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.