Closed-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 proposed in a previous paper involve open-ended flux lines running along the entire $z$ axis, rather than closed loops of flux; consequently, these solutions are annihilated by the Gauss constraint at interior points of the $z$ axis, but not at the two boundary points. The solutions studied in the present paper are based on closed flux loops and satisfy the Gauss constraint for all $z$.