Classical Noncommutative Electrodynamics with External Source

TL;DR

This paper extends noncommutative gauge theory to include external sources, finds solutions for static charges, and reveals that such charges behave as magnetic dipoles with size-dependent magnetic moments.

Contribution

It introduces a gauge-invariant extension of the Seiberg-Witten map with external currents and derives classical solutions showing magnetic dipole behavior of static charges.

Findings

Static charge acts as a magnetic dipole with inverse size magnetic moment.

External magnetic field alters Coulomb potential and electromagnetic form-factors.

Solutions are free of singularities and the SW map ambiguity relates to homogeneous solutions.

Abstract

In a $U(1)_{\star}$-noncommutative (NC) gauge field theory we extend the Seiberg-Witten (SW) map to include the (gauge-invariance-violating) external current and formulate - to the first order in the NC parameter - gauge-covariant classical field equations. We find solutions to these equations in the vacuum and in an external magnetic field, when the 4-current is a static electric charge of a finite size $a$, restricted from below by the elementary length. We impose extra boundary conditions, which we use to rule out all singularities, $1/r$ included, from the solutions. The static charge proves to be a magnetic dipole, with its magnetic moment being inversely proportional to its size $a$. The external magnetic field modifies the long-range Coulomb field and some electromagnetic form-factors. We also analyze the ambiguity in the SW map and show that at least to the order studied here it is equivalent to the ambiguity of adding a homogeneous solution to the current-conservation equation.