Quantum Graphity

Fotini Markopoulou; Lee Smolin; Tomasz Konopka

REVIEW

Quantum Graphity

Tomasz Konopka, Fotini Markopoulou, Lee Smolin

High Energy Physics - Theory · Submitted 17 Nov 2006 · 14 references

Pith's verdict: A clean, honest model-building paper whose central claim — that a regular lattice is the low-temperature ground state — is asserted rather than shown, and the loop-selection criterion has a real gap.

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arXiv IDhep-th/0611197 (version 1)

Pith ID3SHQ5M25, Pith's permanent record of this paper and its review

CitationFotini Markopoulou, Lee Smolin, Tomasz Konopka. "Quantum Graphity." arXiv:hep-th/0611197 (2006). https://pith.science/paper/3SHQ5M25

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Editor's note

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Quantum Graphity is worth your time as a model, not as a result. The construction is economical and genuinely new: a complete graph on N nodes, a four-state edge Hilbert space with one off state and three on states carrying m in {-1,0,+1}, and a permutation-invariant Hamiltonian whose ground state is supposed to break the permutation symmetry down to a lattice translation group. That is a nice piece of design. The reduction in section IV is the part that is actually checked, and it checks out: freeze the graph to a regular lattice, and the m variables become the Levin-Wen rotor model; comparing Eq. 29 to the Kogut-Susskind Hamiltonian gives g^2 and the lattice spacing in terms of the model couplings (Eq. 31). The algebra is clean and parameter-free given the model. The cosmology in section V is qualitative, but the point that an emergent-geometry transition can dissolve the horizon problem is a real conceptual payoff, not decoration. The authors are also candid: section III.B says 'We cannot fully answer this question at this point' and the Conclusions say 'we have conjectured.' I take that at face value and it counts in their favor. The stress-test note lands. Eq. 8 defines the preferred loop length by maximizing B^L/L! alone, and section III.B then asserts the ground state maximizes the number of loops of that length. That step drops N_L(g), the number of minimal loops of length L, which is graph-dependent and grows exponentially with L on generic 4-valent graphs. So the parameter that is supposed to pick the square or diamond lattice is not doing the work claimed; the entropic factor is exactly what would decide the lattice-versus-random-graph comparison the paper leaves open. The corrected criterion is not simply (3B)^L/(2L·L!) either, since the m-product expectation also enters, but the qualitative gap is real and it is the load-bearing one. Two smaller things: the high-temperature valence formula (Eq. 22) rests on factorizing Z_N into Z_1^N, which is a mean-field step presented without comment; and the survival of photon-like excitations in the truncated m spectrum is supported by a private communication [10], which is thin. None of this is hidden, and none of it is fatal to the program. This is a founding paper for a research direction, and it is honest about its gaps. Send it to referees who will push on the loop-counting argument; that is the referee report worth having. I would cite it as the origin of the model, not as evidence that the lattice phase exists.

Referee report

Significance

If the low-temperature lattice phase exists, this is an economical and tractable toy model of geometrogenesis: a permutation-symmetric spin system with no background geometry that produces both a lattice and an emergent gauge field. The reduction in Sec. IV is the strongest part of the paper. Matching Eq. (29) to the Kogut-Susskind Hamiltonian Eq. (30) and reading off Eq. (31) is clean algebra, and I verified that the quoted g^2 = sqrt(4! D/(B0 B4)) and 1/a^2 = D B0 B4/3! follow from the stated coefficients, with Eq. (32) following from Eq. (25). The identification is parameter-free given the model, and the paper is candid that the lattice phase is conjectural (Sec. III.B: 'We cannot fully answer this question at this point'; Conclusions: 'we have conjectured'). The cosmological proposal is in principle falsifiable through a transition temperature, a critical exponent, and a residual non-local-link fraction.

Major objections (2)

[Sec. III.B, Eqs. (5), (7), (8)] The criterion that fixes the preferred loop length, Eq. (8), maximizes B^L/L! as a function of B alone. The quantity that actually enters the energy is the graph-dependent sum B0 sum_L (B^L/L!) N_L(g), where N_L(g) is the number of minimal loops of length L in the candidate graph g. On a generic 4-valent graph N_L grows exponentially, N_L ~ (z-1)^L/(2L) = 3^L/(2L), so the effective weight per loop is (3B)^L/(2L·L!), whose maximum sits near L ≈ 3B rather than L* ≈ B. The same B that the paper says gives L* = 4 (square, Fig. 2a) or L* = 6 (diamond, Fig. 2b) may therefore favor longer loops once N_L is included. This is not the same as the acknowledged gap that the lattice-versus-generic-graph comparison has not been carried out: the issue is that the stated selection criterion is incomplete, because it drops the entropic factor that would decide that comparison. A concrete test would be to compute N_L for the square lattice, the diamond lattice, and a random 4-regular graph at the same B and compare the resulting loop energy per vertex; a frustration/coherence analysis is also needed, since on a random graph the m variables cannot simultaneously make all loop products +1, which may restore the lattice. Relatedly, the argument in Sec. III.B is purely energetic (T = 0), whereas the low-temperature phase at T > 0 is set by the free energy; the entropy of v0-regular graphs is extensive and competes with the O(N) loop energy, so an estimate of the entropy per vertex at the transition is needed before 'the ground state will be such that the number of loops of this length is maximized' can be read as a phase statement.
[Sec. V and abstract] The horizon-problem claim rests on the pre-transition ensemble being in thermal equilibrium across regions that later become the observable universe. The paper infers this from the graph diameter being approximately log N, but gives no estimate of the thermalization rate against the cooling rate, and no estimate of the homogeneity scale that results after the transition. One quantitative step, for example a comparison of the equilibration time on the high-temperature graph with the cooling time set by the couplings, or a statement of the comoving scale that is in causal contact before geometrogenesis, would turn the scenario from a plausibility argument into a supported claim. As written, the abstract's 'thus avoiding the horizon problem' is stronger than Sec. V establishes.

Minor objections (10)

[Eq. (16)] The loop Hamiltonian in Eq. (16) is written as a single product of M± operators and is not Hermitian as it stands; the Hermitian conjugate appears only in Eq. (29). Please add h.c. in Eq. (16) or state the convention explicitly.
[Eqs. (27) and (29)] The coefficient of the loop term changes from 1/4! in Eq. (27) to 1/3! in Eq. (29) without explanation. Since this coefficient propagates into Eq. (31) and Eq. (33) at the level of a factor sqrt(2), please track the factor or state that Eq. (31) is order-of-magnitude only.
[Secs. III.B and IV] Sec. III.B fixes v0 = 4 and illustrates the square and diamond lattices in Fig. 2, while Sec. IV considers a three-dimensional pattern of four-sided plaquettes with v0 = 6. Please flag the change of valence and note that the L* = 4 example of Fig. 2a is two-dimensional whereas the Kogut-Susskind comparison is three-dimensional.
[Sec. II.A] The definition of 'minimal loops' ('loops that cannot be factored into the product of two loops of on edges that contain some of the same edges') is not precise enough to count N_L on a general graph. Since the loop count is central to the selection argument, please give a graph-theoretic definition.
[Eq. (18)] The factorization Z_N ≈ Z_1^N in Eq. (18) ignores the fact that each edge is shared by two nodes. Please state that this is a mean-field approximation valid for N ≫ 1.
[Abstract] 'Broken to the translation group of that lattice' should read 'broken to the automorphism (space) group of the lattice,' since a square-lattice ground state also preserves rotations and reflections.
[Sec. V] The statement that the graph diameter is approximately log N holds for a random graph well above the connectivity threshold; at p ~ T/(NV) with T ~ V the graph is near p ~ 1/N, where the diameter is much larger. Please specify the temperature regime being described.
[Sec. III.A and Eq. (33)] Sec. III.A assumes V ≫ B, C, D, while Eq. (33) gives B0 B4 ~ m_P g^{-2} ≫ D for weak g. Please state the full hierarchy explicitly and confirm that the high-temperature analysis and the string-condensation regime of Sec. IV can be realized with the same choice of couplings.
[References] Ref. [10] is a private communication and Refs. [13] and [14] are preprints in preparation. The claims that lean on them, namely photon-like excitations with m restricted to -1, 0, 1 in Sec. IV and the measurable consequences of the transition in the Conclusions, should either cite published support or be marked as unpublished.
[Throughout] There are several typographical errors: 'parition' (below Eq. 19), 'simplity' (Sec. III), 'plaquetes' (Sec. IV), 'deﬁntions' (footnote 1), 'disordeded' (Sec. I), and 'mimimal' (Sec. II.A).

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The paper

The paper claims

A model in which space itself is a low-temperature phase of a permutation-symmetric graph, with lattice geometry and a U(1) gauge field condensing out as the system cools.

The reading

The paper proposes a model of physics with no geometry built in at all: the fundamental degrees of freedom sit on the edges of a complete graph of $N$ points, and the Hamiltonian is invariant under every permutation of those points. Each edge carries a four-state label — an 'off' state plus a spin-one triplet — so the configuration space contains every graph on $N$ nodes, and the dynamics is written so that the low-temperature phase prefers a fixed valence and a preferred loop length. The authors argue that below some temperature the edges freeze into a regular low-dimensional lattice (a square lattice if the preferred loop length is four, a diamond lattice if it is six), breaking the permutation symmetry down to the translations of that lattice, while the leftover $m$ labels reduce to a rotor model whose excitations behave like photons of an emergent $U(1)$ gauge theory. The payoff, if the argument holds, is a tractable toy model in which classical geometry and a gauge field are not assumed but are consequences of a phase transition. The authors then use this to sketch a cosmology: an early hot disordered phase in which every degree of freedom is a short graph distance from every other, followed by a cooling transition into geometry, which would explain the observed uniformity of the universe without invoking inflation.

What carries the argument

The load-bearing object is the loop term $H_{loops} = -B_0 B^L/L!$ summed over minimal loops, with the factorial weighting that gives a preferred loop length $L^*$. Its job is to break the tie among the many four-valent graphs that the valence term alone cannot distinguish: trees, random regular graphs, and non-planar networks all minimize the valence energy equally, so something must reward closed cycles of one particular size. The second piece of machinery is the reduction of the frozen-graph $m$ sector to the Levin–Wen rotor model, which is what turns a combinatorial graph into a gauge theory: the plaquette operator $W = M^+ M^- M^+ M^-$ around a four-edge loop plays the role of the magnetic flux term, and matching coefficients against the Kogut–Susskind Hamiltonian yields a weak emergent coupling when the string-condensation inequality $2D < B_0 B^4/4!$ holds.

Extended reading notes

Core claim

The central claim is that a permutation-symmetric spin model on a complete graph possesses a low-temperature phase in which a low-dimensional lattice emerges and the permutation symmetry is broken to that lattice's translation group, with the residual edge labels producing an emergent $U(1)$ gauge theory. The mechanism is a competition written into the Hamiltonian: a term $V( v_0 - J )^2$ fixes the valence of every node to $v_0$, and a loop term $-B(L)/L!$ with $B(L) = B_0 B^L$ rewards short closed cycles, with a preferred loop length $L^*$ set by where $B^L/L!$ peaks. Because $B^L/L!$ is small at both very small and very large $L$, the loop term singles out one plaquette size, and the authors argue this selects a regular lattice — square for $L^*=4$, diamond for $L^*=6$ — rather than a tree or a random network. Once the 'on'/'off' pattern freezes, the $m$ labels on the surviving edges are governed by a tension term $D M^2$ and the loop operator, which the authors identify with the Levin–Wen rotor model and hence with the Kogut–Susskind Hamiltonian of $U(1)$ lattice gauge theory in the $A_0 = 0$ gauge, with coupling $g^2 = (2/3!)(D/B_0 B_4)$ up to the conventions they state. The cosmological corollary is that before the transition the graph diameter is only of order $1$, or at most $N$, so all regions were in causal contact, and the horizon problem is dissolved rather than inflated away.

Load-bearing premise

The argument needs the loop term to beat the string tension and to make a regular lattice the lowest-free-energy four-valent graph, but the paper compares only two candidate lattices and never evaluates the free energy of a random, tree-like, or non-planar four-valent graph, a gap the authors themselves flag when they write that they cannot fully answer the question at this point.

Editorial extensions

If this is right

If the low-temperature phase really is a regular lattice, then a model with no metric, no dimension, and no gauge fields at the fundamental level reproduces both a lattice geometry and a $U(1)$ gauge field as collective low-energy behaviour.
The emergent gauge coupling is fixed by the microscopic couplings, $g^2 \sim \sqrt{4!\,D/(B_0 B_4)}$, so the strength of the emergent force is a thermodynamic parameter rather than an input.
Setting the emergent lattice spacing to the Planck length translates the couplings into $B_0 B_4 \sim m_P g^{-2}$ and $D \sim g^2 m_P$, giving a concrete numerical target for where the transition should sit.
Because the high-temperature graph has diameter of order $\log N$, all degrees of freedom equilibrate before geometry appears, which removes the horizon problem without a period of exponential expansion.
Diffeomorphism invariance becomes an emergent, approximate statement: the graph has no embedding, so a deformation of the emergent metric that preserves connectivity leaves all observables unchanged.

Reading between the lines

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

The proposal makes a sharp statistical-mechanics question out of an old quantum-gravity question: the phase diagram of a four-valent graph ensemble with a loop reward. That is a computable object — Monte Carlo on random 4-regular graphs with the $B^L/L!$ weight would either find the square and diamond lattices as free-energy minima or find something else.
The same architecture suggests a natural account of why matter and geometry are not separable here: the $m$ labels that become photons are also what organize the links, so there is no limit in which gravity is switched off while matter survives, unlike the Einstein–Hilbert plus matter split the authors contrast themselves with.
If the transition leaves a residue of non-local links, as the authors speculate, that residue is a candidate for a cosmological signature distinct from inflation — a small population of pairs of regions that are close in the graph but far in the emergent metric.
The construction invites an obvious generalization test: allowing $m$ to range beyond $-1,0,+1$ should turn the emergent $U(1)$ into something richer, and the paper's own remark that the correspondence is inexact for the three-state truncation makes that generalization the natural next check.

Original abstract

We introduce a new model of background independent physics in which the degrees of freedom live on a complete graph and the physics is invariant under the permutations of all the points. We argue that the model has a low energy phase in which physics on a low dimensional lattice emerges and the permutation symmetry is broken to the translation group of that lattice. In the high temperature, or disordered, phase the permutation symmetry is respected and the average distance between degrees of freedom is small. This may serve as a tractable model for the emergence of classical geometry in background independent models of spacetime. We use this model to argue for a cosmological scenario in which the universe underwent a transition from the high to the low temperature phase, thus avoiding the horizon problem.

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.

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The abstract, sentence by sentence

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

We introduce a new model of background independent physics in which the degrees of freedom live on a complete graph and the physics is invariant under the permutations of all the points. We argue that the model has a low energy phase in which physics on a low dimensional lattice emerges and the permutation symmetry is broken to the translation group of that lattice. In the high temperature, or disordered, phase the permutation symmetry is respected and the average distance between degrees of freedom is small. This may serve as a tractable model for the emergence of classical geometry in background independent models of spacetime. We use this model to argue for a cosmological scenario in which the universe underwent a transition from the high to the low temperature phase, thus avoiding the horizon problem.

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	We introduce a new model of background independent physics in which the degrees of freedom live on a complete graph and the physics is invariant under the permutations of all the points.	contribution 0.95	0.86	0.08	asserted 1.00
s1	We argue that the model has a low energy phase in which physics on a low dimensional lattice emerges and the permutation symmetry is broken to the translation group of that lattice.	result 0.85	0.67	0.40	asserted 0.44
s2	In the high temperature, or disordered, phase the permutation symmetry is respected and the average distance between degrees of freedom is small.	result 0.98	0.48	0.07	asserted 1.00
s3	This may serve as a tractable model for the emergence of classical geometry in background independent models of spacetime.	interpretation 0.99	0.21	0.22	hedged possibility 1.00
s4	We use this model to argue for a cosmological scenario in which the universe underwent a transition from the high to the low temperature phase, thus avoiding the horizon problem.	interpretation 0.85	0.70	0.30	asserted 0.68

What kind of paper the classifier reports this is: theoretical result 0.99 · evidence from argument 1.00.

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sources read

arXiv listing abstract v1

not examined

full text
figures and tables
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levels run

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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

Figures

**Figure 1.** FIG. 1: Examples of terms in Hamiltonian

**Figure 2.** FIG. 2: Sample lattice formations with 4-valent vertices. S

**Figure 3.** FIG. 3: Diffeomorphism invariance on the honeycomb lattice

Figures reproduced from arXiv:hep-th/0611197 by the authors.

References

Works cited (14)

[10]

X. G. Wen, private communication

View on Pith
[1]

J. Ambjorn, J. Jurkiewicz and R. Loll, “Quantum gravity , or the art of building spacetime,” arXiv:hep-th/0604212

View on Pith Pith review arXiv
[2]

E. Bianchi, L. Modesto, C. Rovelli and S. Speziale, “Gravi ton propagator in loop quantum gravity ,” Class. Quant. Grav .23, 6989 (2006) [arXiv:gr-qc/0604044]

View on Pith Pith review arXiv 2006
[3]

R. Gambini, J. Pullin, Loops, Knots, Gauge Theories and Qu antum Gravity , Cambridge U. Press, New York (2000); C. Rovelli, Quantum Gravity , Camb ridge U. Press, New York (2004); T. Thiemann, “Introduction to modern canonica l quantum general relativity ,” arXiv:gr-qc/0110034.; A. Ashtekar and J. Lewandowski, “Ba ckground independent quan- tum gravity: A...

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[4]

M. Levin and X. G. Wen, “Fermions, strings, and gauge ﬁelds i n lattice spin models,” Phys. Rev . B67, 245316 (2003) [arXiv:cond-mat/0302460]

View on Pith Pith review arXiv 2003
[5]

M. A. Levin and X. G. Wen, “String-net condensation: A phys ical mechanism for topological phases,” Phys. Rev . B71, 045110 (2005) [arXiv:cond-mat/0404617]

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[6]

M. Levin and X. G. Wen, “Quantum ether: Photons and electro ns from a rotor model,” arXiv:hep-th/0507118

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[7]

F. Markopoulou, “Towards gravity from the quantum,” arXi v:hep-th/0604120

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[8]

D. W. Kribs and F. Markopoulou, “Geometry from quantum part icles,” arXiv:gr-qc/0510052

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[9]

J. B. Kogut and L. Susskind, “Hamiltonian Formulation Of W ilson’s Lattice Gauge Theories,” Phys. Rev . D11, 395 (1975)

View on Pith 1975
[11]

Z. C. Gu and X. G. Wen, “A lattice bosonic model as a quantu m theory of gravity ,” arXiv:gr-qc/0606100

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[12]

S. S. Lee, “Emergence of fully covariant quantum gravit y from a lattice boson model,” arXiv:gr-qc/0609107

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[13]

J. Magueijo and L. Smolin, “Holography and the scale inva riance of density ﬂuctuations,” preprint in preparation (2006)

View on Pith 2006
[14]

F. Markopoulou, C. Prescod-Weinstein and L. Smolin, “Da rk energy as a consequence of disordered locality ,” preprint in preparation (2006)

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How the literature uses it

46 arXiv papers cite this in 91 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 whose main result relates to this one's

Found among the papers closest to this one by title and main claim. Pith's classifier compared each pair's main results and a second model family confirmed the relation; checked against two further families, shown conflicts agreed 118 times in 120 and shown same results 218 times in 239. The quoted sentence is the other paper's main claim.

States the same result 1 paper

We present evidence that the model also has a low-energy phase in which the graph describing the system breaks permutation symmetry and appears to be ordered, low-dimensional and local.

Quantum Graphity: a model of emergent locality · 0.94

Papers citing it · 51 sentences without a confident label

Builds on it 3 papers

To extend this construction to the case in which the ${ j_ij }$ degrees of freedom are not frozen in a particular lattice $Γ_N$ like the square lattice of Fig.[ref](a-b), in [ref] we modified the Hamiltonian of [ref] to the terms: H_loops =-g_L∑_loops∑_L=0^∞r^L/L! ∏_a=1^L M_a^±,

Background independent condensed matter models for quantum gravity · Quantum graphity: A model of emergent geometry and matter · 1.00
In order to connect with the original quantum graphity model [ref] , consider relabeling the states of ([ref]) so that _edge = span { | 0, 0 ⟩, | 1, -1⟩, | 1, 0⟩, | 1, +1⟩} = span { | j, m⟩} so that the off state has $j=0$ and $m=0$, and the on states have $j=1$ and $m=0,±1$.

Quantum Graphity: a model of emergent locality · Extensions with More Degrees of Freedom · 0.81
Both the hamiltonian function driving the transition, and thus the selection of the ground state lattice, and the effective hamiltonian governing the dynamics of quasi-particles around the resulting ground state, the two main ingredients of the model in [ref] , can in principle be derived from any given choice of GFT action, whose dynamical content is indeed the same, after appropriate simplifications.

Group field theory as the microscopic description of the quantum spacetime fluid: a new perspective on the continuum in quantum gravity · Geometrogenesis using GFTs · 0.83

Uses its method or tool 1 paper

In the present paper, we will use the framework of Quantum Graphity [ref] to construct a simple model of quantum foam.

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Compares with it 2 papers

In [ref] a different Hamiltonian based on graphs and close in spirit to [ref] found, instead, a 2-dimensional complex in a low-energy phase.

Curved geometry and Graphs · Introduction · 0.97
To give an example of the latter class we mention the so-called quantum graphity approach (see for example [ref] , [ref] , [ref] ) which we want to compare to our framework being introduced in the following and which may be called Structurally Dynamic Cellular Networks (SDCN) or automata (SDCA) approach to quantum geometry.

The Structurally Dynamic Cellular Network and Quantum Graphity Approaches to Quantum Gravity and Quantum Geometry - A Review and Comparison · Introduction · 0.86

Cites it as support 2 papers

However, the results of [ref] in two dimensions and those of [ref] , support the conjecture that, in general, such a Hamiltonian exists.

Disordered locality and Lorentz dispersion relations: an explicit model of quantum foam · The effect of a quantum superposition of graphs · 0.99
This is corroborated by many other approaches in which the notion of spacetime arises either as a structure for the propagation of collective modes as phonons in BEC [ref] , or because it arises as a statistical concept as in random models [ref] , or because the very topology (in the sense of connectivity) of spacetime looks different at different scales [ref] .

Emergent Models for Gravity: an Overview of Microscopic Models · Challenges and future directions · 0.97

Cites it as background 25 papers

Dynamics in Quantum Graphity is given by a hamiltonian acting on the graph states of the form [ref] : [equation] where $H_V$ keeps track of how many on edges are attached to a single vertex, and $H_L$ counts closed paths in the graph.

Properties of Quantum Graphity at Low Temperature · Quantum Graphity · 0.81
Quantum Graphity models [ref] are spin system toy models for emergent geometry and gravity.

Disordered locality and Lorentz dispersion relations: an explicit model of quantum foam · Introduction · 0.98
Graphity [ref] is a name for a class of models in which states are associated with graphs and the configuration space is the space of all possible simple graphs with a given number of nodes $N$.

Statistical Mechanics of Graphity Models · Graphity Models · 0.96

Discussion

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