QFT on quantum spacetime: A compatible classical framework

We develop a systematic classical framework to accommodate the canonical quantization of geometric and matter perturbations on a quantum homogeneous isotropic flat spacetime. The existing approach of standard cosmological perturbations is indeed proved to be good only up to first order in the inhomogeneities, and only if the background is treated classically. To consistently quantize the perturbations and the background, a new set of classical phase-space variables is required. We show that, in a natural gauge, a set of such Dirac observables exists, and their algebra is of the canonical form. Finally, we compute the physical Hamiltonian that generates the dynamics of such observables with respect to the homogeneous part of a Klein-Gordon “clock” field $T$. The results of this work provide a good starting point to understanding and calculating the effects that quantum cosmological spacetime in the background has on the quantum perturbations of the metric tensor and of matter fields.