On the Configuration Spaces of Homogeneous Loop Quantum Cosmology and Loop Quantum Gravity

TL;DR

The paper investigates the relationship between the configuration spaces of homogeneous loop quantum cosmology and loop quantum gravity, revealing that the embedding of the former into the latter is not continuous due to path dependence issues.

Contribution

It demonstrates that the natural embedding of homogeneous isotropic connections into the full theory's configuration space is not continuous, highlighting fundamental differences in their topological structures.

Findings

Embedding of homogeneous isotropic connections is not continuous.

Parallel transports along non-straight paths depend non-almost periodically.

Results extend to anisotropic cases.

Abstract

The set of homogeneous isotropic connections, as used in loop quantum cosmology, forms a line $l$ in the space of all connections $\cal A$. This embedding, however, does not continuously extend to an embedding of the configuration space $\overline l$ of homogeneous isotropic loop quantum cosmology into that of loop quantum gravity, $\overline{\cal A}$. This follows from the fact that the parallel transports for general, non-straight paths in the base manifold do not depend almost periodically on $l$. Analogous results are given for the anisotropic case.