Symmetry-reduced Loop Quantum Gravity: Plane Waves, Flat Space and the Hamiltonian Constraint

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.