  Here we examine the noncommutative counterpart of QED, which is called as
noncommutative QED. The theory is obtained by examining the consistent minimal
coupling to noncommutative U(1) gauge field. The *-product admits the coupling
of the matter with only three varieties of charges, i.e., 0, +1 and -1.
Ultraviolet divergence can be absorbed into the rescaling of the fields and the
parameters at least at one loop level. To examine the infrared aspect of the
theory the anomalous magnetic dipole moment is calculated. The dependence on
the direction of photon momentum reflects the Lorentz symmetry violation of the
system. The explicit calculation of the finite part of the photon vacuum
polarization shows the singularity ln({q C^TC q}) (C^{\mu\nu} is a
noncommutative parameter.) in the infrared side which also exists in
noncommutative Yang-Mills theory. It is associated with the ultraviolet
behavior of the theory. We also consider the extension to chiral gauge theory
in the present context, but the requirement of anomaly cancellation allows only
noncommutative QED.
