Non-Almost Periodicity of Parallel Transports for Homogeneous Connections

Abstract

Let ${\cal A}$ be the affine space of all connections in an SU(2) principal fibre bundle over ℝ³. The set of homogeneous isotropic connections forms a line l in ${\cal A}$. We prove that the parallel transports for general, non-straight paths in the base manifold do not depend almost periodically on l. Consequently, the embedding $l \hookrightarrow {\cal A}$ does not continuously extend to an embedding $\overline{l} \hookrightarrow \overline{\cal A}$ of the respective compactifications. Here, the Bohr compactification $\overline{l}$ corresponds to the configuration space of homogeneous isotropic loop quantum cosmology and $\overline{\cal A}$ to that of loop quantum gravity. Analogous results are given for the anisotropic case.