A Toy Model Approach to the Canonical Non-perturbative Quantization of the Spatially Flat Robertson-Walker Spacetimes with Cosmological Constant

TL;DR

This paper introduces a toy model for the non-perturbative quantum gravity of flat Robertson-Walker universes with a cosmological constant, revealing quantum fluctuations and volume quantization effects.

Contribution

It provides an exactly solvable model that connects the Wheeler-DeWitt equation with classical Einstein equations, highlighting quantum effects in cosmological expansion.

Findings

Quantum fluctuations of energy and metric field observed.

Volume quantization emerges from quantum fluctuations.

Fock states are not invariant under the Hamiltonian in an expanding universe.

Abstract

We present a toy model approach to the canonical non-perturbative quantization of the spatially-flat Robertson-Walker Universes with cosmological constant, based on the fact that such models are exactly solvable within the framework of a simple Lagrangian formulation. The essential quantum dynamical metric-field and the corresponding Hamiltonian, explicitly derived in terms of annihilation and creation operators, point out that the Wheeler - DeWitt equation is a natural (quantum) generalization of the $G_{44}$ - Einstein equation for the classical De Sitter spacetime and selects the physical states of the quantum De Sitter Universe. As a result of the exponential universal expansion, the usual Fock states (defined as the eigenstates of the number-operator) are no longer invariant under the derived Hamiltonian. They exhibit quantum fluctuation of the energy and of the metric field which lead to a (geometrical) volume quantization.