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- W47465699 abstract "In this thesis, Isham's group theoretic quantisation technique has been appliedto quantise Hall systems with spheres as their underlying configuration spaces.Before doing this, a preliminary mathematical tools needed for this work is givenfollowed by an overview of the above mentioned quantisation scheme.Beginning with the simple sphere in the first stage, it is found that the part ofcanonical group which acts on the configuration space when the magnetic field isabsent is either the group SO(3) or its covering group SU(2). However when theexternal field is present there is an obstruction which necessitates the group SU(2) asthe canonical group. The representations of the group SU(2) are parameterized by aninteger n which could be used to classify the integer Hall states. This however givesonly a description for the case of integer quantum Hall effect.To get the quantisation of a system of a test particle within a many- particleformalism punctures are introduced on the sphere. First, the quantisation problemon 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 thecanonical group at first glance, but with the problem of global definition, the rightchoice of canonical group would be the quotient group SU(2)/H with H as thesubgroup 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 thesymmetry exchange of the punctures. To overcome a small portion of this problemwe use uniformisation theory to get the canonical group directly by Isham'stechnique of the homogeneous space. Within this approach it is possible to adopt thequotient group SL(2,JR) / SO(2) as the canonical group for the case without magneticfield and SL(2, JR) for the case with magnetic field. From another perspective we alsoattempted quantisation on the universal covering, the upper half plane with the hopeof projecting it down to the punctured sphere, and we found SL(2,JR) to be thecanonical group. However the use of representations of SL(2,JR) cannot lead to aclassification of the fractional Hall state and a twisted representation could benecessary to get such classification.At the end of this thesis a different technique of approaching the fractionalquantum Hall classification has been applied to the special case of the thrice puncturedsphere. First we present a link between the principal congruence subgroupof the modular group of prime level 2, r(2) as the isomorphic group to thefundamental group of the thrice-puncture sphere and the braid group of threeparticles on the plane. Then a classification of the Hall states, integer as well asfractional, has been given using the action of the group r(2) on the cusps of thefundamental region defining the punctured sphere on the upper half plane." @default.
- W47465699 created "2016-06-24" @default.
- W47465699 creator A5087965883 @default.
- W47465699 date "2000-06-01" @default.
- W47465699 modified "2023-09-27" @default.
- W47465699 title "Group Theoretic Quantisation on Spheres and Quantum Hall Effect" @default.
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