Nakanishi–Kugo–Ojima quantization of general relativity in Heisenberg picture
High Energy Accelerator Organization (KEK), 305-0801, Tsukuba, Ibaraki, Japan
Accepted: 19 April 2021
Published online: 28 April 2021
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.
© The Author(s), under exclusive licence to Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2021