Electron-positron pair oscillation in spatially inhomogeneous electric fields and radiation

It is known that strong electric fields produce electron and positron pairs from the vacuum, and due to the back-reaction these pairs oscillate back and forth coherently with the alternating electric fields in time. We study this phenomenon in spatially inhomogeneous and bound electric fields by integrating the equations of energy-momentum and particle-number conservations and Maxwell equations. The space and time evolutions of the pair-induced electric field, electric charge- and current-densities are calculated. The results show non-vanishing electric charge-density and the propagation of pair-induced electric fields, that are different from the case of homogeneous and unbound electric fields. The space and time variations of pair-induced electric charges and currents emit an electromagnetic radiation. We obtain the narrow spectrum and intensity of this radiation, whose peak ωpeak locates in the region around 4 keV for electric field strength ∼Ec. We discuss their relevances to both the laboratory experiments for electron and positron pair-productions and the astrophysical observations of compact stars with an electromagnetic structure.