  We write a Ginzburg-Landau Hamiltonian for a charged order parameter
interacting with a background electromagnetic field in 2+1 dimensions. Using
the method of Lund we derive a collective coordinate action for vortex defects
in the order parameter and demonstrate that the vortices are charged. We
examine the classical dynamics of the vortices and then quantize their motion,
demonstrating that their peculiar classical motion is a result of the fact that
the quantum motion takes place in the lowest Landau level. The classical and
quantum motion in two dimensional regions with boundaries is also investigated.
The quantum theory is not invariant under magnetic translations. Magnetic
translations add total time derivative terms to the collective action, but no
extra constants of the motion result.
