Abstract:

This thesis addresses the following questions and discusses the answers: i) What is the limit in which the classical approximation is exact?; ii) When does the theory itself become classical? (Not just the energetics, but the Kinematical description itself); iii) What is that classical theory?; iv) How do we get the LLL classical theory?; v) How do we give a semiclassical description to quantum skyrmions? vi) What is the effect of Landau level mixing on skyrmions?. The Hilbert space of the composite bosons is defined and its coherent state basis is constructed. To obtain the explicit mapping between the electronic Hilbert space and observables and the gauge invariant states and observables in the bosonic theory, the gauge invariant anticommuting operators are constructed. They create and annihilate flux carrying bosons which are then used to represent the electron creation and annihilation operators. The pathintegral representation of the evolution operator is derived and the standard ChernSimon's gauge field theory coupled to matter field is obtained. The composite boson coherent states projected on to the gauge invariant sector and their wave functions are calculated. It is found that the electric charge density is proportional to the topological charge density if and only if the LLL condition is satisfied. The effect of Landau level mixing on the spin charge relation is studied. The quasi particle states  the projected coherent states, are examined and it is found to satisfy the properties of generalized coherent states parameterized by the values of physical observables. This gauge invariant coherent state basis enables a gauge invariant bosonization. A classical theory for composite boson is given, and NLSM with Hopf term is obtained upon imposing the LLL condition in this classical theory. A complete kinematical description of the LLL skyrmions interms of their collective coordinates, is given. The overlaps for these skyrmions are determined and constructed with finite spin skyrmions. 