Canonical quantization of all minisuperspaces with consistent symmetry reductions

TL;DR

This paper develops a comprehensive quantization framework for all symmetry-reduced minisuperspaces in general relativity, ensuring consistency with Einstein's equations and exploring their quantum properties.

Contribution

It introduces a unified quantization approach for all symmetry reductions of the Einstein-Hilbert action, including derivation of Hamiltonians and solutions to the Wheeler-DeWitt equation.

Findings

Successfully quantized various symmetric spacetimes including Schwarzschild, FLRW, and Bianchi models.

Derived Hamiltonians and conformal symmetries for the superspace metrics.

Solved the Wheeler-DeWitt equation with symmetry constraints.

Abstract

We present the quantization of all symmetry reductions of the Einstein--Hilbert Lagrangian that correctly reproduce the reduced Einstein's field equations -- i.e., characterized by the infinitesimal group actions obeying the principle of symmetric criticality. These correspond to the spacetime symmetries of spherical/hyperbolic/planar Schwarzschild/Taub--NUT, BI/BII/BIII-metrics, near-horizon extreme Kerr geometry, swirling universe, closed/open/flat FLRW cosmologies, other FLRW-type metrics, and Bianchi type I, II, VIII, and IX spacetimes. We derive the Hamiltonian and the conformal symmetries of the superspace metrics (the conditional symmetries), promote them to operators, and solve the Wheeler--DeWitt equation with and without imposing these symmetries.