A quantum reduction to spherical symmetry in loop quantum gravity

TL;DR

This paper introduces two quantum reduction methods to impose spherical symmetry in loop quantum gravity, enabling tractable quantum dynamics for spherically symmetric spacetimes including dust collapse.

Contribution

It proposes novel quantum reduction techniques for spherical symmetry in full loop quantum gravity using geometric observables and group averaging, facilitating analysis of spherically symmetric models.

Findings

Group averaging over rotations yields spherically symmetric states.

Full theory Hamiltonians with angle-independent lapse preserve the symmetric sector.

The reduced holonomy-flux algebra simplifies to a 2+1-dimensional structure along the radial coordinate.

Abstract

Based on a recent purely geometric construction of observables for the spatial diffeomorphism constraint, we propose two distinct quantum reductions to spherical symmetry within full 3+1-dimensional loop quantum gravity. The construction of observables corresponds to using the radial gauge for the spatial metric and allows to identify rotations around a central observer as unitary transformations in the quantum theory. Group averaging over these rotations yields our first proposal for spherical symmetry. Hamiltonians of the full theory with angle-independent lapse preserve this spherically symmetric subsector of the full Hilbert space. A second proposal consists in implementing the vanishing of a certain vector field in spherical symmetry as a constraint on the full Hilbert space, leading to a close analogue of diffeomorphisms invariant states. While this second set of spherically symmetric states does not allow for using the full Hamiltonian, it is naturally suited to implement the spherically symmetric midisuperspace Hamiltonian, as an operator in the full theory, on it. Due to the canonical structure of the reduced variables, the holonomy-flux algebra behaves effectively as a one parameter family of 2+1-dimensional algebras along the radial coordinate, leading to a diagonal non-vanishing volume operator on 3-valent vertices. The quantum dynamics thus becomes tractable, including scenarios like spherically symmetric dust collapse.