A proposal for analyzing the classical limit of kinematic loop gravity

TL;DR

This paper investigates the classical limit of kinematic loop quantum gravity, showing limitations in approximating classical variables with quantum states and proposing a lattice-based approach using macroscopic operators for better classical correspondence.

Contribution

It introduces a novel lattice-based framework for analyzing the classical limit in loop quantum gravity, emphasizing macroscopic operators and quasi-classical states.

Findings

No quantum states approximate all classical holonomies along all loops.

A countable set of loops based on physical lattices suffices for classical approximation.

Explicit construction of quasi-classical states in 2D and potential generalization to 3D.

Abstract

We analyze the classical limit of kinematic loop quantum gravity in which the diffeomorphism and hamiltonian constraints are ignored. We show that there are no quantum states in which the primary variables of the loop approach, namely the SU(2) holonomies along {\em all} possible loops, approximate their classical counterparts. At most a countable number of loops must be specified. To preserve spatial covariance, we choose this set of loops to be based on physical lattices specified by the quasi-classical states themselves. We construct ``macroscopic'' operators based on such lattices and propose that these operators be used to analyze the classical limit. Thus, our aim is to approximate classical data using states in which appropriate macroscopic operators have low quantum fluctuations. Although, in principle, the holonomies of `large' loops on these lattices could be used to analyze the classical limit, we argue that it may be simpler to base the analysis on an alternate set of ``flux'' based operators. We explicitly construct candidate quasi-classical states in 2 spatial dimensions and indicate how these constructions may generalize to 3d. We discuss the less robust aspects of our proposal with a view towards possible modifications. Finally, we show that our proposal also applies to the diffeomorphism invariant Rovelli model which couples a matter reference system to the Hussain Kucha{\v r} model.