Embedding loop quantum cosmology without piecewise linearity

TL;DR

This paper develops a new embedding of loop quantum cosmology into loop quantum gravity that does not rely on piecewise linear paths, enabling the definition of more general holonomy operators and strengthening the link between the two theories.

Contribution

It introduces a novel embedding of LQC into LQG that avoids piecewise linearity, allowing for the inclusion of non-piecewise-linear holonomies at the quantum level.

Findings

Embedding is well-defined without fixing graphs or using piecewise linear paths.

Operators for holonomies along non-piecewise-linear paths are defined in LQC.

The embedding satisfies an operator equation implying homogeneity and isotropy.

Abstract

An important goal is to understand better the relation between full loop quantum gravity (LQG) and the simplified, reduced theory known as loop quantum cosmology (LQC), {\em directly at the quantum level}. Such a firmer understanding would increase confidence in the reduced theory as a tool for formulating predictions of the full theory, as well as permitting lessons from the reduced theory to guide further development in the full theory. The present paper constructs an embedding of the usual state space of LQC into that of standard LQG, that is, LQG based on \textit{piecewise analytic paths}. The embedding is well-defined even prior to solving the diffeomorphism constraint, at no point is a graph fixed, and at no point is the piecewise linear category used. This motivates for the first time a definition of operators in LQC corresponding to holonomies along non-piecewise-linear paths, without changing the usual kinematics of LQC in any way. The new embedding intertwines all operators corresponding to such holonomies, and all elements in its image satisfy an operator equation which classically implies homogeneity and isotropy. The construction is made possible by a recent result proven by Fleischhack.