Group-theoretic quantisation and central extensions

Abstract

This work is concerned with the applications of Isham's group-theoretic quantisation programme to simple systems which involve central extensions of some symmetry group. Of particular interest are those systems with a 'Wess-Zumino'-like term in their actions where other nontrivial modifications are necessary. In Chapters 1 and 2, a review of the necessary tools used in this work as well as outlines of the group-theoretic quantisation programme are given to facilitate a smooth discussion in the latter chapters. The programme is first exemplified by the normal quantum mechanics on R". This example also involves central extensions but of a slightly different nature from those which arise from systems with a 'Wess-Zumino'-like term. Chapter 3 forms the core of the whole work. The discussions there provide the basis for further examples. It is concerned with the group-theoretic quantisation of the system of a particle moving on the two-torus in a constant magnetic field with quantised flux. The case without the magnetic field is also given for comparison. The canonical group for the case with the magnetic field is required to be the central extension of the universal cover of the canonical group for the case without the magnetic field. These results are then generalised to the corresponding systems on the n-torus. Chapter 4 is a digression from the main topic of quantisation and central extensions to the discussions of a-models with Wess-Zumino term. The main purpose of this chapter is to provide a parallel between these σ-models and the systems of a particle moving in a magnetic field (as in Chapter 3). The general construction of a Wess-Zumino term is given along with the discussion of an Abelian gauge symmetry that the term provides for the σ-models. The a-models can be interpreted as systems of a 'particle' moving on an infinite-dimensional configuration space in a background 'functional magnetic field'. This interpretation is further reinforced by the discussions of Noether's theorem and topological effects.