  Using a quantum kinetic equation coupled to Maxwell's equation we study the
possibility that focused beams at proposed X-ray free electron laser facilities
can generate electric field strengths large enough to cause spontaneous
electron-positron pair production from the QED vacuum. Our approach yields the
time and momentum dependence of the single particle distribution function.
Under conditions reckoned achievable at planned facilities, repeated cycles of
particle creation and annihilation take place in tune with the laser frequency.
However, the peak particle number density is insensitive to this frequency and
one can anticipate the production of a few hundred particle pairs per laser
period. Field-current feedback and quantum statistical effects are small and
can be neglected in this application of non-equilibrium quantum mean field
theory.
