Group theoretic quantisation on spheres and quantum hall effect

In this thesis, Isham's group theoretic quantisation technique has been applied to quantise Hall systems with spheres as their underlying configuration spaces. Before doing this, a preliminary mathematical tools needed for this work is given followed by an overview of the above mentioned quantisation scheme. Beginning with the simple sphere in the first stage, it is found that the part of canonical group which acts on the configuration space when the magnetic field is absent is either the group SO(3) or its covering group SU(2). However when the external field is present there is an obstruction which necessitates the group SU(2) as the canonical group. The representations of the group SU(2) are parameterized by an integer n which could be used to classify the integer Hall states. This however gives only a description for the case of integer quantum Hall effect. To get the quantisation of a system of a test particle within a "many- particle formalism" punctures are introduced on the sphere. First, the quantisation problem on the punctured sphere is approached using a generalization of the method that works for the simple sphere. This method seems to show that SU(2) is still the canonical group at first glance, but with the problem of global definition, the right choice of canonical group would be the quotient group SU(2)/H with H as the subgroup of SU(2) which takes points on the sphere to the punctures. Unfortunately, such description is not very illuminating and this group doesn't show clearly the symmetry exchange of the punctures. To overcome a small portion of this problem we use uniformisation theory to get the canonical group directly by Isham's technique of the homogeneous space. Within this approach it is possible to adopt the quotient group SL(2,JR) / SO(2) as the canonical group for the case without magnetic field and SL(2, JR) for the case with magnetic field. From another perspective we also attempted quantisation on the universal covering, the upper half plane with the hope of projecting it down to the punctured sphere, and we found SL(2,JR) to be the canonical group. However the use of representations of SL(2,JR) cannot lead to a classification of the fractional Hall state and a twisted representation could be necessary to get such classification. At the end of this thesis a different technique of approaching the fractional quantum Hall classification has been applied to the special case of the thrice punctured sphere. First we present a link between the principal congruence subgroup of the modular group of prime level 2, r(2) as the isomorphic group to the fundamental group of the thrice-puncture sphere and the braid group of three particles on the plane. Then a classification of the Hall states, integer as well as fractional, has been given using the action of the group r(2) on the cusps of the fundamental region defining the punctured sphere on the upper half plane.

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