An effective $\mathrm{U}(1)$ gauge invariant theory is constructed for a noncommutative Schrödinger field coupled to a background $\mathrm{U}(1{)}_{\star }$ gauge field in $2+1$-dimensions using first order Seiberg-Witten map. We show that this effective theory can be cast in the form of usual Schrödinger action with interaction terms of noncommutative origin provided the gauge field is of background type with constant magnetic field. The Galilean symmetry is investigated and a violation is found in the boost sector. We also consider the problem of Hall conductivity in this framework.

• Received 20 August 2004

DOI:https://doi.org/10.1103/PhysRevD.70.107707