Algebra and geometry of Dirac's magnetic monopole
thesisposted on 2013-06-13, 13:47 authored by Graham Kemp
This thesis is concerned with the quantum Dirac magnetic monopole and two classes of its generalisations. The first of these are certain analogues of the Dirac magnetic monopole on coadjoint orbits of compact Lie groups, equipped with the normal metric. The original Dirac magnetic monopole on the unit sphere S^2 corresponds to the particular case of the coadjoint orbits of SU(2). The main idea is that the Hilbert space of the problem, which is the space of L^2-sections of a line bundle over the orbit, can be interpreted algebraically as an induced representation. The spectrum of the corresponding Schodinger operator is described explicitly using tools of representation theory, including the Frobenius reciprocity and Kostant's branching formula. In the second part some discrete versions of Dirac magnetic monopoles on S^2 are introduced and studied. The corresponding quantum Hamiltonian is a magnetic Schodinger operator on a regular polyhedral graph. The construction is based on interpreting the vertices of the graph as points of a discrete homogeneous space G/H, where G is a binary polyhedral subgroup of SU(2). The edges are constructed using a specially selected central element from the group algebra, which is used also in the definition of the magnetic Schrodinger operator together with a character of H. The spectrum is computed explicitly using representation theory by interpreting the Hilbert space as an induced representation.
- Mathematical Sciences