Symmetry Reduction of Loop Quantum Gravity

TL;DR

This paper investigates the topological obstructions preventing the embedding of Loop Quantum Cosmology into full Loop Quantum Gravity, deriving the topology of relevant subspaces and constructing an embedding that aligns with previous work.

Contribution

It identifies and analyzes the topological obstruction, extending the LQC configuration space to enable a consistent embedding into LQG.

Findings

Extended LQC configuration space admits an embedding into LQG.

The constructed embedding coincides asymptotically with standard LQC.

Topological analysis clarifies the relation between LQC and LQG.

Abstract

The relation between standard Loop Quantum Cosmology and full Loop Quantum Gravity fails already at the first nontrivial step: The configuration space of Loop Quantum Cosmology can not be embedded into the configuration space of full Loop Quantum Gravity due to a topological obstruction. We investigate this obstruction in detail, because many topological obstructions are the source of physical effects. For this we derive the topology of a large class of subspaces of the Loop Quantum Gravity configuration space. This allows us to find the extension of the standard Loop Quantum Cosmology configuration space that admits an embedding in agreement with Fleischhack:2010. We then construct the embedding for flat FRW Loop Quantum Cosmology and find that it coincides asymptotically with standard LQC.