On knottings in the physical Hilbert space of LQG as given by the EPRL model

Benjamin Bahr

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On knottings in the physical Hilbert space of LQG as given by the EPRL model

Benjamin Bahr

General Relativity and Quantum Cosmology · Submitted 3 Jun 2010

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

The authors' main claim Using this surprising invariance, we are able to show that in the physical Hilbert space as defined by the sum over all spin foams contains no knotting classes of graphs anymore.

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We consider the EPRL spin foam amplitude for arbitrary embedded two-complexes. Choosing a definition of the face- and edge amplitudes which lead to spin foam amplitudes invariant under trivial subdivisions, we investigate invariance properties of the amplitude under consistent deformations, which are deformations of the embedded two-complex where faces are allowed to pass through each other in a controlled way. Using this surprising invariance, we are able to show that in the physical Hilbert space as defined by the sum over all spin foams contains no knotting classes of graphs anymore.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We consider the EPRL spin foam amplitude for arbitrary embedded two-complexes.	contribution 0.74	0.78	0.02	asserted 0.58
s1	Choosing a definition of the face- and edge amplitudes which lead to spin foam amplitudes invariant under trivial subdivisions, we investigate invariance properties of the amplitude under consistent deformations, which are deformations of the embedded two-complex where faces are allowed to pass through each other in a controlled way.	method 0.88	0.84	0.04	asserted 0.62
s2	Using this surprising invariance, we are able to show that in the physical Hilbert space as defined by the sum over all spin foams contains no knotting classes of graphs anymore.	result 1.00	0.93	0.93	asserted 1.00

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The paper

Original abstract

We consider the EPRL spin foam amplitude for arbitrary embedded two-complexes. Choosing a definition of the face- and edge amplitudes which lead to spin foam amplitudes invariant under trivial subdivisions, we investigate invariance properties of the amplitude under consistent deformations, which are deformations of the embedded two-complex where faces are allowed to pass through each other in a controlled way. Using this surprising invariance, we are able to show that in the physical Hilbert space as defined by the sum over all spin foams contains no knotting classes of graphs anymore.

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References

Forward citations

Cited by 3 Pith papers

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

Coarse graining methods for spin net and spin foam models
gr-qc 2011-09 conditional novelty 6.0 of 10

Two coarse-graining schemes are applied to finite-group spin net and spin foam models, with a new Gauss-constraint-preserving tensor network algorithm and a demonstration that certain cutoff models are equivalent.
Perfect discretization of reparametrization invariant path integrals
gr-qc 2011-01 conditional novelty 5.0 of 10

For one-dimensional reparametrization invariant systems, requiring the discrete path integral to respect vertex translation symmetry forces it to equal the continuum propagator, and an iterative Wilsonian scheme const...
The Spin Foam Approach to Quantum Gravity
gr-qc 2012-05 accept novelty 2.0 of 10

The review presents the spin foam quantization of gravity as a constrained BF path integral, focusing on the EPRL and Freidel-Krasnov models and on 2+1 gravity as a solvable guide.

How the literature uses it

12 arXiv papers cite this in 19 sentences

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Papers citing it · 14 sentences without a confident label

Cites it as support 2 papers

Indeed this notion has been connected to the anomaly–freeness of the path integral measure in spin foams with respect to diffeomorphisms [ref] and has the advantage of fixing uniquely the edge and face measure factors [ref] .

Holonomy Spin Foam Models: Definition and Coarse Graining · Coarse graining in two dimensions · 0.98
These results give new weight to the point suggested in [ref] , which is to build models and select measure factors such that invariance under certain subdivisions of edges and faces is guaranteed.

Quantum group spin nets: refinement limit and relation to spin foams · Reisenberger's principle and intertwiner models · 0.96

Cites it as background 3 papers

Different requirements have been proposed to fix these weights [ref] .

Bubble divergences and gauge symmetries in spin foams · Single bubble contributions for general models · 0.92
Different principles have been suggested to fix this measure [ref] , leading to different proposals.

Coarse graining flow of spin foam intertwiners · The BC construction · 0.86
The role of knotting in the Hilbert space of the EPRL model is studied in [ref] .

The Spin Foam Approach to Quantum Gravity · Boundary Data for the New Models and Relationship with the Canonical Theory · 0.82

What it cites · 18 sentences without a confident label

Compares with 1 paper

See however [ref] for a different amplitude, which in particular has the advantage of reducing the degree of divergence in the sum over spin foams ([ref]).

Face amplitude of spinfoam quantum gravity · Face-, edge- and boundary amplitudes · 0.80

Cites as background 6 papers

Two of the spin foam models that are considered mostly today are the EPRL [ref] and the FK model [ref] .

LQG vertex with finite Immirzi parameter · Introduction · 1.00
Two of the spin foam models that are considered mostly today are the EPRL [ref] and the FK model [ref] .

A New Spin Foam Model for 4d Gravity · Introduction · 1.00
This idea goes back to Rovelli and Reisenberger ( [ref] , see [ref] for a review), and lies at the foundation of spin foam models as understood in the context of Loop Quantum Gravity.[The mathematical concept as abstract state sum model is much older (see e.g. literature in [ref] ), but differs slightly from the way spin foams are understood in this article.]

The spin-foam-representation of loop quantum gravity · Introduction · 0.92

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