Chern-Simons vortices in an open system

A gauge invariant quantum field theory with a spacetime dependent Chern-Simons coefficient is studied. Using a constraint formalism together with the Schwinger action principle it is shown that non-zero gradients in the coefficient induce magnetic-moment corrections to the Hall current and transform vortex singularities into non-local objects. The fundamental commutator for the density fluctuations is obtained from the action principle and the Hamiltonian of the Chern-Simons field is shown to vanish only under the restricted class of variations which satisfy the gauge invariance constraint.