The standard model of particle physics predicts a very small value for the electric dipole moment of fundamental particles. In recent years, extensive efforts has been done to explore the experimental electric dipole moment in atoms and their constituent particles which have been leaded to value for the electric dipole moment, several times larger than the predicted one by the Standard Model. The Nonzero electric dipole moment for the fundamental particles Lead to CP symmetry breaking. Meanwhile, the only source of the CP violation in the Standard Model is related to the phase in the quarks mass matrix elements (CKM matrix) which is too small to explain the peresent bounds on the EDM of leptons. Furthermore, in the standard model extension (SME) there are terms which explicitly violate the CP symmetry. These terms can generate the value for the EDM as large as the experimental bounds. In this thesis we give a brief review on the standard model extension as a suitable framework for the study of the Lorentz and CPT violations. In the QED part of the SME the fermionic currents besides the electric charge and magnetic dipole form factors depend on the electric dipole form factor too. Then assuming interaction of the electric dipole moment of e with photon and by using the obtained value of the electric dipole moment in the SME and different limits that obtained for Lorentz violating parameter , we calculate the total cross-section of the Compton scattering. We compare this cross section with the cross section of the Compton scattering in the standard model. The obtained result indicates that there is a possibility of observing the effects of Lorentz violation in the Compton scattering cross sectio at the energies higher than 100 GeV ; in fact for the energies less than 100 GeV there is no valuable deviation from the ordinary standard model.