C*-dynamics, quantum equilibrium, and symmetries

算子代数理论使我们能够感知和利用量子物理学中具体现象与数论和其他数学领域之间惊人的相似之处，这些领域非常抽象，但却通过互联网安全和密码学等影响着我们的日常生活。一个关键的共同特征是对称在物理学和数论中的作用。另一个原因是，执行连续基本操作的结果取决于它们执行的顺序——这是海森堡测不准原理背后的潜在现象。在量子系统中，非交换操作是对粒子的位置和动量的测量；在数值系统中，它们是加法和乘法这两个基本运算。类似的相似之处也存在于由随机过程和分形物体产生的动力系统中，在这些系统中，相同的形状在较小的尺度上重复自身。本提案旨在利用这些相似之处，通过开发一个框架来转移见解和解决问题，使用算子-代数量子模型来分析数论和动力系统中出现的复杂问题。例如，越来越多的证据表明，物理相变的数论对应，如水冻结成冰，铁磁体的自发磁化，以及电子在极低温度下在二维网格中的运动，是数论中主要开放问题的核心。通过对这些算子代数模型的研究，提出的研究将为长期存在的问题提供一个创新的视角，并将为传统上不同的数学领域之间的例子、技术和思想的迷人交流奠定基础。该计划将扩大算子代数方法的应用范围，将为数学中的开放性问题提供新的亮点，并将产生基础知识。培训青年科学家将是该建议的一个重要组成部分，这将提高加拿大的科学能力。*****

The theory of operator algebras allows us to perceive and exploit a striking parallel between concrete phenomena in quantum physics, and aspects of number theory and other areas of mathematics that are very abstract, but which affect our everyday life through things like internet security and cryptography. One key common feature is the role of symmetries in physics and in number theory. Another one is that the outcome of performing successive fundamental operations depends on the order in which they are performed -this is 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. Similar parallels also exist with dynamical systems arising from stochastic processes and from fractal-like objects, in which the same shape repeats itself at smaller scales. This proposal aims to exploit these parallels by developing a framework to transfer insights and solve problems using the operator-algebraic quantum models to analyze sophisticated problems arising in number theory and dynamical systems. For instance, there is mounting evidence that the number-theoretic counterparts of physical phase transitions such as the freezing of water into ice, the spontaneous magnetization of a ferromagnet, and the motion of electrons in a two dimensional grid at very low temperature, are at the heart of the main open questions in number theory. Through the study of these operator-algebraic models, the proposed research will offer an innovative perspective into long-standing questions and will set the stage for a fascinating exchange of examples, techniques and ideas between traditionally distinct areas of mathematics. The program will expand the scope of application of operator-algebraic methods, will shed new light on open problems in mathematics, and will generate fundamental knowledge. The training of young scientists will be an important component of the proposal, which will enhance Canada's scientific capacity.*****