Mathematical Structure of Loop Quantum Cosmology: Homogeneous Models

TL;DR

This paper develops a more general mathematical framework for homogeneous loop quantum cosmology, addressing limitations of previous Abelian models and improving the connection to the full loop quantum gravity theory.

Contribution

Introduces a new quantization method for non-Abelian homogeneous models, overcoming degeneracies of previous approaches and aligning cosmological models more closely with the full theory.

Findings

New quantization applies to non-Abelian models

Lattice refinement emerges from state-dependent regularizations

Enhanced connection to full loop quantum gravity theory

Abstract

The mathematical structure of homogeneous loop quantum cosmology is analyzed, starting with and taking into account the general classification of homogeneous connections not restricted to be Abelian. As a first consequence, it is seen that the usual approach of quantizing Abelian models using spaces of functions on the Bohr compactification of the real line does not capture all properties of homogeneous connections. A new, more general quantization is introduced which applies to non-Abelian models and, in the Abelian case, can be mapped by an isometric, but not unitary, algebra morphism onto common representations making use of the Bohr compactification. Physically, the Bohr compactification of spaces of Abelian connections leads to a degeneracy of edge lengths and representations of holonomies. Lifting this degeneracy, the new quantization gives rise to several dynamical properties, including lattice refinement seen as a direct consequence of state-dependent regularizations of the Hamiltonian constraint of loop quantum gravity. The representation of basic operators - holonomies and fluxes - can be derived from the full theory specialized to lattices. With the new methods of this article, loop quantum cosmology comes closer to the full theory and is in a better position to produce reliable predictions when all quantum effects of the theory are taken into account.