Minisuperspace Model for Revised Canonical Quantum Gravity

TL;DR

This paper introduces a new approach to canonical quantum gravity using a fixed reference frame, resulting in a Schrödinger equation with a dust fluid interpretation, and applies it to cosmological models like Bianchi IX.

Contribution

The paper reformulates canonical quantum gravity in minisuperspace with a fixed Gaussian frame, leading to a Schrödinger equation and a dust fluid interpretation, extending to generic cosmological solutions.

Findings

Derivation of a Schrödinger equation with non-zero super-Hamiltonian eigenvalues.

Interpretation of the eigenvalues as a dust fluid energy density.

Application to Bianchi IX universe and classical cosmological implications.

Abstract

We present a reformulation of the canonical quantization of gravity, as referred to the minisuperspace; the new approach is based on fixing a Gaussian (or synchronous) reference frame and then quantizing the system via the reconstruction of a suitable constraint; then the quantum dynamics is re-stated in a generic coordinates system and it becomes dependent on the lapse function. The analysis follows a parallelism with the case of the non-relativistic particle and leads to the minisuperspace implementation of the so-called {\em kinematical action} as proposed in \cite{M02} (here almost coinciding also with the approach presented in \cite{KT91}). The new constraint leads to a Schr\"odinger equation for the system. i.e. to non-vanishing eigenvalues for the super-Hamiltonian operator; the physical interpretation of this feature relies on the appearance of a ``dust fluid'' (non-positive definite) energy density, i.e. a kind of ``materialization'' of the reference frame. As an example of minisuperspace model, we consider a Bianchi type IX Universe, for which some dynamical implications of the revised canonical quantum gravity are discussed. We also show how, on the classical limit, the presence of the dust fluid can have relevant cosmological issues. Finally we upgrade our analysis by its extension to the generic cosmological solution, which is performed in the so-called long-wavelength approximation. In fact, near the Big-Bang, we can neglect the spatial gradients of the dynamical variables and arrive to implement, in each space point, the same minisuperspace paradigm valid for the Bianchi IX model.