A quantum reduction to spherical symmetry in loop quantum gravity

Understand

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.