Relating loop quantum cosmology to loop quantum gravity: Symmetric sectors and embeddings

TL;DR

This paper establishes a rigorous connection between loop quantum cosmology and loop quantum gravity by formulating embeddings of LQC states into LQG, introducing new b-embeddings, and analyzing their properties and operators.

Contribution

It introduces a rigorous embedding framework for LQC states into LQG and constructs new b-embeddings using holomorphic representations, advancing the understanding of their relationship.

Findings

Bojowald-Kastrup embedding is rigorously formulated.

New b-embeddings are constructed via holomorphic representations.

Operators preserving embeddings are identified and shown to be consistent.

Abstract

In this paper we address the meaning of states in loop quantum cosmology (LQC), in the context of loop quantum gravity. First, we introduce a rigorous formulation of an embedding proposed by Bojowald and Kastrup, of LQC states into loop quantum gravity. Then, using certain holomorphic representations, a new class of embeddings, called b-embeddings, are constructed, following the ideas of [1]. We exhibit a class of operators preserving each of these embeddings, and show their consistency with the LQC quantization. In the b-embedding case, the classical analogues of these operators separate points in phase space. Embedding at the gauge and diffeomorphism invariant level is discussed briefly in the conclusions.