Symmetries in quantum mechanics and cor epresentation theory

本研究将考虑离散对称性在量子力学系统中的作用，重点是经典动力学为混沌的系统。在数学上，对称性是用正或反正的算子来描述的，它们与系统的哈密顿量交换或反交换。考虑到没有与哈密顿量交换的酉对称的情况，这已经允许将量子系统分类为Altland和Zirnbauer的十个普世性类。如果系统的经典动力学是混沌的，它的谱统计量和在某些情况下的平均能级密度与根据对称类选择的随机矩阵集合的预测一致。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不同行为相关的某些子谱的特征。先前的研究表明，这可能会导致新的物理现象。例如，它可以用来在没有自旋的系统中获得高斯辛系综（GSE）的随机矩阵统计量，这导致了最近在物理系统中对GSE统计量的第一次实验观察。查理会更系统地发展这种方法。对正酉对称性和反正酉对称性之间相互作用的一般描述需要在共表示理论的背景下观察对称性。在这个方向上的第一步将是根据共表示对具有可交换算子和不可交换算子的最简单对称群的行为进行分类。

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.