Quantum equilibrium and symmetries from number theory

数论中的量子平衡和对称性

• 批准号: 249696-2012
• 负责人: Laca, Marcelo
• 金额: $ 1.82万
• 依托单位: University of Victoria
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=570692
• 关键词: Quantum; equilibrium; symmetries; number; theory

The proposal seeks to develop a recently established connection between operator algebras, originally developed to model concrete quantum mechanical phenomena, and abstract algebraic structures arising in number theory. This remarkable connection is based on the fact that numbers and sub-atomic particles share some common features that make them tractable with the same mathematical tools. One of these common features is the prominent role that symmetries play in their study; another one is the relevance of operations that do not commute with each other --the underlying phenomenon behind Heisenberg's uncertainty principle. In quantum systems the non-commuting operations are the measurements of position and momentum of particles; in numerical systems, they are the two basic operations of addition and multiplication. This line of thought reveals that concrete phenomena such as the freezing of water into ice, or the spontaneous magnetization of a ferromagnet, have surprisingly sophisticated abstract analogues. It turns out that when we let number systems 'freeze' and then analyze the equilibrium and the symmetries that are spontaneously broken in the process, or when we let them 'boil' and then analyze the ensuing chaotic behavior, we are led to key questions in number theory viewed from a very innovative perspective. This sets up a fascinating exchange of examples, techniques and insight between traditionally distinct areas of mathematics. The program falls within the general theory of operator algebras, in which Canada is an international leader. The expected outcomes will broaden the scope of their application, will shed light on long-standing problems, and will generate fundamental knowledge, thus enhancing Canada's presence in the international scientific arena.

该提案旨在发展最近建立的算子代数之间的联系，最初是为了模拟具体的量子力学现象而开发的，而抽象的代数结构出现在数论中。 这种显著的联系是基于这样一个事实，即数字和亚原子粒子具有一些共同的特征，使它们易于使用相同的数学工具。这些共同的特征之一是对称性在他们的研究中发挥的突出作用;另一个是相互不对易的操作的相关性-海森堡不确定性原理背后的基本现象。在量子系统中，非对易操作是粒子的位置和动量的测量;在数值系统中，它们是加法和乘法的两个基本操作。这一思路揭示了具体的现象，如水冻结成冰，或铁磁体的自发磁化，有令人惊讶的复杂抽象的类似物。事实证明，当我们让数字系统“冻结”，然后分析在这个过程中自发打破的平衡和对称性时，或者当我们让它们“沸腾”，然后分析随之而来的混沌行为时，我们会从一个非常创新的角度看待数论中的关键问题。这在传统的不同数学领域之间建立了一个有趣的例子，技术和见解的交流。 该计划属于福尔斯一般理论的算子代数，在加拿大是一个国际领导者。预期成果将扩大其应用范围，将阐明长期存在的问题，并将产生基本知识，从而加强加拿大在国际科学竞技场的存在。