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.

经典广义相对论是洛伦兹特征的时空度量的动力学理论。特别是，经典度量场在时空中没有退化的地方。在其相对于柯西表面的初始值公式中，诱导度量是欧几里得特征的，并且在其上没有任何地方退化。只有在这种诱导度量不简并的假设下，才能在初始值约束之间推导出超曲面变形代数，这从需要归纳度量的倒数来闭合代数的事实来看是绝对透明的。此语句与可能希望为空间度量配备的密度权重无关。因此，在量子引力中超表面变形代数的非反常表示的定义必须解决经典理论中所需的诱导度量的非简并问题。在环量子引力 （LQG） 中采用的希尔伯特空间表示中，最强调的是定义自旋网络状态的密集域上的逆度量运算符，尽管它们表示几乎到处都是退化的诱导量子几何。毫不奇怪，在这个域上证明约束代数的闭合遇到了困难，因为它是量子理论的一个部门，在经典超曲面变形代数的定义范围之外。已经提出了各种解决该问题的建议，例如非标准运算符拓扑、对偶空间（栖息地）和密度权重，以解决 LQG 量子动力学方面的这个问题。 在本文中，我们总结了这些发展，并认为在 LQG 表示中坚持非简并态的密集域可能会自然地解决这个问题，从而可能避免上述非标准构造。