Axial anomaly of QED in a strong magnetic field and noncommutative anomaly

The Adler-Bell-Jackiw (ABJ) anomaly of a $3+1$ dimensional QED is calculated in the presence of a strong magnetic field. It is shown that in the regime with the lowest Landau level (LLL) dominance a dimensional reduction from $D=4$ to $D=2$ dimensions occurs in the longitudinal sector of the low energy effective field theory. In the chiral limit, the resulting anomaly is therefore comparable with the axial anomaly of a two-dimensional massless Schwinger model. It is further shown that the ${U}_{A}(1)$ anomaly of QED in a strong magnetic field is closely related to the nonplanar axial anomaly of a conventional noncommutative $U(1)$ gauge theory.