Gauge Invariant Bosonization of Quantum Hall Systems and Skyrmions : Kinematics

We develop a systematic semiclassical approximation scheme for quantum Hall skyrmions near filling factors $\nu = {1 \over 2n+1}$, which is exact in the long wavelength limit. We construct a coherent state basis for the Hilbert space of Chern-Simons gauge fields and composite bosons with spin. These states are projected to the physical gauge invariant subspace and their wavefunctions explicitly evaluated. The lowest Landau level (LLL) condition is shown to be equivalent to an analyticity condition on the parameters. The matrix elements of physical observables between these states are shown to be calculable in the limit of small amplitude long wavelength density fluctuations. The electric charge density is shown to be proportional to the toplological charge density if and only if the LLL condition is satisfied. We then show that these states themselves form a generalised coherent state basis, parameterised by the values of physical observables. The theory can therefore be written in terms of these gauge invariant bosonic fields in the long wavelength regime. The off diagonal matrix elements of observables in these coherent states are computed and shown to vanish in the long wavelength limit. Thus we are able to prove that the classical description of the skyrmion is exact in the limit of large skyrmions.