The Chern–Weil topological theory is applied to a classical formulation of general relativity in four-dimensional spacetime. Einstein–Hilbert gravitational action is shown to be invariant with respect to a novel translation (co-translation) operator up to the total derivative; thus, a topological invariant of a second Chern class exists owing to Chern–Weil theory. Using topological insight, fundamental forms can be introduced as a principal bundle of the spacetime manifold. Canonical quantization of general relativity is performed in a Heisenberg picture using the Nakanishi–Kugo–Ojima formalism in which a complete set of quantum Lagrangian and BRST transformations including auxiliary and ghost fields is provided in a self-consistent manner. An appropriate Hilbert space and physical states are introduced into the theory, and the positivity of these physical states and the unitarity of the transition matrix are ensured according to the Kugo–Ojima theorem. The nonrenormalizability of quantum gravity is reconsidered under the formulation proposed herein.

Chern-Weil拓扑理论适用于二维时空中广义相对论的经典表述。对于新的翻译（共翻译）算符，爱因斯坦-希尔伯特的引力作用是不变的，直到总导数为止。因此，由于Chern-Weil理论，存在第二个Chern类的拓扑不变量。利用拓扑洞察力，基本形式可以作为时空流形的主要束引入。使用Nakanishi–Kugo–Ojima形式主义，在Heisenberg图片中对广义相对论进行规范化量化，其中以自洽的方式提供了包括辅助场和重影场在内的一整套量子Lagrangian和BRST变换。理论中引入了适当的希尔伯特空间和物理状态，根据Kugo-Ojima定理，可以确保这些物理状态的正性和过渡矩阵的统一性。根据本文提出的公式，重新考虑了量子引力的不可归一化性。