  The metric for plane gravitational waves is quantized using the Ashtekar
field variables. The z axis (direction of travel of the waves) is taken to be
the entire real line. Solutions to the constraints are proposed; they involve
open-ended flux lines running along the entire z axis. These solutions are
annihilated by the constraints except at the two boundary points, where the
Gauss constraint does not annihilate the solutions. This result is in sharp
contrast to the situation in the general, 3+1 dimensional case without planar
symmetry, where the Gauss constraints do not contribute at boundaries because
the Lagrange multipliers for the Gauss constraints vanish there. The
constraints annihilate the solutions if classical matter is included (so that
flux lines are terminated on the matter). The SU(2) holonomy matrices used in
the solutions are (2j+1) dimensional, where j may be any spin, not necessarily
j = 1/2. In this respect the solutions resemble the spin network states
recently constructed by Rovelli and Smolin in loop space.
