Open-flux solutions to the quantum constraints for plane gravity waves

The metric for plane gravitational waves is quantized within the Hamiltonian framework, using a Dirac constraint quantization and the self-dual field variables proposed by Ashtekar. The z axis (direction of travel of the waves) is taken to be the entire real line rather than the torus [manifold coordinatized by $\mathrm{}(z,t)\mathrm{}$ is $R\times R$ rather than $S_1\times R$]. Solutions to the constraints are proposed; they involve open-ended flux lines running along the entire $z$ axis, rather than closed loops of flux. These solutions are annihilated by the constraints at all interior points of the $z$ axis. At the two boundary points, the Gauss constraint does not annihilate the solutions, because of the presence of open-ended flux lines at the boundaries. 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 the boundaries because the Lagrange multipliers for the Gauss constraints fall to zero at spatial infinity. In the planar symmetry case, the Lagrange multiplier for rotations about the internal $Z$ axis survives at the boundary. The constraints are found to annihilate the solutions when classical matter terms are added to the Hamiltonian (so that flux lines are terminated on the matter). The holonomy matrices used in the solutions are generated by the $\mathrm{}(2j+1)\mathrm{}$-dimensional representation of SU(2), where $j$ may be any spin, not necessarily $j=1/2\mathrm{}$. In this respect the solutions resemble the symmetric states, or spin network states recently constructed by Rovelli and Smolin in loop space. The Rovelli-Smolin area operator for areas in the $\mathrm{xy}$ plane is constructed and applied to the solutions, with puzzling results.