QFT on quantum spacetime: a compatible classical framework

TL;DR

This paper develops a classical framework for the canonical quantization of geometric and matter perturbations on a quantum homogeneous isotropic flat spacetime, extending beyond the first-order approximation of standard cosmological perturbation theory.

Contribution

It introduces a new set of classical phase space variables and Dirac observables that allow consistent quantization of both background and perturbations in quantum cosmology.

Findings

Established a set of Dirac observables with canonical algebra.

Derived the physical Hamiltonian governing the dynamics of perturbations.

Provided a foundation for analyzing quantum spacetime effects on cosmological perturbations.

Abstract

We develop a systematic classical framework to accommodate 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 \emph{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 K-G "clock" field $T$. The results of this work provide a good starting point to understanding and calculating effects that quantum cosmological spacetime in the background has on the quantum perturbations of the metric tensor and of matter fields.