Non-degenerate metrics, hypersurface deformation algebra, non-anomalous representations and density weights in quantum gravity

Classical General Relativity is a dynamical theory of spacetime metrics of Lorentzian signature. In particular the classical metric field is nowhere degenerate in spacetime. In its initial value formulation with respect to a Cauchy surface the induced metric is of Euclidian signature and nowhere degenerate on it. It is only under this assumption of non-degeneracy of the induced metric that one can derive the hypersurface deformation algebra between the initial value constraints which is absolutely transparent from the fact that the inverse of the induced metric is needed to close the algebra. This statement is independent of the density weight that one may want to equip the spatial metric with. Accordingly, the very definition of a non-anomalous representation of the hypersurface deformation algebra in quantum gravity has to address the issue of non-degeneracy of the induced metric that is needed in the classical theory. In the Hilbert space representation employed in Loop Quantum Gravity (LQG) most emphasis has been laid to define an inverse metric operator on the dense domain of spin network states although they represent induced quantum geometries which are degenerate almost everywhere. It is no surprise that demonstration of closure of the constraint algebra on this domain meets difficulties because it is a sector of the quantum theory which is classically forbidden and which lies outside the domain of definition of the classical hypersurface deformation algebra. Various suggestions for addressing the issue such as non-standard operator topologies, dual spaces (habitats) and density weights have been proposed to address this issue with respect to the quantum dynamics of LQG. In this article we summarise these developments and argue that insisting on a dense domain of non-degenerate states within the LQG representation may provide a natural resolution of the issue thereby possibly avoiding the above mentioned non-standard constructions.