  There are definite plans for the construction of X-ray free electron lasers
(FEL), both at DESY, where the so-called XFEL is part of the design of the
electron-positron linear collider TESLA, as well as at SLAC, where the
so-called Linac Coherent Light Source (LCLS) has been proposed. Such an X-ray
laser would allow for high-field science applications: One could make use of
not only the high energy and transverse coherence of the X-ray beam, but also
of the possibility of focusing it to a spot with a small radius, hopefully in
the range of the laser wavelength. Along this route one obtains very large
electric fields, much larger than those obtainable with any optical laser of
the same power. In this letter we discuss the possibility of obtaining an
electric field so high that electron-positron pairs are spontaneously produced
in vacuum (Schwinger pair production). We find that if X-ray optics can be
improved to approach the diffraction limit of focusing, and if the power of the
planned X-ray FELs can be increased to the terawatt region, then there is ample
room for an investigation of the Schwinger pair production mechanism.
