Symmetry-reduced loop quantum gravity: plane waves, flat space and the Hamiltonian constraint

Loop quantum gravity (LQG) methods are applied to a symmetry-reduced model with homogeneity in two dimensions, derived from a Gowdy model. The conditions for the propagation of unidirectional plane gravitational waves at exactly the speed of light are set up in the form of two null Killing equations in terms of Ashtekar variables and imposed as operators on the quantum states of the system. Owing to symmetry reduction the gauge group of the system reduces formally from SU(2) to U(1). Under the assumption of equal spacing of the holonomy eigenvalues, the solutions are not normalizable in the sense of the usual inner product on U(1). 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 (non-commuting) Killing conditions have to be renormalized. Two kinds of renormalization are presented. The combination of the occurrence of non-commuting Killing operators and the necessity of renormalization indicates fluctuations of the propagation speed, i. e. dispersion of gravitational waves. Finally the same methods are applied to the Hamiltonian constraint with the same result concerning normalizability. After renormalization the constraint is not exactly satisfied any more, which suggests the presence of some kind of interacting matter.