Quantum equilibrium and symmetries from number theory

该提议寻求在算子代数和数论中出现的抽象代数结构之间建立一种最近建立的联系，算子代数最初是为了模拟具体的量子力学现象而开发的。 这种非凡的联系是基于这样一个事实，即数字和亚原子粒子有一些共同的特征，这使得它们可以用同样的数学工具来处理。这些共同特征之一是对称性在他们的研究中扮演着突出的角色；另一个是不相互交换的运算的相关性--海森伯格不确定性原理背后的潜在现象。在量子系统中，非对易运算是对粒子位置和动量的测量；在数值系统中，它们是加法和乘法这两种基本运算。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，有着令人惊讶的复杂的抽象类似物。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，或者当我们让它们‘煮沸’然后分析随之而来的混沌行为时，我们从一个非常创新的角度引出了数论中的关键问题。这在传统上不同的数学领域之间建立了一种令人着迷的例子、技术和见解的交流。 该计划属于算子代数的一般理论，加拿大在这方面处于国际领先地位。预期成果将扩大其应用范围，揭示长期存在的问题，并将产生基础知识，从而加强加拿大在国际科学舞台上的存在。

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.