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在量子力学系统中的作用，重点放在经典动力学是混沌的系统上。在数学上，对称性是由与系统的哈密顿量交换或反交换的酉算子或反酉算子来描述的。考虑到没有与哈密顿量交换的么正对称性的情况，这已经允许将量子系统归入Altland和Zirnbauer的十个普适性类别。如果系统的经典动力学是混沌的，它的谱统计，在某些情况下，平均能级密度也与根据对称类选择的随机矩阵系综的预测一致。这种分类推广到也具有与哈密顿量交换的么正对称性的系统，然而，在这种情况下，要考虑的光谱特征是与酉对称下的不同行为相关的某些子谱的光谱特征。先前的研究表明，这可能会导致新的物理现象。例如，它可以用来获得无自旋系统中高斯辛系综(GSE)的随机矩阵统计，这导致了最近在物理系统中首次实验观察到GSE统计。查理将更系统地发展这类方法。对酉对称和反酉对称之间相互作用的一般描述需要在共表示理论的背景下看待对称。在这个方向上的第一步将是将最简单的对称群的行为归类为具有交换和交换算子的联合表示。