One-dimensional fermions as two-dimensional droplets via Chern-Simons theory

Based on the observation that particle motion in one dimension maps to two-dimensional motion of a charged particle in a uniform magnetic field, constrained in the lowest Landau level, we formulate a system of one-dimensional nonrelativistic fermions by using a Chern-Simons field theory in 2+1 dimensions. Using a hydrodynamical formulation we obtain a two-dimensional droplet picture of one-dimensional fermions. The dynamics involved is that of the boundary between a uniform density of particles and vortices. We use the sharp boundary approximation. In order to test our approach we apply it to a system of fermions in a harmonic oscillator potential. In the case of well separated boundaries we derive the one-dimensional collective field hamiltonian. Symmetries of the theory are also discussed as properties of curves in two dimensions. (orig.)