Symmetries in quantum mechanics and cor epresentation theory

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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.