Noncommutativity vs. transversality in QED in a strong magnetic field

Quantum electrodynamics (QED) in a strong constant magnetic field is investigated from the viewpoint of its connection with noncommutative QED. It turns out that within the lowest Landau level (LLL) approximation the 1-loop contribution of fermions provides an effective action with the noncommutative U(1)(NC) gauge symmetry. As a result, the Ward identities connected with the initial U(1) gauge symmetry are broken down in the LLL approximation. On the other hand, it is shown that the sum over the infinite number of the higher Landau levels (HLL's) is relevant despite the fact that each contribution of the HLL is suppressed. Owing to this nondecoupling phenomenon the transversality is restored in the whole effective action. The kinematic region where the LLL contribution is dominant is also discussed.