U(1)gauge invariant noncommutative Schrödinger theory and gravity

Abstract

We consider the complex, massive Klein-Gordon field living in the noncommutative space, and coupled to noncommutative electromagnetic fields. After employing the Seiberg-Witten map to first order, we analyze the noncommutative Klein-Gordon theory as $c$, which corresponds to the velocity of light, goes to infinity. We show that the theory exhibits a regular ``magnetic'' limit only for certain forms of magnetic fields. The resulting theory is nothing but the Schr\"odinger theory in a gravitational background generated by the gauge fields.