical Mechanics and Quantum Theory

9705779 国王 在这个建议中描述了三个项目，都与相互作用的多体系统的物理学有关。第一个是“光子的自发和受激发射”;这里的问题是找到一个有用的近似的时间演化算子在最简单的相互作用量子场论，即一个两态原子耦合到量子辐射场，并调查受激光子发射在这个模型。 第二个项目的标题是"相互避免自避免随机游走";目标是定位网格上不相交随机游走模型的精确相边界，并确定每个相的相关函数。 第三个项目是“临界模型中相关性的标度行为”，其中提出在二维环面上的非相交随机回路模型中找到序参数的标度和大偏差性质。 该项目的主要目标是通过从微观公式中得出数学结论来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。挑战在于预测这些定律的宏观后果。 该项目侧重于两个领域：第一，光子与原子的相互作用，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，宏观行为是长程关联的突然发生。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塞）。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.