U(1)noncommutative gauge fields and magnetogenesis

The connection between the Lorentz invariance violation in the Lagrangian context and the quantum theory of noncommutative fields is established for the $U(1)$ gauge field. The modified Maxwell equations coincide with other derivations obtained using different procedures. These modified equations are interpreted as describing macroscopic ones in a polarized and magnetized medium. A tiny magnetic field (seed) emerges as particular static solution that gradually increases once the modified Maxwell equations are solved as a self-consistent equations system.