On loop quantum gravity kinematics with non-degenerate spatial background

TL;DR

This paper extends and refines loop quantum gravity kinematic representations with a non-degenerate background, demonstrating well-defined geometric operators and exploring their potential for effective large-scale theories.

Contribution

It generalizes Koslowski's non-degenerate background representations, showing regularized operators are well-defined and detailing the implementation of constraints.

Findings

Standard geometric operators are well-defined in new representations.

The standard representation is a special case of the generalized constructions.

These representations may aid in developing effective large-scale theories.

Abstract

In a remarkable paper, T. Koslowski introduced kinematical representations for loop quantum gravity in which there is a non-degenerate spatial background metric present. He also considered their properties, and showed that Gauss and diffeomorphism constraints can be implemented. With the present article, we streamline and extend his treatment. In particular, we show that the standard regularization of the geometric operators leads to well defined operators in the new representations, and we work out their properties fully. We also give details on the implementation of the constraints. All of this is done in such a way as to show that the standard representation is a particular (and in some ways exceptional) case of the more general constructions. This does not mean that these new representations are as fundamental as the standard one. Rather, we believe they might be useful to find some form of effective theory of loop quantum gravity on large scales.