Spin Foam Models

John C. Baez

RECORD

Spin Foam Models

John C. Baez

General Relativity and Quantum Cosmology · Submitted 21 Sep 1997

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

The authors' main claim We conclude by presenting a spin foam model of 4-dimensional Euclidean quantum gravity, closely related to the state sum model of Barrett and Crane, but not assuming the presence of an underlying spacetime manifold.

how it is stated · asserted 1.00 read the record →

Comment (0) Read on arXiv ↗

Cite ▾

Claim paper

The compiled record

The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Open the compiled record

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 6,315 input tokens, $0.0003 reported.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.25 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.03 · compares with prior results 0.74 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v3. Results and the main claim are underlined; show every role.

While the use of spin networks has greatly improved our understanding of the kinematical aspects of quantum gravity, the dynamical aspects remain obscure. To address this problem, we define the concept of a `spin foam' going from one spin network to another. Just as a spin network is a graph with edges labeled by representations and vertices labeled by intertwining operators, a spin foam is a 2-dimensional complex with faces labeled by representations and edges labeled by intertwining operators. Spin foams arise naturally as higher-dimensional analogs of Feynman diagrams in quantum gravity and other gauge theories in the continuum, as well as in lattice gauge theory. When formulated as a `spin foam model', such a theory consists of a rule for computing amplitudes from spin foam vertices, faces, and edges. The product of these amplitudes gives the amplitude for the spin foam, and the transition amplitude between spin networks is given as a sum over spin foams. After reviewing how spin networks describe `quantum 3-geometries', we describe how spin foams describe `quantum 4-geometries'. We conclude by presenting a spin foam model of 4-dimensional Euclidean quantum gravity, closely related to the state sum model of Barrett and Crane, but not assuming the presence of an underlying spacetime manifold.

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	While the use of spin networks has greatly improved our understanding of the kinematical aspects of quantum gravity, the dynamical aspects remain obscure.	gap 0.84	0.06	0.03	asserted 1.00
s1	To address this problem, we define the concept of a `spin foam' going from one spin network to another.	contribution 0.95	0.89	0.04	asserted 0.96
s2	Just as a spin network is a graph with edges labeled by representations and vertices labeled by intertwining operators, a spin foam is a 2-dimensional complex with faces labeled by representations and edges labeled by intertwining operators.	contribution 0.64	0.37	0.04	asserted 1.00
s3	Spin foams arise naturally as higher-dimensional analogs of Feynman diagrams in quantum gravity and other gauge theories in the continuum, as well as in lattice gauge theory.	background 0.53	0.12	0.04	asserted 0.99
s4	When formulated as a `spin foam model', such a theory consists of a rule for computing amplitudes from spin foam vertices, faces, and edges.	method 0.85	0.24	0.06	asserted 1.00
s5	The product of these amplitudes gives the amplitude for the spin foam, and the transition amplitude between spin networks is given as a sum over spin foams.	method 0.97	0.25	0.07	asserted 1.00
s6	After reviewing how spin networks describe `quantum 3-geometries', we describe how spin foams describe `quantum 4-geometries'.	contribution 0.59	0.63	0.05	asserted 0.91
s7	We conclude by presenting a spin foam model of 4-dimensional Euclidean quantum gravity, closely related to the state sum model of Barrett and Crane, but not assuming the presence of an underlying spacetime manifold.	contribution 0.98	0.87	0.73	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.67 · evidence from argument 0.77.

What was examined, and what was not

sources read

arXiv listing abstract v3

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper

Original abstract

While the use of spin networks has greatly improved our understanding of the kinematical aspects of quantum gravity, the dynamical aspects remain obscure. To address this problem, we define the concept of a `spin foam' going from one spin network to another. Just as a spin network is a graph with edges labeled by representations and vertices labeled by intertwining operators, a spin foam is a 2-dimensional complex with faces labeled by representations and edges labeled by intertwining operators. Spin foams arise naturally as higher-dimensional analogs of Feynman diagrams in quantum gravity and other gauge theories in the continuum, as well as in lattice gauge theory. When formulated as a `spin foam model', such a theory consists of a rule for computing amplitudes from spin foam vertices, faces, and edges. The product of these amplitudes gives the amplitude for the spin foam, and the transition amplitude between spin networks is given as a sum over spin foams. After reviewing how spin networks describe `quantum 3-geometries', we describe how spin foams describe `quantum 4-geometries'. We conclude by presenting a spin foam model of 4-dimensional Euclidean quantum gravity, closely related to the state sum model of Barrett and Crane, but not assuming the presence of an underlying spacetime manifold.

Human review

How Pith checks papers → · Full audit of this paper →

References

Forward citations

Cited by 15 Pith papers

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

Relativistic spin networks and quantum gravity
gr-qc 1997-09 conditional novelty 8.0 of 10

A classical theorem characterizing 4-simplices by bivectors on their triangles is used to propose a four-dimensional spin-network state sum for quantum gravity.
Edge modes of gravity. Part III. Corner simplicity constraints
hep-th 2020-07 conditional novelty 7.0 of 10

The corner area element equals the Poincaré spin Casimir, and the continuum simplicity constraints split into two second class and one first class component, leaving the full Lorentz algebra as Dirac observables.
Deep learning spinfoam vertex amplitudes: the Euclidean Barrett-Crane model
gr-qc 2025-05 conditional novelty 6.0 of 10

A proof-of-principle that simple neural networks can learn Euclidean Barrett-Crane 10j vertex amplitudes: classification generalizes to higher cutoffs, regression works only within the trained low-spin domain.
Studying replica wormholes and the Page curve with simplicial quantum gravity
hep-th 2025-04 conditional novelty 6.0 of 10

A coarse simplicial quantum gravity model produces semiclassical saddles whose swap entropy rises and then falls, reproducing the Page transition at n→1+.
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.

Show 10 more citing papers

Diffeomorphisms in group field theories
hep-th 2011-01 conditional novelty 6.0 of 10

The colored Boulatov group field theory is invariant under a Drinfeld-double vertex-translation symmetry, which reproduces, in metric, group, and spin representations, the discrete diffeomorphism invariance of 3d simp...
Improved and Perfect Actions in Discrete Gravity
gr-qc 2009-07 conditional novelty 6.0 of 10

In three dimensions with a cosmological constant, the perfect action of Regge calculus is the Regge action of constantly curved tetrahedra whose curvature equals the cosmological constant, and it restores the vertex d...
Phase space descriptions for simplicial 4d geometries
gr-qc 2008-07 conditional novelty 6.0 of 10

On a fixed triangulation, the gauge-invariant phase space of loop quantum gravity is strictly larger than that of length or area Regge calculus and coincides with area-angle Regge calculus prior to the gluing constraints.
MacDowell-Mansouri gravity and Cartan geometry
gr-qc 2006-11 accept novelty 4.0 of 10

MacDowell-Mansouri gravity is a Cartan geometry that describes spacetime by rolling a homogeneous model spacetime (such as de Sitter space) along it, unifying the connection and coframe field into a single geometric object.
Group Field Theory: An overview
hep-th 2005-05 conditional novelty 4.0 of 10

Group field theory's tree-level two-point function is proposed as a triangulation-independent physical inner product for loop quantum gravity, with the full two-point function giving a sum over spacetime topologies.
An Invitation to Higher Gauge Theory
hep-th 2010-03 conditional novelty 3.0 of 10

Higher gauge theory describes string parallel transport as a 2-functor from a path 2-groupoid to a Lie 2-group, with the 1-form and 2-form tied by t(B) = dA + A wedge A, and six families of Lie 2-groups cover the know...
Les Houches lectures on Spinfoam Path Integrals
gr-qc 2026-07 accept novelty 2.0 of 10

A pedagogical review of spinfoam path integrals, from 1d quantum mechanics and 2d BF theory through Ponzano-Regge/Turaev-Viro to the 4d EPRL model, accurate but with no new results.
Quantum gravity phenomenology at the dawn of the multi-messenger era -- A review
hep-ph 2021-11 accept novelty 2.0 of 10

The review argues that multi-messenger astronomy now supplies workable empirical constraints on Planck-scale quantum gravity, chiefly through energy-dependent propagation effects accumulated over cosmological distances.
Zakopane lectures on loop gravity
gr-qc 2011-02 accept novelty 2.0 of 10

Loop quantum gravity is presented as a covariant framework in which discrete spin-network quanta of space evolve through spinfoam amplitudes whose semiclassical limit is claimed to reproduce general relativity.
Loop Quantum Gravity
gr-qc 1997-10 conditional novelty 2.0 of 10

Loop quantum gravity, as of 1997, offers a well-defined background-independent quantum kinematics with discrete area and volume spectra, while its dynamics remains unsettled.

How the literature uses it

58 arXiv papers cite this in 77 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 · 32 sentences without a confident label

Compares with it 1 paper

Similar invariance properties were considered in [ref] , as an implementation of diffeomorphism invariance and in analogy with cylindrical consistency.

Local spin foams · Introduction · 0.80

Cites it as background 39 papers

In the absence of a cosmological constant, non-perturbative quantization in the canonical approach, the loop quantum gravity (LQG) based on the BF formulation of gravity [ref] , and the covariant approach, based on spin foams and more precisely the Ponzano-Regge model [ref] , give consistent results.

Spinor Representation of the Hamiltonian Constraint in 3D LQG with a Non-zero Cosmological Constant · Introduction · 0.96
In this way, spin networks give the quantum states of three-dimensional geometries, while the spin foams give cobordisms between spin networks and are used to define partition functions and transition amplitudes as “sums over histories” [ref] .

Spin Foams and Noncommutative Geometry · Spin networks and foams enriched with topological data · 0.94
Spin foam models [ref] form a promising approach to background independent, non-perturbative quantization and have been mainly developped as the structure emerging when writing transition amplitudes between spin network states in loop quantum gravity [ref] .

Spin foam models for quantum gravity from lattice path integrals · Introduction · 0.99

Discussion

Know something about this paper? Leave a comment, or highlight any passage above to comment on it.