A solution to Ward - Takahashi - identity in QED

Abstract We discuss the properties of fermion propagator in three dimensional QED with dressed photon based on spectral representation with LSZ reduction formula.We evaluate the spectral function by the sum of infinite numbers of rainbow type diagram for intermediate states.In this method we have the solution as one derived by renormalization group in four-dimension.Massless vacuum polarization adds higher order correction for the finite mass shift and position dependent mass which we get in quenched approximation. Renormalization constant vanishes for arbitrary coupling and confining properties is not alterd by vacuum polarization.Vacuum expectation value of fermion pair condensate becomes finite above the a critical coupling constant.In the long distance the screening effect is significant to suppress mass shift in our approximation.