Mathematical Sciences: Dynamical Behaviour of Vortices and Monopoles in Classical Gauge Theories

数学科学：经典规范理论中涡旋和磁单极子的动力学行为

• 批准号: 9214067
• 负责人: David Stuart
• 金额: $ 5.88万
• 依托单位: University of California-Davis
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1993
• 资助国家: 美国
• 起止时间: 1993-03-15 至 1996-02-29
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9214067&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Dynamical; Behaviour; Vortices

This project is on the time-dependent Yang-Mills-Higgs equations. These are nonlinear hyperbolic partial differential equations which are invariant under the Lorentz group and the infinite dimensional gauge group. The investigator has proved that under certain situations it is possible to construct solutions to the equations which correspond to motion of the vortices or monopoles according to a finite dimensional Hamiltonian system on the configuration space. These approximations are known to be valid on long but finite time intervals. The present aim of this investigation is to extend this understanding to more general situations, in particular to monopole scattering which is described by geodesics with respect to the Atiyah-Hitchin metric on the moduli space. A longer term aim is to extend the understanding to infinite time -- it is conceivable that asymptotically the solution approaches a combination of vortices and monopoles moving according to the Hamiltonian system on the configuration space in the maximum norm. The reason for this conjecture is the dispersion of radiation. Another long term aim is to extend the understanding to the equations of general relativity in which black holes take the place of vortices and monopoles. The work has possible applications to superconductivity and particle physics in addition to being of mathematical interest.

该项目是关于时间相关的杨米尔斯希格斯玻色子 方程这些是非线性双曲型偏微分 在洛伦兹群下不变的方程， 无限维规范群 调查员已经证明 在某些情况下， 方程的解对应于运动的 涡流或 根据有限维的单极子 位形空间上的哈密顿系统。 这些 已知近似在长而有限的时间上是有效的 的间隔 本次调查的目的是扩大 这种理解更普遍 情况，特别是 由测地线描述的关于 模空间上的Atiyah-Hitchin度量 较长期 目的是将理解扩展到无限的时间-它是 可以想象，渐近的解决方案接近一个 涡旋和单极子的组合， 最大位形空间上的Hamilton系统 norm. 其原因 猜想是分散的 辐射 另一个长期目标是扩大对 在广义相对论的方程中， 漩涡和单极子的位置。 这项工作有可能应用于超导和 粒子物理学除了是数学的兴趣。