对称类
Symmetry Classes
摘要：
表现出随机或混沌行为的物理系统通常可以用随机矩阵模型来处理。在决定一个好的模型选择，随机矩阵物理的约束和指导对称性的考虑。“对称性类”（symmetry class）的概念（不要与“普适性类”混淆）表达了对称性作为一种组织原则的相关性。戴森在他1962年的论文《三重方法》中给出了基于量子力学对称性的随机矩阵集合的主要分类。本文从现代的角度对戴森的三重道路进行了回顾。然后，我们描述了一个最小的扩展戴森的设置，将手征狄拉克费米子和无序超导体的物理。在这个极小扩展的设置中，希尔伯特空间被配备了粒子-空穴共轭的反酉操作的福克空间所取代，对称类与黎曼对称空间的大家族一一对应。
英文摘要：
Physical systems exhibiting stochastic or chaotic behavior are often amenable to treatment by random matrix models. In deciding on a good choice of model, random matrix physics is constrained and guided by symmetry considerations. The notion of 'symmetry class' (not to be confused with 'universality class') expresses the relevance of symmetries as an organizational principle. Dyson, in his 1962 paper referred to as the Threefold Way, gave the prime classification of random matrix ensembles based on a quantum mechanical setting with symmetries. In this article we review Dyson's Threefold Way from a modern perspective. We then describe a minimal extension of Dyson's setting to incorporate the physics of chiral Dirac fermions and disordered superconductors. In this minimally extended setting, where Hilbert space is replaced by Fock space equipped with the anti-unitary operation of particle-hole conjugation, symmetry classes are in one-to-one correspondence with the large families of Riemannian symmetric spaces.
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