U(1) gauge invariant noncommutative Schr\"odinger theory and gravity

TL;DR

This paper explores the noncommutative Klein-Gordon field coupled with electromagnetic fields, revealing a connection to Schr"odinger theory in a gravitational background in the non-relativistic limit.

Contribution

It demonstrates how noncommutative Klein-Gordon theory reduces to Schr"odinger theory in a gravitational background using the Seiberg-Witten map and non-relativistic limit analysis.

Findings

Regular magnetic limit exists only for specific magnetic field configurations.

The noncommutative Klein-Gordon theory converges to Schr"odinger theory in a gravitational background.

The approach links noncommutative gauge theories to gravitational effects in the non-relativistic regime.

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 $, 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.