Space-Time from the Spectral Point of View: Towards Quantum Gravity

理论物理学中最具挑战性的问题是如何将量子物理定律与爱因斯坦的广义相对论统一起来。在这个方向上，最受关注的思想是超弦理论和非交换几何。这两种方法密切相关。非交换几何为研究黎曼几何标准方法无法处理的空间提供了一个新的框架。它为规范理论的研究和希格斯场的作用提供了一个新的视角，并且是描述普朗克能量下几何形状的一个有希望的候选者。在过去的几年里，人们已经确定，时空是由一个连续的四维流形与有限维空间的一个定义良好的乘积近似的。本计划的主要重点是构造一个在亚普朗克尺度下有效的真正非交换有限维空间，其在较低能量下的极限是标准模型的非交换空间。几何将被简化为几乎可交换矩阵的几何，并且有可能得到量化几何的处理。它还将研究谱作用的新渐近极限以及宇宙学意义。该项目的其他部分将研究引力的各个方面，如BPS单极子的稳定性，一致的质量引力理论的公式，右手中微子存在对时空几何的影响，存在奇点时的光谱行为以及普朗克长度尺度上动力学场的非摄动演化。项目的广泛影响：由于其跨学科性质，这项涉及非交换几何前沿研究的工作将对物理学家和数学家都有好处。本提案中的新思想和新方法将对高能理论和现象学、量子引力和数学界的知识做出贡献，并在较小程度上对天体物理学界做出贡献。贝鲁特美国大学（AUB）在中东地区推广美国教育体系方面发挥着非常重要的作用。从历史上看，重点是自由教育。新的目标是为促进研究提供一个充满活力的环境。2009年10月启动的理论物理学博士项目就是朝这个方向迈出的一步，并吸引了高素质的学生。

The most challenging problem in theoretical physics is the unification of the laws of quantum physics with Einstein's general theory of relativity. The ideas in this direction that have been pursued most forcefully are superstring theory and noncommutative geometry. These two approaches are intimately related. Noncommutative geometry has provided a new framework to study spaces which could not be handled by the standard methods of Riemannian geometry. It offered a new perspective in the study of gauge theories and the role of the Higgs field and is a promising candidate to describe geometries at Planckian energies. In the past years it has been established that space-time is approximated by a well defined product of a continuous four-dimensional manifold by a finite dimensional space. The main focus of this program is to construct a genuinely noncommutative finite dimensional space valid at sub-planckian scales and whose limit at lower energies is the noncommutative space of the standard model. The geometry will be reduced to those of almost commuting matrices, and it will be possible to get a handle on quantizing the geometry. It will also study new asymptotic limits of the spectral action as well as cosmological implications. Other parts of this project will study various aspects of gravity such as stability of BPS monopoles, and formulation of consistent massive theories of gravity, implications of the presence of right-handed neutrinos on the geometry of spacetime, the behavior of spectral actions in the presence of singularities and non-perturbative evolution of the dynamical fields at scales of the order of the Planck length. Broader impact of Project: Due to its interdisciplinary nature, this work which involves research on the frontiers of noncommutative geometry, will be of benefit to both physicists and mathematicians. The new ideas and methods in this proposal will contribute to the knowledge of the high energy theory and phenomenology, quantum gravity and mathematics communities, and to a lesser extent to the astro-physics community. The American University of Beirut (AUB) plays a very important role in promoting the American educational system in the Middle East. Historically, the emphasis was on liberal education. The new aim is to provide in addition a vibrant environment for the promotion of research. The launching of the Ph.D. program in theoretical physics in October 2009 is a step in that direction and is attracting students of high caliber.