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

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.

30 pages; 2 proofs added, notation changed, minor corrections and changes; organizational changes, discussion of sectors and embeddings better distinguished, discussion of interpretation added, another proof added, title changed; minor corrections, final version published in CQG

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