Symmetries in quantum mechanics and cor epresentation theory

Lead Research Organisation: University of Bristol
Department Name: Mathematics

Abstract

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 either
commute or anticommute with the Hamiltonian of the system. Considering the case without
unitary symmetries that commute with the Hamiltonian, this already allows to classify
quantum systems into Altland's and Zirnbauer's ten universality classes. If the classical
dynamics of the system is chaotic, its spectral statistics and in some cases also the average
level density agrees with predictions from ensembles of random matrices chosen according
to the symmetry class. This classification generalises to systems that also have unitary
symmetries commuting with the Hamiltonian, however in this case the spectral
characteristics to be considered are those of certain subspectra associated to different
behaviour under the unitary symmetries.
Previous research has shown that this can lead to new physical phenomena. For example, it
can 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 GSE
statistics in a physical system. Charlie will develop this type of approach more
systematically.
A general description of the interplay between unitary and antiunitary symmetries requires to
view symmetries in the context of corepresentation theory. A first step in this direction will be
to classify the behaviour of the simplest symmetry groups with commuting and
nticommuting operators in terms of corepresentations.

Publications

10 25 50

Studentship Projects

Project Reference Relationship Related To Start End Student Name
EP/N509619/1 01/10/2016 30/09/2021
1793787 Studentship EP/N509619/1 01/10/2016 31/03/2020 Charlotte Johnson