Symmetries in quantum mechanics and cor epresentation theory

这项研究将考虑离散对称性在量子力学系统中的作用，重点是经典动力学是混沌的系统。在数学上，对称性是由算符描述的，这些算符要么是酉的，要么是反酉的，要么与系统的哈密顿量可换或反可换。考虑到没有幺正对称与哈密尔顿算子交换的情况，这已经允许将量子系统分类为Altland和Zirnbauer的十个普适性类。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，平均水平密度同意从合奏的随机矩阵选择根据对称类的预测。这种分类推广到也具有与哈密顿量交换的幺正对称的系统，然而在这种情况下，要考虑的谱特征是与幺正对称下的不同行为相关的某些子谱的谱特征。例如，它可以用来获得无自旋系统中高斯辛包络（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种类型的方法。对酉和反酉对称性之间相互作用的一般描述需要在共表征理论的背景下看待对称性。在这个方向上的第一步将是分类的最简单的对称群的行为与交换和非交换运营商在corepresentation。

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.