Symmetry Reduction of Loop Quantum Gravity

Johannes Brunnemann; Tim A. Koslowski

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Symmetry Reduction of Loop Quantum Gravity

Johannes Brunnemann, Tim A. Koslowski

General Relativity and Quantum Cosmology · Submitted 30 Nov 2010

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

The authors' main claim The relation between standard Loop Quantum Cosmology and full Loop Quantum Gravity fails already at the first nontrivial step: The configuration space of Loop Quantum Cosmology can not be embedded into the configuration space of full Loop Quantum Gravity due to a topological obstruction.

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What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.25 · human participants 0.01 · controlled comparison 0.07 · randomized assignment 0.01 · states quantitative result 0.04 · compares with prior results 0.90 · mentions code or data availability 0.02.

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The relation between standard Loop Quantum Cosmology and full Loop Quantum Gravity fails already at the first nontrivial step: The configuration space of Loop Quantum Cosmology can not be embedded into the configuration space of full Loop Quantum Gravity due to a topological obstruction. We investigate this obstruction in detail, because many topological obstructions are the source of physical effects. For this we derive the topology of a large class of subspaces of the Loop Quantum Gravity configuration space. This allows us to find the extension of the standard Loop Quantum Cosmology configuration space that admits an embedding in agreement with Fleischhack:2010. We then construct the embedding for flat FRW Loop Quantum Cosmology and find that it coincides asymptotically with standard LQC.

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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	The relation between standard Loop Quantum Cosmology and full Loop Quantum Gravity fails already at the first nontrivial step: The configuration space of Loop Quantum Cosmology can not be embedded into the configuration space of full Loop Quantum Gravity due to a topological obstruction.	background 0.50	0.35	0.72	asserted 1.00
s1	We investigate this obstruction in detail, because many topological obstructions are the source of physical effects.	contribution 0.63	0.32	0.04	asserted 0.56
s2	For this we derive the topology of a large class of subspaces of the Loop Quantum Gravity configuration space.	method 0.90	0.94	0.04	asserted 0.99
s3	This allows us to find the extension of the standard Loop Quantum Cosmology configuration space that admits an embedding in agreement with Fleischhack:2010.	result 0.89	0.82	0.32	asserted 0.99
s4	We then construct the embedding for flat FRW Loop Quantum Cosmology and find that it coincides asymptotically with standard LQC.	result 0.97	0.96	0.30	asserted 1.00

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Original abstract

The relation between standard Loop Quantum Cosmology and full Loop Quantum Gravity fails already at the first nontrivial step: The configuration space of Loop Quantum Cosmology can not be embedded into the configuration space of full Loop Quantum Gravity due to a topological obstruction. We investigate this obstruction in detail, because many topological obstructions are the source of physical effects. For this we derive the topology of a large class of subspaces of the Loop Quantum Gravity configuration space. This allows us to find the extension of the standard Loop Quantum Cosmology configuration space that admits an embedding in agreement with Fleischhack:2010. We then construct the embedding for flat FRW Loop Quantum Cosmology and find that it coincides asymptotically with standard LQC.

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8 arXiv papers cite this in 10 sentences

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Papers citing it · 4 sentences without a confident label

Cites it as background 6 papers

The issue of performing a symmetry reduction in loop quantum gravity directly at the quantum level has received much attention in recent years [ref] .

An embedding of loop quantum cosmology in (b, v) variables into a full theory context · Introduction · 1.00
Within loop quantum gravity (LQG), there has been a lot of recent interest in this subject, see e.g. [ref] .

A quantum reduction to Bianchi I models in loop quantum gravity · Introduction · 0.99
Some of those have already been mentioned in the main text, but there are many more that we could not cover in these lectures, among them new connections between the full theory and symmetry reduced models [ref] , coherent states for quantum geometry [ref] , an interpretation of quantum geometry in terms of polyhedra in flat space [ref] , and corresponding stunning results about the quantum volume [ref] and the use of spinor techniques [ref] .

From Classical To Quantum Gravity: Introduction to Loop Quantum Gravity · Summary and open problems · 0.99

What it cites · 13 sentences without a confident label

Uses the method or tool of 2 papers

The construction strategy for the quantum embedding follows [ref] : Let $A(c)$ be the embedding of the symmetry reduced configuration space, parametrized by $c$ into the space of all connections, so $A$ can be used to pull-back any configuration observable (in particular cylindrical functions) $T(A)$ to a function $T(A(c))$ on the symmetry reduced configuration space, which defines equivalence classes of configuration observables in the full theory whose pull-back to the symmetry reduced configuration space coincides.

Reduction of a Quantum Theory · An Explicit Quantum Symmetry Reduction · 1.00
Using a general result on symmetric connections [ref] , one obtains a parametrization of isotropic Bianchi I connections by $A=c·δ^I_a dx^a⊗τ_I$.

On Symmetric Gauge Fields for arbitrary Gauge and Symmetry Groups · Holonomy ODE for Homogeneous Isotropic Cosmological Model · 0.92

Cites as support 1 paper

In [ref] it was shown, that the configurations space of LQC, given by $R_Bohr$, the Bohr compactification of the real line cannot be continuously embedded into the configuration space of LQG, the space $A$ of generalized connections.

On the Configuration Spaces of Homogeneous Loop Quantum Cosmology and Loop Quantum Gravity · Non-Embedability of Configuration Spaces · 0.99

Cites as background 3 papers

A phenomenological investigation of this issue has been avoided in the LQC framework and the Spin Foam Cosmology (SFC) framework (see e.g. [ref] ).

Stepping out of Homogeneity in Loop Quantum Cosmology · Introduction · 0.80
A phenomenological investigation of this issue has been avoided in the LQC framework and the Spin Foam Cosmology (SFC) framework (see e.g. [ref] ).

Towards Spinfoam Cosmology · Introduction · 0.80
A particularly fruitful program is Loop Quantum Gravity (LQG) [ref] , as it represents a mathematically well developed quantum field theoretic framework.

Background Independent Quantum Gravity: A Status Report · Introduction · 0.99

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