The generator of spatial diffeomorphisms in the Koslowski- Sahlmann representation

TL;DR

This paper develops a method to construct the diffeomorphism constraint operator in the Koslowski-Sahlmann representation of Loop Quantum Gravity, highlighting new background exponential operators that modify smooth spatial geometries.

Contribution

It introduces a construction of the diffeomorphism constraint operator in the KS representation and reveals new background exponential operators affecting spatial geometry labels.

Findings

Construction of the diffeomorphism constraint operator in KS representation.

Identification of background exponential operators that change spatial geometry labels.

Demonstration that these operators are quantizations of connection-dependent functions.

Abstract

A generalization of the representation underlying the discrete spatial geometry of Loop Quantum Gravity, to accomodate states labelled by smooth spatial geometries, was discovered by Koslowski and further studied by Sahlmann. We show how to construct the diffeomorphism constraint operator in this Koslowski- Sahlmann (KS) representation from suitable connection and triad dependent operators. We show that the KS representation supports the action of hitherto unnoticed "background exponential" operators which, in contrast to the holonomy- flux operators, change the smooth spatial geometry label of the states. These operators are shown to be quantizations of certain connection dependent functions and play a key role in the construction of the diffeomorphism constraint operator.