Dynamics of electron-positron pairs in a vacuum polarized by an external radiation field

The polarization of the vacuum under the action of an external classical field of electromagnetic radiation is investigated in the stationary regime. The electron-positron pairs interact both with the external field and with their own polarization field. For a macroscopic piece of vacuum the pairs are condensed on the low-momenta states and tend to form a quasi-localized electron-positron plasma of pairs, with single-particle states labeled by the position vector. In the polarization process under the action of a classical field of radiation the electron-positron and photon dynamics can be treated by means of classical fields. Under these circumstances, the corresponding coupled non-linear equations of motion are solved. It is shown that the pair dynamics consists of quasi-stationary single-particle states, while the polarization field reduces to a static magnetic field. The single-particle 'energy' (temporal phase) due to a monochromatic external field exhibits a spatial distribution characteristic of a stationary wave. Both the pair energy and the polarization energy are computed. Their values are extremely small, even for highly focused, reasonably high, external fields. The number of pairs is determined by the external energy. Under the action of a classical field the polarized vacuum is magnetized, and the corresponding (very low) magnetic susceptibility (the refractive index of the vacuum) is computed.