REVIEW
The continuum limit of loop quantum gravity - a framework for solving the theory
General Relativity and Quantum Cosmology · Submitted 4 Sep 2014 · 78 references
Pith's verdict: A clear, honest roadmap for Dittrich's own coarse-graining program — useful as a map, not as a result, and the one derivation it leans on is heuristic by its own admission.
T0 · AI review
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Referee report
The paper
The paper claims
The paper argues that completing loop quantum gravity requires constructing its continuum limit, and that this is the same task as constructing the physical Hilbert space of the theory.
The reading
What carries the argument
The load-bearing device is cylindrical consistency with dynamical embedding maps. In the kinematical construction the embedding maps encode a fixed vacuum, and in the Ashtekar–Lewandowski case that vacuum is a totally squeezed state peaked on degenerate spatial geometry, which is why states remain distributional. The proposal replaces those maps by the refining evolution $K_{alpha alpha'}$, whose consistency conditions follow from path independence of evolution — Kuchar's requirement, equivalent to the constraint algebra being first class. Equation (10) of the paper, $A^{imp}_alpha = langle psi_emptyset | (K_{emptyset alpha'})^dagger | K_{alpha alpha'} psi_alpha rangle$, is the iterative step: the amplitude on a coarse boundary is improved by transporting it back from a finer boundary using the same dynamics. The practical implementation is tensor network coarse graining, where a singular value decomposition of the reshaped amplitude matrix, $M_{A alpha beta} = sum_i U_{A i} lambda_i V_{i alpha beta}$, truncates the doubled boundary data back to a fixed bond dimension, with the truncation chosen by the dynamics rather than by hand. Decorated tensor networks are invoked to keep the original spin foam boundary variables — the intertwiners — visible along the flow.
Extended reading notes
Core claim
The central claim is an identification: the continuum limit of the dynamics and the physical Hilbert space are one and the same object, and both are to be obtained as an inductive limit. Concretely, one seeks a family of amplitude maps $A_alpha$ on boundary Hilbert spaces $H_alpha$ that is cylindrically consistent, $A_{alpha'}(iota_{alpha alpha'}, psi_alpha) = A_alpha(psi_alpha)$, where the embedding maps $iota_{alpha alpha'} = K_{alpha alpha'}$ are the refining time evolution of the theory itself rather than the fixed Ashtekar–Lewandowski embeddings. Because that evolution acts as a projector onto physical states, a physical state pushed to a finer boundary should come back equivalent to itself, and this is exactly what makes the amplitude family consistent. The paper turns this into an algorithm: build amplitudes on a coarse boundary, glue to a finer one, evolve back with the dynamical embedding map, and iterate to a fixed point; the fixed point amplitude is then the amplitude of the improved discretization, and the whole family, once consistent, replaces the Wilsonian trajectory $A_a$ of a theory with a background scale. The amplitude map is read twice over — as the dynamics, and as a physical vacuum generated by refining from the empty discretization, i.e. a no-boundary state.
Load-bearing premise
The construction rests on one premise the author states outright: that a cylindrically consistent family of amplitude maps actually exists as a fixed point of the coarse-graining flow, with diffeomorphism symmetry restored there.
Editorial extensions
If this is right
- If a fixed point of the flow exists, a single cylindrically consistent family of amplitudes defines the theory at all scales at once, replacing the trajectory $A_a$ of a Wilsonian flow with a family indexed by boundary coarseness $alpha$.
- Diffeomorphism symmetry, realized discretely as invariance under vertex displacements, would imply triangulation independence, so the choice of bulk triangulation loses physical significance and problems of triangulation dependence dissolve.
- Discretization ambiguities become irrelevant couplings with respect to the auxiliary coarse-graining flow, which sharply narrows the space of admissible spin foam models.
- Genuine first class Hamiltonian and diffeomorphism constraints exist only in the refinement limit; on a single discrete structure one gets at best pseudo-constraints that depend weakly on neighbouring time steps.
- The amplitude map obtained by refining the no-boundary state supplies a notion of physical vacuum for quantum gravity, and with it a natural organizing principle for excitations.
Reading between the lines
- The proposal sets a concrete numerical target: the two-dimensional spin net models already coarse grained with finite groups and $SU(2)_k$ are the nearest place to check whether a non-trivial fixed point with restored vertex-displacement symmetry is reachable at all, before any four-dimensional spin foam is attempted.
- If such a fixed point exists and is unique, cylindrical consistency combined with the semiclassical requirement $A_B(X) \sim \cos(S_H(X))$ at mesoscopic scales acts as a selection principle strong enough to eliminate most candidate spin foam models without any experimental input.
- Identifying the amplitude map with a no-boundary vacuum suggests the cosmological sector is where a low truncation order could already be predictive, since coarse boundary data is precisely what cosmology supplies.
- Because the paper replaces a background scale with boundary coarseness while warning against equating the two, any comparison with asymptotic safety should be read as an analogy until a geometric notion of scale is reconstructed from the boundary data.
Original abstract
The construction of a continuum limit for the dynamics of loop quantum gravity is unavoidable to complete the theory. We explain that such a construction is equivalent to obtaining the continuum physical Hilbert space, which encodes the solutions of the theory. We present iterative coarse graining methods to construct physical states in a truncation scheme and explain in which sense this scheme represents a renormalization flow. We comment on the role of diffeomorphism symmetry as an indicator for the continuum limit.
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.
The abstract, sentence by sentence
The construction of a continuum limit for the dynamics of loop quantum gravity is unavoidable to complete the theory. We explain that such a construction is equivalent to obtaining the continuum physical Hilbert space, which encodes the solutions of the theory. We present iterative coarse graining methods to construct physical states in a truncation scheme and explain in which sense this scheme represents a renormalization flow. We comment on the role of diffeomorphism symmetry as an indicator for the continuum limit.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | The construction of a continuum limit for the dynamics of loop quantum gravity is unavoidable to complete the theory. | background 0.62 | 0.09 | 0.36 | asserted 1.00 |
| s1 | We explain that such a construction is equivalent to obtaining the continuum physical Hilbert space, which encodes the solutions of the theory. | interpretation 0.64 | 0.47 | 0.23 | asserted 1.00 |
| s2 | We present iterative coarse graining methods to construct physical states in a truncation scheme and explain in which sense this scheme represents a renormalization flow. | method 0.64 | 0.92 | 0.19 | asserted 1.00 |
| s3 | We comment on the role of diffeomorphism symmetry as an indicator for the continuum limit. | interpretation 0.70 | 0.50 | 0.06 | hedged possibility 0.62 |
What was examined, and what was not
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- relations to other papers
- P abstract
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Figures
Figures reproduced from arXiv:1409.1450 by the authors.
References
How the literature uses it
83 arXiv papers cite this in 134 sentences
Papers citing it · 96 sentences without a confident label
Uses its method or tool 3 papers
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Thus the results presented in this article can be checked for consistency. * The simplicity of quantum cuboids, both in their geometric interpretation and algebraic complexity, allows for a first implementation of the RG methods outlined in [ref] .
Hypercuboidal renormalization in spin foam quantum gravity · Cuboidal spin foams · 0.89
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Luckily the coarse–graining schemes developed in [ref] , which aim at an understanding of the continuum limit, provide such an approximation scheme.
Towards a phase diagram for spin foams · Introduction · 0.86
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This ambiguity can only be resolved by having a better control over the microscopic dynamics and the coarse-graining properties of the quantum physical states, for instance by following an approach similar to the one advocated in [ref] for the construction of physical states through iterative coarse graining methods.
Asymptotically de Sitter Universe inside a Schwarzschild black hole · Emergent cosmological constant · 0.89
Cites it as support 2 papers
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Indeed, partition functions for finite regions are central in formulating and understanding the theory [ref] and do encode the complete dynamics of the theory [ref] .
3D holography: from discretum to continuum · Introduction · 0.97
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But there is another argument to include such Euclidean configuration even within Regge calculus: This argument relies on the demand that the amplitudes associated to a simplex should be ideally invariant under refinement [ref] .
Effective Spin Foam Models for Lorentzian Quantum Gravity · The path integral · 0.97
Cites it as background 30 papers
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Attempts to obtain the continuum and semi-classical limit have already been studied [ref] , with the latest result is reported elsewhere [ref] .
Statistical discrete geometry · Introduction · 0.96
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There have been many works which introduce and study such methods, both in the canonical [ref] and covariant settings [ref] .
Coarse-grained models for loop quantum gravity and their renormalization · Introduction · 0.94
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Understanding the role the maps $Φ_b'b$ play, prominently by Dittrich and others [ref] , has been one of the corner stones of advances in this area.
Numerical evidence for a phase transition in 4d spin foam quantum gravity · Coarse graining hypercuboids · 0.93
What it cites · 71 sentences without a confident label
Builds on 1 paper
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We will in particular provide an extension of aspects developed in the work [ref] from the AL embedding maps to dynamical embedding maps.
On background-independent renormalization of spin foam models · Renormalization flow and scale in background independent theories · 1.00
Uses the method or tool of 2 papers
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The group symmetry protecting variant of the algorithm developed in [ref] allows to keep track of the behaviour of intertwiner degrees of freedom, which signify the status of the simplicity constraints – the ingredient of spin foams that distinguishes them from standard lattice gauge theories.
Coarse graining methods for spin net and spin foam models · (Decorated) Tensor network renormalization for spin nets and spin foams · 0.83
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The group symmetry protecting variant of the algorithm developed in [ref] allows to keep track of the behaviour of intertwiner degrees of freedom, which signify the status of the simplicity constraints – the ingredient of spin foams that distinguishes them from standard lattice gauge theories.
Quantum group spin nets: refinement limit and relation to spin foams · (Decorated) Tensor network renormalization for spin nets and spin foams · 0.83
Compares with 3 papers
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Here we want to discuss the relations and differences of the framework developed in section [ref], where the construction of cylindrical consistent amplitudes is central, to the understanding of renormalization flow in systems with a notion of background (scale) [ref] .
Renormalization of Discrete Models without Background · Renormalization flow and scale in background independent theories · 0.93
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Here we want to discuss the relations and differences of the framework developed in section [ref], where the construction of cylindrical consistent amplitudes is central, to the understanding of renormalization flow in systems with a notion of background (scale) [ref] .
On background-independent renormalization of spin foam models · Renormalization flow and scale in background independent theories · 0.93
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This philosophy is similar to the asymptotic safety scenario [ref] where one hopes to extrapolate to the UV starting from the IR.
Quantum Einstein Gravity · Renormalization flow and scale in background independent theories · 0.89
Cites as support 16 papers
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Thus one reason to take the refinement limit is actually to restore the diffeomorphism symmetry [ref] , as is also used in the perfect action program [ref] for lattice QCD with regard to Lorentz symmetry.
Perfect lattice action for asymptotically free theories · Diffeomorphism symmetry in the discrete, constraints and divergencies · 0.98
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Thus one reason to take the refinement limit is actually to restore the diffeomorphism symmetry [ref] , as is also used in the perfect action program [ref] for lattice QCD with regard to Lorentz symmetry.
Improved and Perfect Actions in Discrete Gravity · Diffeomorphism symmetry in the discrete, constraints and divergencies · 0.98
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Indeed in this case the truncation can be even made exact [ref] .
Time evolution as refining, coarse graining and entangling · (Decorated) Tensor network renormalization for spin nets and spin foams · 0.97
Cites as background 5 papers
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Keeping the Ashtekar–Lewandowski (AL) representation the construction of coherent states has been explored [ref] , also coarse graining in the kinematical Hilbert space have been considered [ref] .
Curvature function and coarse graining · Kinematical understanding of the continuum limit · 0.91
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For instance in Regge calculus [ref] the variables are given by the lengths of the edges in a triangulation.[ Alternatively one can use areas and angles [ref] , which is nearer to the variables used in spin foams.] These geometric data of the discrete elements allow to determine the relative position of the vertices with respect to each other.
Area-angle variables for general relativity · Diffeomorphism symmetry in the discrete, constraints and divergencies · 0.83
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Another alternative representation, that is based on a (space–time) diffeomorphism invariant vacuum has been recently proposed [ref] and can be understood as dualization of the AL representation.
A new vacuum for Loop Quantum Gravity · Kinematical understanding of the continuum limit · 0.81
Discussion
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