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Backward Compton Scattering in Strong Uniform Magnetic Field

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.