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On loop quantum gravity kinematics with non-degenerate spatial background

Hanno Sahlmann

TL;DR

This paper addresses how loop quantum gravity kinematics can be extended to include a non-degenerate background spatial geometry. It demonstrates that standard geometric operators can be defined with the same regularization and split into a vacuum part and a background contribution, e.g., $\widehat{A}_S = \widehat{A}_S^{vac} + A_S^{(0)} id$ and $V_R = V_R^{vac} + V_R^{(0)} id$. It also shows that diffeomorphism and gauge transformations can be implemented via an enlarged Hilbert space bundling background geometries, yielding an anomaly-free representation of the bundle automorphism group. Finally, the paper clarifies constraint implementation, showing that the diffeomorphism constraint can be handled as in the vacuum case while the Gauß constraint requires gauge-invariant starting states to avoid divergences, and it constructs invariant spaces for both diffeomorphisms and gauge transformations. These results imply a viable semiclassical or effective framework for LQG on large scales, with limitations relative to the fundamental vacuum representation.

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.

On loop quantum gravity kinematics with non-degenerate spatial background

TL;DR

This paper addresses how loop quantum gravity kinematics can be extended to include a non-degenerate background spatial geometry. It demonstrates that standard geometric operators can be defined with the same regularization and split into a vacuum part and a background contribution, e.g., and . It also shows that diffeomorphism and gauge transformations can be implemented via an enlarged Hilbert space bundling background geometries, yielding an anomaly-free representation of the bundle automorphism group. Finally, the paper clarifies constraint implementation, showing that the diffeomorphism constraint can be handled as in the vacuum case while the Gauß constraint requires gauge-invariant starting states to avoid divergences, and it constructs invariant spaces for both diffeomorphisms and gauge transformations. These results imply a viable semiclassical or effective framework for LQG on large scales, with limitations relative to the fundamental vacuum representation.

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.

Paper Structure

This paper contains 11 sections, 2 theorems, 69 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The new kinematical representations
  3. Geometric operators
  4. Area
  5. Volume
  6. Unitary implementation of bundle automorphisms
  7. Implementation of the constraints
  8. Diffeomorphism constraint
  9. Gauß constraint
  10. States invariant under bundle automorphisms
  11. Outlook

Key Result

Proposition 3.1

The standard regularization and quantization procedures for area and volume operators can be applied to the new representations and lead to well defined operators. We find with $V^{\text{vac}}, A^{\text{vac}}$ the geometric operators in the vacuum representation, and $V^{(0)}_R,A^{(0)}_S$ the classical values in the background geometry.

Figures (2)

  • Figure 1: Surface with single vertex $v$ as described in the text.
  • Figure 2: The different types of cubes considered in the text.

Theorems & Definitions (3)

  • Proposition 3.1
  • Lemma 5.1
  • proof