Projective Structures in Loop Quantum Cosmology

Abstract: Projective structures have successfully been used for the construction of measures in the framework of loop quantum gravity. In the present paper, we establish such structures for the configuration space $\mathbb{R}\sqcup \mathbb{R}{\mathbb{Bohr}}$, recently introduced in the context of homogeneous isotropic loop quantum cosmology. In contrast to the traditional space $\mathbb{R}{\mathbb{Bohr}}$, the first one is canonically embedded into the quantum configuration space of the full theory. In particular, for the embedding of states into a corresponding symmetric sector of loop quantum gravity, this is advantageous. However, in contrast to the traditional space, there is no Haar measure on $\mathbb{R}\sqcup \mathbb{R}{\mathbb{Bohr}}$ defining a canonical kinematical $L^2$-Hilbert space on which operators can be represented. The introduced projective structures allow to construct a family of natural measures on $\mathbb{R}\sqcup \mathbb{R}{\mathbb{Bohr}}$ whose corresponding $L^2$-Hilbert spaces we finally investigate.