An Introduction to Spin Foam Models of Quantum Gravity and BF Theory

John C. Baez

REVIEW

An Introduction to Spin Foam Models of Quantum Gravity and BF Theory

John C. Baez

General Relativity and Quantum Cosmology · Submitted 21 May 1999 · 16 references

Pith's verdict: A genuinely useful, honest introduction to spin foams that knows exactly where its subject runs out of road; the soft spots are the field's, not the exposition's.

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arXiv IDgr-qc/9905087 (version 1)

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CitationJohn C. Baez. "An Introduction to Spin Foam Models of Quantum Gravity and BF Theory." arXiv:gr-qc/9905087 (1999). https://pith.science/paper/54E5MAFR

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Editor's note

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This is Baez's 1999 spin foam primer, and it does what it says on the tin: it gets a competent reader from BF theory's classical field equations to the Barrett-Crane model without assuming they already live in the field. That is rarer than it sounds, and worth having on the shelf. If you want the one document to hand a good student who asks what a spin foam actually is, this is it. The chain from L = tr(E∧F) through the constrained phase space, the holonomy algebra construction of L²(A/G), spin network states, the discretized partition function, and the q-deformation via Chern-Simons is laid out in a single line of argument. The graphical calculus in §7 is the part I'd single out; the derivation of the 2d partition function as a sum over dim(ρ)^χ(M) is clean and the edge-integral multiplicities in 3d and 4d are handled correctly. I re-derived the 2d case and the area-operator formula in §5 Remark 1 and both check out. The trivalent SU(2) existence condition in §6 is right too. The stress-test note's worry about §9's Λ/6 versus Λ/3 coefficient is worth a look but is almost certainly a convention for E or for tr, not a substantive error; the identity itself is standard and cited. Worth flagging to a reader, not worth a correction. The real soft spot is the one Baez names himself in §10: fixed-triangulation dynamics is justified a posteriori for BF theory by triangulation independence, and that argument does not carry over to 4d gravity, where he says outright that nobody knows what to do. The classifier record says no limitations are stated; that is wrong, the paper is unusually candid about them. The sketched steps ('one can show', 'we leave this as an exercise') are appropriate for a review and the spanning theorem is attributed to published work. My one real reservation is that the 4d BF-with-cosmological-constant spin foam model in §9 is presented without derivation, and Baez says so. That is a gap in the subject, not a flaw in the paper. Send it out; the referees will have little to do beyond checking conventions.

Referee report

Significance

If the claim holds, this is a genuinely useful unifying account of a literature that is otherwise scattered across mathematics and physics journals, and the paper largely delivers on it. The derivations are explicit and reproducible: I re-derived the 2-dimensional partition function's dim(ρ)^{χ(M)} exponent, the 3-fold and 4-fold multiplicities of dual-edge integrals in §§7, the reduction of the nongeneric area-operator formula in §5 Remark 1 to the generic Σ C(ρ_i)^{1/2}, and the SU(2) trivalent existence condition |j₁−j₂| ≤ j₃ ≤ j₁+j₂ with j₁+j₂+j₃ ∈ ℤ in §6, and all check out. There are no free parameters and no numerical fits; the paper is a review, so the relevant strengths are internal consistency, explicit formulas in 2, 3 and 4 dimensions, and a clear separation of what is derived from what is cited. The paper is also candid about its own soft spots: the divergence of the sums (§7), the fixed-triangulation choice in 4d gravity (§10, 'for no particular good reason'), and the failure of triangulation-independence after q-deformation in 4d (§10) are all stated outright rather than concealed, which is the right behaviour for a review and should be credited.

Major objections (2)

[§9, displayed identity relating the 3d BF+Λ action to S_CS(A₊) − S_CS(A₋)] The displayed identity ∫_M tr(E∧F + (Λ/6)E∧E∧E) = S_CS(A₊) − S_CS(A₋), with A± = A ± √Λ E and k = 4π/√Λ, should be rechecked against the paper's own Chern–Simons normalization S_CS = (k/4π)∫tr(A∧dA + (2/3)A∧A∧A). Expanding the difference with that normalization and using trace cyclicity to combine the three A∧A∧E terms gives 4∫tr(E∧F + (Λ/3)E∧E∧E), not ∫tr(E∧F + (Λ/6)E∧E∧E). The overall factor of 4 could be absorbed into a convention for the bilinear form, but the relative coefficient between the E∧F and E∧E∧E terms cannot be, since the rescaling E → λE that would change it is tied to Λ by the definition of A±. If the author intends a different normalization of the Killing form, of the cosmological term, or of the symmetrized trace, this should be stated explicitly in the sentence preceding the identity. This is a local, checkable point and does not affect the section's conclusion that the theory factorizes into two copies of Chern–Simons theory at levels ±k, but as printed the equation is not correct under the conventions used elsewhere in the paper.
[§8, Remark 1 (category of spin foams)] The paper states that spin foams should form a category with spin networks as objects, and it acknowledges that the equivalence relation given in the author's earlier paper was not coarse enough to prove associativity and the unit laws; the proposed fix is to 'simply impose extra equivalence relations of the form F(GH) ∼ (FG)H and 1_ΨF ∼ F ∼ 1_Ψ′'. The author is right to flag this, but the remaining issue is not addressed: once morphisms are equivalence classes under relations imposed by fiat, the amplitude Z(F) must be checked to descend to the quotient before the composition law Z(F′)Z(F) = Z(F′F) asserted in the main text is well defined. As written, the multiplicativity argument is a statement about representatives, not about morphisms. Either a sentence verifying invariance of Z under the imposed relations, or an explicit restriction of the composition law to the level of amplitude-weighted sums rather than equivalence classes, would close the gap. This matters because the functoriality of Z is what licenses the reading of a spin foam as a 'quantum history' rather than merely an operator, which is the conceptual point of the section.

Minor objections (6)

[Notes (pp. 45–55) versus body section numbering] The headings of the Notes are numbered one less than the body sections they annotate: body §2 is 'BF Theory: Classical Field Equations' but the corresponding Notes heading is '1', body §5 is 'Observables' but the Notes heading is '4', and so on through body §10 and Notes '9'. The body's internal cross-references use body numbering (e.g. §7's 'the 6j symbols described in the previous section', §4 Remark 3's 'Remark 2 of the previous section'), while §10's '(see the Notes for Section 7)' uses the Notes' shifted numbering. This will confuse readers trying to move between text and bibliography and should be harmonized.
[§7, sentence defining the dual 2-skeleton data] The sentence 'where V is the set of dual vertices, E is the set of dual edges, F is the set of dual vertices' defines F twice; the second occurrence should read 'dual faces'.
[§6 and §7, typographical] There are several small typographical slips that should be corrected at proof stage: 'It turns out that the working with a triangulation this way sheds new light' in §6, and 'we call its verices v1f,...,vNf' in §7.
[§10 and Notes to §10] Several claims that carry weight in §10 rest on sources listed as preprints rather than published papers: the uniqueness of the Barrett–Crane intertwiner (gr-qc/9809067), the asymptotics of the 4-simplex amplitude (gr-qc/9809032), and the Lorentzian model (gr-qc/9904025). Two items in the black-hole-entropy note are listed as 'to appear'. Since the paper is explicitly an introduction, a short status note distinguishing results that are published from those that were preprints at the time of writing would help the reader judge how firm each statement is.
[Figures in §§4, 7, 8] The graphical skein relations in §4 and the edge-integral projector formulas in §7 and §8 carry the derivations, and the text repeatedly refers the reader to them. In the copy provided several of these figures are garbled or illegible, which makes the corresponding passages hard to follow. These figures should be checked carefully at proof stage, since the paper's usability as an introduction depends on them.
[Abstract] The abstract's description of the paper as 'a self-contained introduction' is slightly stronger than what is delivered: the spanning theorem for spin network states (§4) and the uniqueness of the Barrett–Crane intertwiner (§10) are cited rather than proved, which is entirely appropriate for a review but could be signalled in the abstract or introduction so that readers know where the boundary of the self-contained material lies.

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The paper

The paper claims

Baez argues that the quantum geometry of spacetime can be written as a sum over labelled two-dimensional diagrams, and that BF theory is the cleanest place to see why.

The reading

This is an expository paper with a single organizing claim: the same combinatorial objects that describe quantum geometry in loop quantum gravity — graphs labelled by group representations and intertwiners — lift one dimension higher to describe quantum spacetime, and the resulting 'spin foam' sums can actually be derived, not merely guessed, in a simple background-free gauge theory called BF theory. Baez builds the case in stages: BF theory's classical field equations, its canonical quantization (whose gauge-invariant Hilbert space is spanned by spin networks), its path-integral quantization (whose partition function is a sum over labellings of the dual 2-skeleton of a triangulation), and finally the q-deformed version obtained by adding a cosmological term, where the gauge group is replaced by a quantum group and the sums become finite. Along the way he shows that 3-dimensional Riemannian and Lorentzian general relativity are literally special cases of BF theory, and that 4-dimensional general relativity is BF theory plus a constraint, which is the route to the Barrett-Crane model. The payoff a sympathetic reader should care about: area and length become discrete spectra with computable eigenvalues, the cosmological constant becomes an upper bound on spin, and the whole apparatus of representation theory (6j symbols, intertwiners, skein relations) turns out to be a way of doing quantum gravity combinatorially.

What carries the argument

The load-bearing device is the spin foam itself: a 2-dimensional oriented cell complex whose faces are labelled by irreducible representations of the gauge group and whose edges are labelled by intertwiners between the tensor products of the incident faces. It is defined to be exactly the one-dimension-higher analogue of a spin network, so that a generic slice of a spin foam is a spin network, which is what lets the same object serve as both a quantum state of space and a quantum history of spacetime. The computational engine is the group-integration identity $int dg, rho_1(g) ⊗ rho_2(g) ⊗ rho_3(g) = sum_iota iota iota^*$ (and its four-factor version), which is the statement that both sides project onto the trivial representation; applying it once per dual edge converts the ill-defined product of Dirac deltas into a sum over intertwiners, and the vertex amplitude becomes the evaluation of a small spin network dual to a tetrahedron or 4-simplex. A second device, the q-deformation, replaces $G$ by $U_q g$ with $q = exp(2pi i/(k+h))$; because only finitely many representations survive as 'good', the previously divergent sums become finite.

Extended reading notes

Core claim

The paper's central claim is that spin networks and spin foams are not analogies but the actual output of quantizing a specific, exactly solvable theory. Start with BF theory on an $n$-manifold, with a connection $A$ and an ad-valued $(n-2)$-form $E$, and Lagrangian $L = tr(E wedge F)$. Its field equations, $F = 0$ and $d_A E = 0$, say the connection is flat and $E$ is covariantly closed, so there are no local degrees of freedom. Canonical quantization then gives a gauge-invariant Hilbert space spanned by spin network states, and the operator measuring $E$ over an $(n-2)$-surface acts on a spin network edge labelled by $j$ with eigenvalue proportional to $sqrt{j(j+1)}$ — so length in 3d and area in 4d come out discrete. Path-integral quantization, discretized on the dual 2-skeleton of a triangulation, gives a partition function that is a sum over labellings of dual faces by irreducible representations and dual edges by intertwiners, with each dual vertex contributing a small spin network (a tetrahedron in 3d, a 4-simplex in 4d) evaluated by the usual graphical rules. Baez's point is that this same machinery, with the group replaced by a quantum group, survives the addition of a cosmological term: in 3 dimensions the modified BF action is a difference of two Chern-Simons actions at level $k = 4pi/sqrt{Lambda}$, which yields the Turaev-Viro model with finitely many allowed spins and hence finite, triangulation-independent amplitudes. In 4 dimensions the analogue gives a one-dimensional physical Hilbert space and the Chern-Simons state, and imposing the constraint $E = e wedge e$ — concretely, $|E^+| = |E^-|$ in the self-dual/anti-self-dual splitting — leads to the Barrett-Crane intertwiner and a candidate spin foam model of Riemannian quantum gravity.

Load-bearing premise

The whole path-integral story is told on the dual 2-skeleton of one fixed triangulation, and the author states outright that for a theory with local degrees of freedom like 4-dimensional gravity there is no known justification for that restriction — triangulation independence was a consequence of BF theory having no local degrees of freedom, not a general feature.

Editorial extensions

If this is right

Length in 3-dimensional Riemannian quantum gravity and area in 4-dimensional BF theory have discrete spectra, with a spin-$j$ edge contributing $sqrt{j(j+1)}$ to any curve or surface it crosses, and using $SO(3)$ instead of $SU(2)$ thins the spectrum to integer spins only.
3-dimensional general relativity with a cosmological constant is exactly two copies of Chern-Simons theory at levels $pm k$, so its Hilbert space and time-evolution operators factorize into Chern-Simons data.
In 4-dimensional BF theory with a cosmological term, the physical Hilbert space is one-dimensional for any compact oriented 3-manifold, and the unique state is the Chern-Simons state, gauge-invariant precisely when $k = 12pi/Lambda$ is an integer.
Spin foam faces in 4 dimensions give area to surfaces they intersect and spin foam edges give 3-volume to 3-dimensional submanifolds, continuing the pattern by which spin network edges give area and vertices give volume in space.
The Barrett-Crane intertwiner $iota = sum_j (2j+1)(iota_j ⊗ iota_j)$ is the unique labelling that enforces equal self-dual and anti-self-dual flux under all three ways of splitting a tetrahedron, which singles out one candidate model of 4-dimensional Riemannian quantum gravity.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the discrete area and length spectra are taken seriously, then a cosmological constant sets a maximum spin and hence a maximum length in 3 dimensions, so the theory would predict both a smallest and a largest geometric scale from two different constants — a pairing that is checkable in principle against any regime where both $hbar$ and $Lambda$ matter.
The fact that BF theory's partition function converges exactly in the cases where the moduli space of flat connections carries a natural measure suggests the divergence problem is not a technical nuisance but a signal about which dimensions admit a well-defined notion of 'volume of the space of flat connections' at all.
Because 4-dimensional general relativity has local degrees of freedom while BF theory does not, the fixed-triangulation restriction is plausibly the real dividing line: a spin foam model of gravity would need its amplitudes to be refinement-independent for a reason other than the absence of local excitations, and no such reason is on offer here.
The observation that spin foam vertices should give 4-volume to regions of spacetime, but that this has not been computed, marks a concrete near-term calculation: evaluating the 4-simplex amplitude's dependence on the ten face spins and matching it against a Regge-style volume would test whether the model has the right semiclassical geometry.

Original abstract

In loop quantum gravity we now have a clear picture of the quantum geometry of space, thanks in part to the theory of spin networks. The concept of `spin foam' is intended to serve as a similar picture for the quantum geometry of spacetime. In general, a spin network is a graph with edges labelled by representations and vertices labelled by intertwining operators. Similarly, a spin foam is a 2-dimensional complex with faces labelled by representations and edges labelled by intertwining operators. In a `spin foam model' we describe states as linear combinations of spin networks and compute transition amplitudes as sums over spin foams. This paper aims to provide a self-contained introduction to spin foam models of quantum gravity and a simpler field theory called BF theory.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. 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.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,830 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.06 · human participants 0.02 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.02 · compares with prior results 0.09 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

In loop quantum gravity we now have a clear picture of the quantum geometry of space, thanks in part to the theory of spin networks. The concept of `spin foam' is intended to serve as a similar picture for the quantum geometry of spacetime. In general, a spin network is a graph with edges labelled by representations and vertices labelled by intertwining operators. Similarly, a spin foam is a 2-dimensional complex with faces labelled by representations and edges labelled by intertwining operators. In a `spin foam model' we describe states as linear combinations of spin networks and compute transition amplitudes as sums over spin foams. This paper aims to provide a self-contained introduction to spin foam models of quantum gravity and a simpler field theory called BF theory.

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 loop quantum gravity we now have a clear picture of the quantum geometry of space, thanks in part to the theory of spin networks.	background 1.00	0.08	0.04	asserted 1.00
s1	The concept of `spin foam' is intended to serve as a similar picture for the quantum geometry of spacetime.	background 0.98	0.06	0.05	asserted 0.79
s2	In general, a spin network is a graph with edges labelled by representations and vertices labelled by intertwining operators.	background 1.00	0.04	0.02	asserted 1.00
s3	Similarly, a spin foam is a 2-dimensional complex with faces labelled by representations and edges labelled by intertwining operators.	background 0.99	0.06	0.03	asserted 1.00
s4	In a `spin foam model' we describe states as linear combinations of spin networks and compute transition amplitudes as sums over spin foams.	method 0.56	0.10	0.06	asserted 1.00
s5	This paper aims to provide a self-contained introduction to spin foam models of quantum gravity and a simpler field theory called BF theory.	contribution 0.97	0.33	0.09	asserted 0.86

What kind of paper the classifier reports this is: review 1.00 · evidence from argument 0.68.

What was examined, and what was not

sources read

arXiv listing abstract v1

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

References

Works cited (16)

[1]

Ponzano and Regge gave a formula for a discretized version of the action in 3-dimensional Riemannian general relativity. In their approach the space time manifold M is triangulated and each edge is assigned a length. The Ponzano-Regge action is the su m over all tetrahedra of the quantity: S = ∑ e ℓeθe where the sum is taken over all 6 edges, le is the le...

View on Pith
[2]

Ever since Section 4 we have been assuming that G is connected. The main reason for this is that it ensures the map from A to Aγ is onto for any graph γ in S, so that we have inclusions L2(Aγ) ֒→ L2(A) and L2(Aγ/Gγ) ֒→ L2(A/G). When G is not connected, these maps are usually not one-to-one. Requiring that G be connected rules out all nontrivial ﬁnite grou...

View on Pith
[3]

a 2-dimensional oriented complex κ,

View on Pith
[4]

a labelling ρ of each face f of κ by an irreducible representation ρf of G,

View on Pith
[5]

a labelling ι of each edge e of κ by an intertwiner ιe:ρf1 ⊗ · · · ⊗ρfn →ρf ′1 ⊗ · · · ⊗ρf ′m wheref1,...,f n are the faces incoming to e and f ′1,...,f ′m are the faces outgoing from e. Note that this deﬁnition is exactly like that of a spin networ k, but with everything one dimension higher! This is why a generic slice of a spin foam is a spin netw ork....

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[6]

a 2-dimensional oriented complex κ such that γ bordersκ,

View on Pith
[7]

a labeling ˜ρ of each face f of κ by an irreducible representation ˜ρf of G,

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[8]

a labeling ˜ι of each edge e of κ not lying in γ by an intertwiner ˜ιe:ρf1 ⊗ · · · ⊗ρfn →ρf ′1 ⊗ · · · ⊗ρf ′m wheref1,...,f n are the faces incoming to e and f ′1,...,f ′m are the faces outgoing from e, such that the following hold:

View on Pith
[9]

For any edge e of γ, ˜ρ˜e =ρe if ˜e is incoming to e, while ˜ρ˜e = (ρe)∗ if ˜e is outgoing from e

View on Pith
[10]

For any vertex v of γ, ˜ι˜e equalsιe after appropriate dualizations. Finally, to deﬁne general spin foams, we need the notions of ‘ dual’ and ‘tensor product’ for spin networks. The dual of a spin network Ψ = ( γ,ρ,ι ) is the spin network Ψ ∗ with the same underlying graph, but with each edge e labelled by the dual representation ρ∗e, and each vertex v la...

View on Pith
[11]

The notation F : Ψ → Ψ′ is meant to suggest that there is a category with spin network s as objects and spin foams as morphisms. For this, we should be ab le to compose spin foams F : Ψ → Ψ′ and F ′: Ψ′ → Ψ′′ and obtain a spin foam F ′F : Ψ → Ψ′′. This composition should be associative, and for each spin network Ψ we want a spin foam 1 Ψ: Ψ → Ψ serving as...

View on Pith
[12]

The physical meaning of the time evolution operators Z(M ):L2(Aγ/Gγ) →L2(Aγ′/Gγ′) is somewhat subtle in a background-independent theory. For example, when M = S × [0, 1] is a cylinder cobordism from S to itself, we should have Z(M )2 = Z(M ). In this case Z(M ) should represent the projection from the gauge-invariant Hilbert space to the space of physical...

View on Pith
[13]

Regge gave a formula for a discretized version of the actio n in 4-dimensional Riemannian general relativity. In his approach, spacetime is triangulated and each edge is assigned a length. The Regge action is the sum over all 4-simplices of: S = ∑ t Atθt where the sum is taken over the 10 triangular faces t, At is the area of the face t, and θt is the dih...

View on Pith
[14]

Our heuristic explanation of the Barrett-Crane model may make it seem more ad hoc than it actually is. For a more thorough treatment one should see the references in the Notes. At present our best understanding of this model comes from a 4-dimensio nal analogue of the theory of the quantum tetrahedron discussed in Remark 1 of Section 6. In pa rticular, th...

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[15]

If X ∈κ and Y ≤X then Y ∈κ

View on Pith
[16]

If X,Y ∈κ then X ∩Y ≤X,Y . In this paper we restrict our attention to complexes with ﬁni tely many cells. A complex is ‘ k-dimensional’ if it has cells of dimension k but no higher. A complex is ‘ori- ented’ if every cell is equipped with an orientation, with al l 0-cells being equipped with the positive orientation. The union of the cells of a complex κ ...

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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 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.
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.
Introduction to Modern Canonical Quantum General Relativity
gr-qc 2001-10 conditional novelty 2.0 of 10

A structured, self-contained survey of canonical quantum general relativity in connection variables, presenting the classical Ashtekar-Barbero formulation, the distributional-connection Hilbert space, and the status o...

How later papers used it

114 later papers cite it

Evidence: substantial. 114 of the citing papers share no author with it, and 6 of those carry a confident label from Pith's citation classifier.

Pith read 19 of the 167 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

3 later papers by other authors are listed below: 2 used its method or tool and 1 compared its own results with it.

Uses its method or tool 2 of 3 papers with this label

Used this paper's standard computations of group integrals to obtain the spin foam picture of its model from its formula.

A New Class of Group Field Theories for 1st Order Discrete Quantum Gravity (2007) · 0.93 · paraphrased: the licence does not allow quoting
Used a discretization involving spin foams from this paper to define its path integral.

Group field theories (2012) · 0.96 · paraphrased: the licence does not allow quoting

Compares with it 1 of 3 papers with this label

Showed how its saddle point method recovers the results presented in this paper in the simplest case of BF theory in dimension 2.

Quantum Corrections in the Group Field Theory Formulation of the EPRL/FK Models (2010) · 0.90 · paraphrased: the licence does not allow quoting

How the literature uses it

116 arXiv papers cite this in 169 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 · 74 sentences without a confident label

Uses its method or tool 3 papers

Since the boundary BF theory coupled to a bulk gravity can be decomposed classically into a closed form just like the $B$ field in a pure BF theory and a fixed, non-closed form determined by the coupling, one may first quantize the SO$(1,1)$ pure BF theory in loop-quantization formalism [ref] , setting up the complete basis of the boundary Hilbert space, and then use the quantum version of coupling term [the first equality of ([ref])] to choose suitable bulk spin network states via triangulations on a section of the IH defined by boundary `spin-network' states.

The entropy of isolated horizons in non-minimally coupling scalar field theory from BF theory · The second approach · 0.94
Going through the standard computations [ref] of group integrals, we can obtain from this formula the spin foam picture of our model.

A New Class of Group Field Theories for 1st Order Discrete Quantum Gravity · The new models: action and Feynman amplitudes · 0.93
To actually define the path integral we use a discretization involving spin foams [ref] .

Group field theories · Discretization of BF theories · 0.96

Compares with it 3 papers

As a largely simplified explanation, a trivially satisfied constraint equation on the allowed irreducible representation labels in a closed subgraph of the foam admits an unbounded sum over all possible labelings of edges by half-integer representations [ref] .

Network Gravity · Bubble Divergences · 0.83
The peculiar feature of the divergence of $c_j^(n)$ is reminiscent of the infrared divergence encountered in many spin-foam models [ref] and the regularization may correspond to a nonzero cosmological constant [ref] .

Loop quantum cosmology with higher order holonomy corrections · Summary and discussion · 0.81
To close this section, let us see how the saddle point method allows us to recover the results presented in [ref] in the simplest case of BF theory in dimension 2.

Quantum Corrections in the Group Field Theory Formulation of the EPRL/FK Models · Two dimensional case · 0.90

Cites it as background 72 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
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.]

On knottings in the physical Hilbert space of LQG as given by the EPRL model · Introduction · 0.92
Let us start with a brief review of the relevant properties of BF theory [ref] , a topological field theory formulated on the principal bundle of a group $G$ over the spacetime manifold $ℳ$.

Topological Features of the Quantum Vacuum · Constructing the action · 0.99

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

Ponzano and Regge gave a formula for a discretized version of the action in 3-dimensional Riemannian general relativity. In their approach the space time manifold M is triangulated and each edge is assigned a length. The Ponzano-Regge action is the su m over all tetrahedra of the quantity: S = ∑ e ℓeθe where the sum is taken over all 6 edges, le is the le...

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[2] [2]

Ever since Section 4 we have been assuming that G is connected. The main reason for this is that it ensures the map from A to Aγ is onto for any graph γ in S, so that we have inclusions L2(Aγ) ֒→ L2(A) and L2(Aγ/Gγ) ֒→ L2(A/G). When G is not connected, these maps are usually not one-to-one. Requiring that G be connected rules out all nontrivial ﬁnite grou...

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[3] [3]

a 2-dimensional oriented complex κ,

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[4] [4]

a labelling ρ of each face f of κ by an irreducible representation ρf of G,

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[5] [5]

a labelling ι of each edge e of κ by an intertwiner ιe:ρf1 ⊗ · · · ⊗ρfn →ρf ′1 ⊗ · · · ⊗ρf ′m wheref1,...,f n are the faces incoming to e and f ′1,...,f ′m are the faces outgoing from e. Note that this deﬁnition is exactly like that of a spin networ k, but with everything one dimension higher! This is why a generic slice of a spin foam is a spin netw ork....

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[6] [6]

a 2-dimensional oriented complex κ such that γ bordersκ,

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[7] [7]

a labeling ˜ρ of each face f of κ by an irreducible representation ˜ρf of G,

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[8] [8]

a labeling ˜ι of each edge e of κ not lying in γ by an intertwiner ˜ιe:ρf1 ⊗ · · · ⊗ρfn →ρf ′1 ⊗ · · · ⊗ρf ′m wheref1,...,f n are the faces incoming to e and f ′1,...,f ′m are the faces outgoing from e, such that the following hold:

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[9] [9]

For any edge e of γ, ˜ρ˜e =ρe if ˜e is incoming to e, while ˜ρ˜e = (ρe)∗ if ˜e is outgoing from e

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[10] [10]

For any vertex v of γ, ˜ι˜e equalsιe after appropriate dualizations. Finally, to deﬁne general spin foams, we need the notions of ‘ dual’ and ‘tensor product’ for spin networks. The dual of a spin network Ψ = ( γ,ρ,ι ) is the spin network Ψ ∗ with the same underlying graph, but with each edge e labelled by the dual representation ρ∗e, and each vertex v la...

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[11] [11]

The notation F : Ψ → Ψ′ is meant to suggest that there is a category with spin network s as objects and spin foams as morphisms. For this, we should be ab le to compose spin foams F : Ψ → Ψ′ and F ′: Ψ′ → Ψ′′ and obtain a spin foam F ′F : Ψ → Ψ′′. This composition should be associative, and for each spin network Ψ we want a spin foam 1 Ψ: Ψ → Ψ serving as...

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[12] [12]

The physical meaning of the time evolution operators Z(M ):L2(Aγ/Gγ) →L2(Aγ′/Gγ′) is somewhat subtle in a background-independent theory. For example, when M = S × [0, 1] is a cylinder cobordism from S to itself, we should have Z(M )2 = Z(M ). In this case Z(M ) should represent the projection from the gauge-invariant Hilbert space to the space of physical...

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[13] [13]

Regge gave a formula for a discretized version of the actio n in 4-dimensional Riemannian general relativity. In his approach, spacetime is triangulated and each edge is assigned a length. The Regge action is the sum over all 4-simplices of: S = ∑ t Atθt where the sum is taken over the 10 triangular faces t, At is the area of the face t, and θt is the dih...

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[14] [14]

Our heuristic explanation of the Barrett-Crane model may make it seem more ad hoc than it actually is. For a more thorough treatment one should see the references in the Notes. At present our best understanding of this model comes from a 4-dimensio nal analogue of the theory of the quantum tetrahedron discussed in Remark 1 of Section 6. In pa rticular, th...

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[15] [15]

If X ∈κ and Y ≤X then Y ∈κ

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[16] [16]

If X,Y ∈κ then X ∩Y ≤X,Y . In this paper we restrict our attention to complexes with ﬁni tely many cells. A complex is ‘ k-dimensional’ if it has cells of dimension k but no higher. A complex is ‘ori- ented’ if every cell is equipped with an orientation, with al l 0-cells being equipped with the positive orientation. The union of the cells of a complex κ ...

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