3d Quantum Gravity and Effective Non-Commutative Quantum Field Theory

Etera R. Livine; Laurent Freidel

REVIEW

3d Quantum Gravity and Effective Non-Commutative Quantum Field Theory

Laurent Freidel, Etera R. Livine

High Energy Physics - Theory · Submitted 9 Dec 2005 · 17 references

Pith's verdict: A short letter doing real work: Ponzano-Regge amplitudes are the Feynman rules of one braided non-commutative field theory, with the heavy derivations parked in a published companion.

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arXiv IDhep-th/0512113 (version 2)

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CitationEtera R. Livine, Laurent Freidel. "3d Quantum Gravity and Effective Non-Commutative Quantum Field Theory." arXiv:hep-th/0512113 (2005). https://pith.science/paper/NFN5S6EW

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Editor's note

read the letter

The thing to know: this letter claims an exact equivalence — matter coupled to 3d quantum gravity, quantized by Ponzano-Regge, has Feynman diagram evaluations that are literally those of a single braided non-commutative field theory with kappa-Poincare symmetry. The effective action is Eq. (19), and the kappa to 0 limit, Eq. (32), lands on ordinary commutative QFT Feynman rules. That limit is the part I trust most, because it is an external benchmark rather than an internal consistency check. Deriving the deformed symmetry instead of postulating it is the real payoff for the deformed-special-relativity crowd, who have been fitting kappa-Poincare to phenomenology for years without a first-principles origin in a matter theory. Credit where due: the star product of Eq. (11) is not arbitrary, it is dual to SO(3) convolution, and the group Fourier transform is an isometry. That is clean work, and the momentum-space action (20) is a genuinely usable object. The braiding matches the earlier two-particle results of [5] and [10], which is a real independent check on the R-matrix (29). Now the soft spots, in proportion. The stress-test is right that the multi-crossing factorization is asserted in one sentence and delegated to [1], and that the kappa to 0 check is blind to it because the braiding goes to trivial momentum conservation. But [1] is published in Classical and Quantum Gravity, so this is a letter summarizing a companion paper, not an unsupported claim. I would not hold it against them. I do disagree with the reader's location of the weak point: Eq. (8) is not the soft step. Integrating over X_v just produces the delta functions of (7); it is a Fourier representation, not an assumption. Two smaller things worth naming. The Lorentzian extension is asserted by signature change, 'straightforwardly', with no worked check that the star product and propagator survive the non-compact group. And the kappa-Poincare 'symmetry' of Eq. (19) is shown by writing down how generators act on multi-particle states (23)-(24), not by an explicit invariance proof of the action. Both are minor for a letter. The acknowledged non-unitarity is stated openly, so it is a limitation, not a hidden flaw. Who should read it: anyone in spin foams, deformed special relativity, or non-commutative geometry. It deserves a serious referee, mainly to check that the Lorentzian and cosmological-constant extensions in [1] hold up.

Referee report

Significance

If the identification holds, this is a benchmark result for quantum gravity: an exact statement, in a controlled setting, that matter coupled to 3d gravity is a braided non-commutative field theory, with the deformation parameter fixed in terms of Newton's constant rather than introduced by hand. Three features deserve explicit credit. First, the kappa -> 0 limit of Eq. (32) is a genuine external benchmark: it ties the spin-foam amplitude to standard commutative Feynman evaluations, so the construction is not self-referential at leading order. Second, the braiding of Eq. (26) reproduces the two-particle results previously obtained in [5] and [10], which is an independent consistency check on the most distinctive part of the claim. Third, the resulting structure is concrete enough to be checked and used: bounded momenta, a specific deformed conservation law at the vertex, and a specific non-commutative delta-function all follow, and the deformed special relativity program thereby acquires a first-principles derivation in 3d together with a template for the 4d programme outlined at the end. The letter is a summary of a longer published work, which is appropriate for its format.

Major objections (1)

[Eq. (25) and the paragraph immediately following it] The sentence 'the mass gets renormalized m -> sin(kappa m)/kappa' needs qualification. The propagator of Eq. (25) depends on the mass only through sin^2(kappa m)/kappa^2, and the allowed range of the deficit angle theta = kappa m is 0 < theta < 2 pi, as stated above Eq. (5). On that range the map m -> sin(kappa m)/kappa is not monotonic: it rises to 1/kappa at kappa m = pi/2 and falls back to zero at kappa m = pi. Consequently the effective theory cannot distinguish a particle of mass m from one of mass pi/kappa - m, and for kappa m > pi/2 the 'renormalized' mass decreases as the mass increases. This is presumably a genuine feature of the conical geometry, since deficit angles theta and 2 pi - theta give conjugate holonomies and the same geometry, but the letter presents the effect as a simple renormalization. Please state the identification explicitly, or restrict the statement to kappa m <= pi/2, and confirm that this is the reading intended in [1]. This point does not affect the identification of the amplitudes with the non-commutative field theory, but it does affect the physical interpretation of Eq. (25).

Minor objections (10)

[Eq. (6)] The inline formulas for K_theta(j) and Re K_theta in the text following Eq. (6) appear garbled in the version supplied ('i2kappa2 e^{-id_j(theta - i epsilon)} cos theta' and 'cos theta 2 kappa^2 sin theta chi_j(theta)'). Please check these against Eq. (10) and against [1] at proof stage, since the Feynman-versus-Hadamard distinction is used later.
[Introduction and Eqs. (21)-(24)] The invariance of the action (19) under the deformed translations is asserted but not exhibited. One line showing that the star product is invariant under g -> Lambda g Lambda^{-1} and that the deformed coproduct of Eq. (24) leaves the vertex factor delta(g1 g2 g3) of Eq. (20) invariant would make the symmetry claim checkable within the letter rather than only in [1].
[Eq. (10)] The integral over T in R_+ of a pure phase requires an explicit i epsilon prescription, as is done for K_theta(j) above. Please state the prescription and its relation to the choice between the Feynman kernel K_theta and the Hadamard kernel Re K_theta, since the two are distinguished only by the integration range.
[Eqs. (8) and (18)] The derivation of Eq. (8) is explicitly restricted to trivial topology M ~ S^3 with planar Gamma, whereas the sum in Eq. (18) runs over all trivalent graphs. Please state explicitly that Eq. (18) includes non-planar graphs and that the crossing weight of Fig. 1 is what defines them; as written, a reader may take Eq. (8) to cover all Gamma and then wonder where the braiding enters.
[Fig. 1] The caption lists the planar vertices but does not label the crossing weight delta(g1 g2 g1'^{-1} g2'^{-1}) delta(g2 g2'^{-1}) against the corresponding lines in the figure. Since this weight is the distinctive content of the braided statistics, it should be identified unambiguously in the figure.
[Notation, Eqs. (13), (21)-(24), (30)] Here kappa = 4 pi G has dimensions of length, with momenta bounded by 1/kappa, whereas in most of the kappa-Poincare literature kappa denotes a mass scale and the deformation parameter is 1/kappa. A sentence fixing conventions at the outset would prevent confusion, particularly in the addition law of Eq. (13) and the commutators of Eq. (30).
[Final paragraph before the references] The remark that a unitary theory is not expected and that ghosts should appear at the Planck energy is presented as an expectation without any ghost analysis. Please mark it explicitly as a conjecture, or soften it, so that it is not read as a derived result.
[Abstract and introduction] The scope restrictions actually used in the derivation (Riemannian signature first, trivial topology M ~ S^3, planar Gamma for Eq. (8)) are not visible in the abstract or the opening paragraphs. A single sentence stating them would help the reader calibrate the claim.
[Eq. (31)] The normalization of delta_0(X) is with respect to the measure d^3X/(8 pi kappa^3) rather than the usual Lebesgue measure. Please state this explicitly, since the stated width of the distribution is otherwise easy to misread.
[Front matter] The PACS numbers field is empty and should be filled in before publication.

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The paper

The paper claims

Matter coupled to 3d quantum gravity is exactly a non-commutative field theory with a curved momentum space and a deformed Poincaré symmetry.

The reading

In three spacetime dimensions gravity has no local degrees of freedom: space is flat everywhere except where matter sits, and a point particle simply cuts a wedge out of space with deficit angle $m=4piG$ times its mass. The authors ask what becomes of matter Feynman diagrams once those gravitational degrees of freedom are integrated out exactly, and answer that each diagram is evaluated with a curved momentum space and a deformed rule for adding momenta, rather than with ordinary flat momentum space. Their claim is that the sum of all such diagrams is the perturbative expansion of one explicit non-commutative field theory, whose $star$-product is dual to convolution on the rotation group $SO(3)$ in the Riemannian case and on $SO(2,1)$ in the Lorentzian case. If the claim holds, quantum-gravity corrections to scattering amplitudes become computable order by order in Newton's constant, and a field theory invariant under the $kappa$-deformed Poincaré group is derived from first principles rather than postulated.

What carries the argument

The load-bearing device is a group Fourier transform $mathcal{F}: C(SO(3)) o C_kappa(mathbb{R}^3)$ that maps functions on the group to functions on $mathbb{R}^3$ with momenta bounded by $1/kappa$, together with a $star$-product defined so that $e^{frac{1}{2kappa}mathrm{tr}(Xg_1)} star e^{frac{1}{2kappa}mathrm{tr}(Xg_2)} = e^{frac{1}{2kappa}mathrm{tr}(Xg_1g_2)}$. Under this transform the $star$-product is dual to group convolution, which is exactly what turns the product of edge holonomies meeting at a vertex in the spin-foam amplitude into a product of fields. The transform is an isometry between $L^2(SO(3))$ and the space $C_kappa(mathbb{R}^3)$ with the norm built from $varphi star varphi$, so the field-theory Hilbert space and the gravity amplitude sit in the same place. The same $star$-product also generates the non-commuting coordinates $[X_i,X_j] = i kappa epsilon_{ijk} X_k$ and a smeared delta function $delta_0(X) = 2kappa J_1(|X|/kappa)/|X|$, which is where the minimal-length interpretation comes from.

Extended reading notes

Core claim

The central claim is an identification: the Ponzano-Regge spin-foam amplitude for 3d gravity with particle insertions, $I_Gamma$, can be rewritten as a vertex integral over one Lie-algebra element $X_v$ per vertex and one group element $g_e$ per edge, and that integral is precisely the Feynman evaluation of a non-commutative field theory with action $S = int d^3x/(8pi kappa^3),[ tfrac12 (partial_i varphi star partial_i varphi) - tfrac12 (sin^2(kappa m)/kappa^2)(varphi star varphi) + tfrac{lambda}{3!}(varphi star varphi star varphi) ]$. Three consequences are drawn. The mass is renormalized to $sin(kappa m)/kappa$; momentum space is the curved group manifold rather than flat $mathbb{R}^3$, so momenta are bounded by $1/kappa$; and momentum addition at a vertex is the non-linear law $vec P_1 oplus vec P_2$ of the deformed Poincaré group, so $vec P_1+vec P_2+vec P_3 eq 0$ and part of the collision energy is absorbed by gravity. Non-planar diagrams carry a braiding weight, expressed as the exchange relation $tilde varphi(g_1)tilde varphi(g_2) = tilde varphi(g_2)tilde varphi(g_2^{-1}g_1g_2)$, identified as the $R$-matrix of the $kappa$-deformation. The authors also show that the no-gravity limit $kappa o 0$ reproduces ordinary commutative Feynman rules, and that the construction carries over to Lorentzian signature with $AdS_3$ momentum space and to non-zero cosmological constant with $U_q(SU(2))$.

Load-bearing premise

The step that rewrites the Ponzano-Regge amplitude as the vertex integral over one $X_v$ per vertex and one $g_e$ per edge is asserted in this letter with a pointer to a companion paper, is restricted to trivial topology $M simeq S^3$ with planar $Gamma$, and the braiding needed for crossing edges is introduced by the bare sentence that a careful analysis shows it; if either step holds only for planar graphs or only up to triangulation-dependent factors, the field-theory identification does not follow.

Editorial extensions

If this is right

Total momentum in a collision can never exceed the Planck scale: the deformed addition law $vec P_1 oplus vec P_2$ saturates, so gravitational effects cannot be ignored at high energy and the effective theory supplies its own cutoff.
The particle mass is renormalized to $sin(kappa m)/kappa$, which caps the mass at $m_P = (2G)^{-1}$ where the deficit angle reaches $2pi$.
Because the $kappa o 0$ limit reproduces ordinary commutative Feynman evaluations, the deformation is a controlled expansion in Newton's constant rather than a replacement of QFT.
Space-time coordinates fail to commute, $[X_i,X_j] = i kappa epsilon_{ijk} X_k$, and the non-commutative delta function has a non-zero width, giving an explicit minimal length scale of order $kappa$.
A non-zero cosmological constant replaces $SO(3)$ by $U_q(SU(2))$ and the same Feynman rules go through, so the construction is not tied to vanishing curvature.
The authors expect the effective theory to be non-unitary, with ghosts appearing at the Planck energy, precisely because the gravitational degrees of freedom have been traced out.

Reading between the lines

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

If the identification is exact, the bounded momentum space acts as a physical ultraviolet regulator built into the kinematics, so loop integrals in this theory should be finite or milder than their commutative counterparts; checking a one-loop diagram explicitly would test whether the deformation really regularizes rather than merely reshuffles divergences.
The braided exchange relation means the theory is not a local field theory in the ordinary sense, so standard results such as spin-statistics and cluster decomposition would need to be re-derived inside the braided category before the theory can be treated as a conventional QFT with deformed symmetry.
The deformation parameter is Newton's constant itself, so the first correction to any given diagram is a definite computable number; this makes the proposal a target for numerical comparison against spin-foam amplitudes rather than only a structural statement.
The authors' own suggestion that 4d Feynman graphs might be expectation values of a 4d topological spin-foam model is the natural next step, but the 4d theory has local degrees of freedom, so the exact rewriting available in 3d would become at best a semiclassical expansion.

Original abstract

We show that the effective dynamics of matter fields coupled to 3d quantum gravity is described after integration over the gravitational degrees of freedom by a braided non-commutative quantum field theory symmetric under a kappa-deformation of the Poincare group.

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arXiv listing abstract v2. Results and the main claim are underlined; show every role.

We show that the effective dynamics of matter fields coupled to 3d quantum gravity is described after integration over the gravitational degrees of freedom by a braided non-commutative quantum field theory symmetric under a kappa-deformation of the Poincare group.

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s0	We show that the effective dynamics of matter fields coupled to 3d quantum gravity is described after integration over the gravitational degrees of freedom by a braided non-commutative quantum field theory symmetric under a kappa-deformation of the Poincare group.	result 0.95	0.92	0.92	asserted 1.00

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Figures

**Figure 1.** FIG. 1: Feynman rules for particles propagation in the

Figures reproduced from arXiv:hep-th/0512113 by the authors.

References

Works cited (17)

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L Freidel, Nucl.Phys.Proc.Suppl. 88 (2000) 237-240, [arXiv:gr-qc/0102098]; S Davids, [arXiv:gr-qc/0110114]

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How later papers used it

121 later papers cite it

Evidence: substantial. 102 of the citing papers share no author with it, and 13 of those carry a confident label from Pith's citation classifier.

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1 later paper by other authors is listed below: 1 used its method or tool. 1 more paper by the same authors is listed: 1 used its method or tool.

Uses its method or tool 2 of 9 papers with this label · 1 by the same authors

Used this paper's group Fourier transform to obtain the Lie algebra representation of a function of one group variable.

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By the same authors

Followed this paper to expand this function on the group easily in representations.

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How the literature uses it

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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 · 121 sentences without a confident label

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[ref] overlook the case of non-unimodular groups, which we treat with extra care.

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The new $(2)$ Feynman propagator amounts to modifying the kinetic term of our original NCQFT, S[ϕ] = 1/2 ∫_G g (g) ϕ(g) ϕ(g^-1) + λ/3 ∫_G [g]^3 δ(g_1 g_2 g_3) ϕ(g_1)ϕ(g_2)ϕ(g_3) from the original theory written in [ref] , _(3)(g) =P^2(g)-M^2 =^2(sin^2θ-sin^2) =-^24[χ_1(g)-χ_1()] to a new version based on the character of spin $12$ instead, _(2)(g) =-^2[χ_12(g)-χ_12()] =-2^2(cosθ-cos) .

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Using this momentum $(g)$, one can perform a change of variables, from group elements $g$ to usual vectors $$, and write our field theory [ref] in term of standard momenta variables [ref] :

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This represents the first example of concrete non-commutativity arising as a consequence of a well defined mathematical framework that unites the gravitational and quantum realms.[Perhaps the most notable exception are the intriguing results by Freidel and collaborators [ref] ] Needless to say, the story is far from over and the subject deserves further attention.

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Additional elements of motivation for such studies are provided by the fact that $2+1$-dimensional quantum gravity, the only quantum-gravity we have managed to solve [ref] , does indeed lead to DSR-relativistic symmetries and curvature of momentum space.

Anti-de Sitter Momentum Space in 3D and 4D Quantum Gravity · Introduction · 0.97
Furthermore, the MLUR implies that energy-momentum space will be a fluctuating dynamical entity as well [ref] .

On the Minimal Length Uncertainty Relation and the Foundations of String Theory · Need for a UV/IR relation and a Dynamical Energy-Momentum Space · 0.99

Cites it as background 32 papers

The goal of that work, motivated by earlier results in three-dimensional spin foam gravity [ref] , was to try to bridge the gap between the algebraic framework of spin foams and the standard formalism of quantum field theory formulated on a background spacetime with a fixed metric.

A 2-categorical state sum model · Introduction · 0.84
We set out to explore this largely overlooked aspect of quantum spacetime research within the framework of non-commutative geometry [ref] , a leading candidate for the effective description of spacetime emerging from quantum gravity at length scales above the Planck length [ref] .

Indefinite probabilities in quantum spacetime: A deepening of unpredictability · Introduction · 0.91
Various issues in constructing quantum field theories in $κ$-space-time are addressed and resolved in [ref] .

Yang-Mills Field in the $\kappa$-space-time · Introduction · 0.94

Discussion

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