Symmetries in quantum mechanics and cor epresentation theory

量子力学中的对称性和表示理论

• 批准号: 1793787
• 金额: --
• 依托单位: University of Bristol
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2016
• 资助国家: 英国
• 起止时间: 2016 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-1793787
• 关键词: Symmetries; quantum; mechanics; cor; epresentation

This research will consider the role of discrete symmetries in quantum mechanical systems,with an emphasis on systems whose classical dynamics is chaotic. Mathematically,symmetries are described by operators that are either unitary or antiunitary and eithercommute or anticommute with the Hamiltonian of the system. Considering the case withoutunitary symmetries that commute with the Hamiltonian, this already allows to classifyquantum systems into Altland's and Zirnbauer's ten universality classes. If the classicaldynamics of the system is chaotic, its spectral statistics and in some cases also the averagelevel density agrees with predictions from ensembles of random matrices chosen accordingto the symmetry class. This classification generalises to systems that also have unitarysymmetries commuting with the Hamiltonian, however in this case the spectralcharacteristics to be considered are those of certain subspectra associated to differentbehaviour under the unitary symmetries.Previous research has shown that this can lead to new physical phenomena. For example, itcan be used to obtain random matrix statistics of the Gaussian Symplectic Ensemble (GSE)in systems without spin, and this led recently to the first experimental observation of GSEstatistics in a physical system. Charlie will develop this type of approach moresystematically.A general description of the interplay between unitary and antiunitary symmetries requires toview symmetries in the context of corepresentation theory. A first step in this direction will beto classify the behaviour of the simplest symmetry groups with commuting andnticommuting operators in terms of corepresentations.

这项研究将考虑离散对称性在量子机械系统中的作用，重点是经典动态混乱的系统。从数学上讲，对称性是由统一或反对的操作员描述的，并且与系统的哈密顿量进行了对称或反对。考虑到与哈密顿人通勤的没有独立对称的案件，这已经允许将系统分类为Altland和Zirnbauer的十个普遍性类别。如果该系统的经典动力学是混乱的，则其光谱统计数据，在某些情况下，平均密度也与根据对称类别选择的随机矩阵的集合的预测一致。这种分类概括地将与哈密顿量交通的系统也具有单位对称性，但是在这种情况下，要考虑的光谱量是在单一的对称性下与不同行为相关的某些物质的光谱。例如，IT可用于在没有自旋的系统中获得高斯象征合奏（GSE）的随机矩阵统计，这最近导致了物理系统中对GSESTATISTIS的首次实验性观察。查理（Charlie）将在系统上发展这种方法。对统一和反对对称性之间的相互作用的一般描述需要在核心代理理论的背景下进行符号对称。朝这个方向迈出的第一步将用核心代理来将最简单的对称组与通勤和NTICOMPONTING OPERATER分类。