On loop quantum gravity kinematics with non-degenerate spatial background

Hanno Sahlmann

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On loop quantum gravity kinematics with non-degenerate spatial background

Hanno Sahlmann

General Relativity and Quantum Cosmology · Submitted 2 Jun 2010

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim 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.

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What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.93 · human participants 0.01 · controlled comparison 0.09 · randomized assignment 0.01 · states quantitative result 0.03 · compares with prior results 0.93 · mentions code or data availability 0.02.

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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.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	In a remarkable paper, T.	background 1.00	0.03	0.01	unclear 0.83
s1	Koslowski introduced kinematical representations for loop quantum gravity in which there is a non-degenerate spatial background metric present.	background 1.00	0.03	0.05	asserted 1.00
s2	He also considered their properties, and showed that Gauss and diffeomorphism constraints can be implemented.	background 0.97	0.04	0.04	asserted 1.00
s3	With the present article, we streamline and extend his treatment.	contribution 0.99	0.46	0.06	asserted 0.98
s4	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.	result 0.57	0.95	0.27	asserted 1.00
s5	We also give details on the implementation of the constraints.	contribution 0.98	0.92	0.04	asserted 0.98
s6	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.	interpretation 0.49	0.83	0.43	asserted 1.00
s7	This does not mean that these new representations are as fundamental as the standard one.	limitation 0.79	0.35	0.09	asserted 0.96
s8	Rather, we believe they might be useful to find some form of effective theory of loop quantum gravity on large scales.	interpretation 0.70	0.24	0.28	speculative 0.79

What kind of paper the classifier reports this is: theoretical result 0.98 · evidence from proof or derivation 1.00.

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The paper

Original 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.

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References

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

A new vacuum for Loop Quantum Gravity
gr-qc 2014-01 conditional novelty 6.0 of 10

A flat-connection (BF) vacuum dual to the Ashtekar-Lewandowski vacuum is constructed for 2+1 dimensional loop quantum gravity, together with a modified holonomy-flux algebra that is claimed to be consistent under Pach...
Quantum Cosmology from Group Field Theory Condensates: a Review
gr-qc 2016-02 accept novelty 2.0 of 10

The review argues that a macroscopic universe can be modeled as a group field theory condensate whose effective dynamics reproduces general-relativistic and loop-quantum-cosmology equations in suitable limits.

How the literature uses it

15 arXiv papers cite this in 57 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 42 sentences without a confident label

Contests it 1 paper

It is pertinent to note that in the case of generic triads of rank 2,3 there are no internal rotations which preserve the triad so that the argument of the previous paragraph cannot be applied to this case [This was not realised in the pioneering work of Reference [ref] . ].

The Koslowski- Sahlmann representation: Gauge and diffeomorphism invariance · On the restriction to gauge invariant spin nets. · contest 0.88

Builds on it 1 paper

Before tackling this problem however, let's sum up Koslowski's and Sahlmann's approach in [ref] and adapt it to our case.

Abelian 2+1D Loop Quantum Gravity Coupled to a Scalar Field · Ashtekar-Lewandowski representation peaked on a classical vacuum · 0.94

Uses its method or tool 2 papers

It turns out that, in fact, the similarities to the representations in [ref] are sufficient to adopt their procedure and especially some crucial details to our situation.

Towards Gaussian states for loop quantum gravity · Revisiting the area operator · 0.88
The idea is to use the work done by Koslowski and Sahlmann [ref] and to develop a representation peaked on a classical non-degenerate spatial metric.

Abelian 2+1D Loop Quantum Gravity Coupled to a Scalar Field · First approach · 0.81

Compares with it 2 papers

In the context of the KS representation in [ref] , there is a similar augmentation to the flux representation of a classical flux.

Towards Gaussian states for loop quantum gravity · Revisiting the area operator · 0.87
The main difference between our work and [ref] is that we do not use any background state above which we consider a compact set of excitations.

A note on conformally compactified connection dynamics tailored for anti-de Sitter space · Comments · 0.99

Cites it as background 3 papers

The gauge symmetry group will be referred to as the bundle automorphism group [ref] or, in short, as the automorphism group.

The Koslowski- Sahlmann representation: Gauge and diffeomorphism invariance · Preliminaries · 0.82
Now we will discuss the representations of $$ with nondegenerate background geometry [ref] .

Loop Quantum Gravity Vacuum with Nondegenerate Geometry · Representations of the holonomy-flux algebra with nondegenerate background geometry · 0.93
It is noteworthy that some other representations have been discussed already in Loop Quantum Gravity, most notably [ref] .

Abelian 2+1D Loop Quantum Gravity Coupled to a Scalar Field · Ashtekar-Lewandowski representation peaked on a classical vacuum · 0.84

What it cites · 25 sentences without a confident label

Builds on 1 paper

The goal of the present article is to round off the picture that is emerging in [ref] , by showing that geometric operators can be defined in the new representations exactly as in the standard representation, and that they have a very simple structure.

Dynamical Quantum Geometry (DQG Programme) · Introduction · 0.93

Uses the method or tool of 4 papers

In this section, we will show that area and volume operators can be defined in the new representations with exactly the same regularization and quantization strategy ( [ref] ) as in the vacuum representation.

Quantum Theory of Geometry II: Volume operators · Geometric operators · 0.90
In this section, we will show that area and volume operators can be defined in the new representations with exactly the same regularization and quantization strategy ( [ref] ) as in the vacuum representation.

Discreteness of area and volume in quantum gravity · Geometric operators · 0.90
In this section, we will show that area and volume operators can be defined in the new representations with exactly the same regularization and quantization strategy ( [ref] ) as in the vacuum representation.

Quantum Theory of Gravity I: Area Operators · Geometric operators · 0.90

Compares with 1 paper

We now come to the difference with [ref] .

Dynamical Quantum Geometry (DQG Programme) · Gauß constraint · 0.86

Cites as support 1 paper

But this is not the case, as was realized in [ref] .

Dynamical Quantum Geometry (DQG Programme) · Implementation of the constraints · 0.97

Cites as background 2 papers

The possibility of such representations was known to some experts before [ref] (they were mentioned for example in passing in [ref] ) but they were not considered interesting, because they did not alleviate the asymmetry in fluctuations between the canonical variables, and because gauge transformations and diffeomorphisms could not be implemented unitarily.

Dynamical Quantum Geometry (DQG Programme) · Introduction · 0.90
The possibility of such representations was known to some experts before [ref] (they were mentioned for example in passing in [ref] ) but they were not considered interesting, because they did not alleviate the asymmetry in fluctuations between the canonical variables, and because gauge transformations and diffeomorphisms could not be implemented unitarily.

When Do Measures on the Space of Connections Support the Triad Operators of Loop Quantum Gravity? · Introduction · 0.90

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