On Time-Evolution in Quantum Gravity

TL;DR

We address the problem of time evolution in quantum gravity by employing BRST quantization of General Relativity as an EFT that unifies the path-integral and canonical pictures. The main result is that the bulk Hamiltonian is BRST-exact, $\hat{H}=M_{\rm pl}\int d^3x\{\hat{Q}, i\hat{\Pi}^c_0\}$, up to boundary terms, which makes the Hamiltonian flow act as a time reparameterization on physical correlators. This structure preserves nontrivial bulk dynamics and the $S$-matrix (via BRST-invariant asymptotic states and IR-dressing) while isolating time evolution to BRST-exact pieces and boundary contributions in asymptotically flat spacetimes. The framework provides a resolution to DeWitt's puzzle by showing how BRST cohomology yields time dependence for gauge-variant observables without trivializing the gravitational dynamics, and it points to explicit constructions of BRST-invariant DeWitt-like states and a boundary-aware S-matrix. Significance lies in a consistent quantum-gravity setup that reconciles time evolution with BRST symmetry and unitarity, suitable for both Minkowski backgrounds and asymptotically flat settings.

Abstract

We derive an explicit BRST-exact operator identity for the bulk Hamiltonian in quantum gravity, working within a BRST-invariant quantization of General Relativity, treated as a low-energy effective field theory. We show that, up to a boundary term, the Hamiltonian can be written elegantly as the anticommutator of the BRST charge and the temporal ghost field. This form makes manifest that the Hamiltonian flow acts as a time-reparameterization on the correlation functions of the physical degrees of freedom. We demonstrate that the BRST-exactness of the bulk Hamiltonian does not trivialize the time evolution of gravitational backgrounds or bulk correlators, nor does it trivialize scattering amplitudes.