  In strong uniform magnetic field, the vacuum Non-Commutative Plane (NCP)
caused by the lowest Landau level(LLL) effect and the QED with NCP (QED-NCP)
are studied. Being similar to the theory of Quantum Hall effect, an effective
filling factor $f(B)$ is introduced to character the possibility that the
electrons stays on LLL. The backward Compton scattering amplitudes of QED-NCP
are derived, and the differential cross sections for the process with polarized
initial electrons and photons are calculated. The existing Spring-8's data has
been analyzed primitively and some hints for QED-NCP effects are shown. We
propose to precisely measure the differential cross sections of the backward
Compton scattering in perpendicular magnetic field experimentally, which may
lead to reveal the effects of QED-NCP.
  PACS number: 12.20.Ds; 11.10.Nx; 71.70.Di; 73.43.Fj.
