  Loop quantum gravity methods are applied to a symmetry-reduced model with
homogeneity in two dimensions, derived from a Gowdy model [5,6]. The conditions
for propagation of unidirectional plane gravitational waves at exactly the
speed of light are set up in form of null Killing equations in terms of
Ashtekar variables and imposed as operators on quantum states of the system.
Due to the effective one-dimensionality, holonomies and holonomy operators
appear as simple phase factors. In correspondence, state functions might be
considered as U(1) elements with the usual inner product. Under the assumption
of equal spacing of the eigenvalues of geometrical quantities the solutions are
not normalizable in this sense. With decreasing spacing for growing
eigenvalues, as introduced for example in [11], the situation becomes worse.
Taking over the inner product from the genuine gauge group SU(2) of LQG renders
the obtained states normalizable, nevertheless fluctuations of geometrical
quantities remain divergent. In consequence, the solutions of the Killing
conditions are modified, which means allowing for small fluctuations of the
propagation speed, i. e. dispersion of gravitational waves. Vacuum fluctuations
of Minkowski space are sketched. Finally the same methods are applied to the
Hamiltonian constraint with the same result concerning normalizability. With
such a modification also the constraint is not exactly satisfied any more,
which indicates the necessary presence of some kind of interacting matter.
