Noncommutativity vs. Transversality in QED in a strong magnetic field

TL;DR

This paper explores how strong magnetic fields in QED lead to noncommutative gauge symmetry at the lowest Landau level, but higher Landau levels restore transversality and gauge invariance in the full theory.

Contribution

It demonstrates the importance of higher Landau levels in restoring gauge invariance and transversality in QED under strong magnetic fields, highlighting a nondecoupling phenomenon.

Findings

LLL approximation yields noncommutative U(1) gauge symmetry.

Higher Landau levels restore transversality and gauge invariance.

Nondecoupling of HLLs is crucial for full effective action.

Abstract

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-Takahashi 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.