Representation Theory of Analytic Holonomy C* Algebras

Abhay Ashtekar; Jerzy lewandowski

REVIEW

Representation Theory of Analytic Holonomy C* Algebras

Abhay Ashtekar, Jerzy Lewandowski

General Relativity and Quantum Cosmology · Submitted 7 Nov 1993 · 16 references

Pith's verdict: The paper that closes out the Ashtekar–Isham program: a clean algebraic characterization of the spectrum and the first faithful, diffeomorphism-invariant measure on it.

T0 · AI review

How this review was made: An AI referee (selfhosted/deepseek-v4.1-flash) read the paper on November 11, 2026 and answered a fixed list of questions about it. T0 is the starting level every paper gets. The levels describe how deeply Pith has checked the paper, never the authors.

T0 AI review An AI read the paper and answered a fixed list of questions.
T1 Audit Automatic checks that the references and files the paper relies on exist and match.
T2 Compute The paper's code or calculations re-run with the same results.
T3 Formal statement The claims written out in a formal math language a computer can read.
T4 Proof Those claims proved, and the proof checked by computer.

How Pith checks papers →

· 1 major objection · 12 minor · Ask about this paper · The compiled record · Read the full review →

Challenge this review Comment (0) Read on arXiv ↗

Cite ▾

arXiv IDgr-qc/9311010 (version 2)

Pith ID7DDURZY3, Pith's permanent record of this paper and its review

CitationAbhay Ashtekar, Jerzy lewandowski. "Representation Theory of Analytic Holonomy C* Algebras." arXiv:gr-qc/9311010 (1993). https://pith.science/paper/7DDURZY3

Cite this reviewBibTeX for Pith's review of this paper

Claim paper

Why Pith says this

Highlight any text to comment on it.

Editor's note

read the letter

Short version: this finishes the Ashtekar–Isham program, and the two results that carry the weight are Theorem 3.5 (the Gel'fand spectrum is exactly Hom(HG, SU(2)) modulo conjugation) and Theorem 4.4 (a faithful, Diff(Σ)-invariant regular measure built from the Haar measure on SU(2)). Both hold up on reading. The measure is the one everybody now calls the Ashtekar–Lewandowski measure, and this is where it comes from. Worth your time if you care about rigorous quantization of connections, and worth citing if you work in the area at all. One correction to the record: the classifier says the authors state no limitations. They do — Appendix A is explicit that the whole construction leans on piecewise analyticity and that the extension to piecewise smooth loops is open for non-Abelian groups. They prove a U(1) analogue and say plainly the argument does not carry over. That is honest and it is the right place to put the caveat. The engine of the paper is the loop decomposition in Section 3.1: any finite set of piecewise analytic loops splits into finitely many strongly independent ones, and Lemma 3.1 then realizes arbitrary SU(2) holonomies on them exactly. That exactness is what makes inequality (3.8) work and hence what makes the converse of the spectrum theorem go through. The stress-test note is right that an epsilon-approximation would not suffice for (3.8), but it does not land here: for analytic loops the realization is exact, and the epsilon version appears only in the Abelian appendix where a different (dense-line) argument is used. So the concern is real but narrow and does not touch the stated results. Genuine soft spots, in order of weight. First, the remark after Lemma 4.1 that the only set of independent hoops generating a given S* is the one you started with is asserted, not proved, and it is what makes the projection π(S*) canonical and the measure basis-independent. It is probably true, but it deserves a line of argument. Second, the strip/momentum operators are promised for a separate paper; the claim that they are densely defined and symmetric on L²(A/G, dµ) is not supported here. Third, Appendix B is a sketch — bundle independence and the U(N)/SU(N) extension are argued in outline with details explicitly omitted. Fourth, the 'generalized knot invariant' is degenerate: Γ vanishes on smoothly embedded loops, as the authors themselves note, so do not read it as a knot invariant in the usual sense. None of these is load-bearing for the main theorems. This is for people doing rigorous canonical quantization of gauge theories and for mathematicians interested in C*-algebras of holonomies. It deserves a serious referee, mainly to check the decomposition argument and the naturality remark.

Referee report

Major objections (1)

[§3.2, Eq. (3.8)] The converse of the spectrum theorem (Lemma 3.4) rests on the sup-norm bound (3.8), which in turn rests on the exact realization of Lemma 3.1. The proof of Lemma 3.1 is correct and uses the strong independence of the β_i: each β_i contains an open interval traced exactly once and not shared, so the connections A(i) can be chosen with essentially disjoint supports and summed to give A_o with H(β_i, A_o) = g_i exactly. The concern that an approximate realization (as in Appendix A's Lemma A.3) would not suffice is therefore not a gap in the main text; the main text gives exact realization. However, the paper should make explicit that exactness is what is being used, since the Appendix A discussion of approximation could mislead a reader into thinking the main argument is approximate.

Minor objections (12)

[§3.3] The sentence 'This is an one illustration of the non-triviality of the completion procedure' contains a typo: 'an one' should be 'one'.
[§4.2] The sentence 'It thus defined a generalized knot invariant' should read 'It thus defines a generalized knot invariant'.
[§4.2] The sentence 'It's action is non-trivial only when the loop is re-traced' should read 'Its action is non-trivial only when the loop is re-traced'.
[Appendix A.1] The sentence 'As in common in the physics literature' should read 'As is common in the physics literature', and 'inspite of topological non-trivialities' should read 'in spite of topological non-trivialities'.
[Appendix B.2] The sentence 'the A-I result that there is a natural embedding of A/G into the spectrum A/G of goes through as before' has a missing word after 'of'.
[§5] In the displayed action of T_γ̃ on connection states, the right side is written as T_γ̃(Ā) · ψ(Ā); it should be Ψ(Ā), since the argument is a connection state, not a hoop state.
[§4.1, Remark 1] The claim that 'the only set of independent hoops which generates (precisely) S* is the set {β_1,...,β_n} we began with' is asserted without proof. It may be true for strongly independent sets, but the paper should either supply an argument or soften the claim, since the canonicity of the projection π(S*) is used in Remark 2.
[§3.3] The claim that 'there is no generalized connection with support at x which depends only on the zeroth order behavior of the curve' is asserted without proof. A short argument or a reference would help.
[Appendix A.1] The assertion that the SU(2) hoop group is torsion-free, stated after Lemma A.2, is not proved. Since the U(1) proof uses Abelianness, the non-Abelian case deserves at least a sketch or a reference.
[§4.2] The claims that Γ(α) vanishes on all smoothly embedded loops and that the integral of t_α vanishes unless every loop β_i is repeated an even number of times are stated without proof. These are interesting and checkable; a short derivation would strengthen the section.
[§5] The statement that the loop transform is made 'rigorous' is stronger than what is delivered: the transform is defined by Eq. (5.1), but the image and inner product of hoop states are explicitly left open in the following paragraph. The wording should be adjusted to match the acknowledged open problems.
[Throughout] The notation 'Ad' and 'ad' is used inconsistently for the adjoint quotient (e.g., '[SU(2)]^n/Ad' versus '[SU(2)]^n/ad'). A single convention would improve readability.

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

The paper

The paper claims

The paper argues that gauge-equivalent connections and closed loops are exact duals: every point of the completed connection space is a rule assigning group elements to loops, and every such rule is a point.

The reading

The paper sets out to make integration over the space $A/G$ of gauge-equivalent connections — the configuration space of gauge theories including general relativity — into a workable mathematical object. Its strategy is algebraic: build the Abelian $C^*-$algebra generated by Wilson loops (traces of holonomies around closed loops), then read the space of quantum states off its Gel'fand spectrum. The central claim is that this spectrum has a complete and purely algebraic description: each of its points is a homomorphism from the group of hoops (holonomically equivalent loops) into $SU(2)$, and conversely every such homomorphism gives a point, with the correspondence one-to-one up to conjugation. On top of that the authors construct an explicit measure on the spectrum from the Haar measure on $SU(2)$, argue it is faithful (strictly positive on nonzero elements) and invariant under diffeomorphisms of the underlying manifold, and thereby make the Rovelli–Smolin loop transform into a genuine map between Hilbert spaces. If this holds, quantization of diffeomorphism-invariant connection theories acquires a controlled home: states are square-integrable functions on a compact Hausdorff space, Wilson loops act by multiplication, and diffeomorphism-invariant measures correspond to generalized knot and link invariants.

What carries the argument

The load-bearing device is the loop decomposition of Section 3.1: because the loops are piecewise analytic, any two overlap in finitely many intervals and intersect in finitely many points, so any finite set of hoops can be replaced by a finite set of 'strongly independent' loops — each containing an open segment traced exactly once and shared with no other loop — that generates a hoop subgroup containing the original hoops. Strong independence yields Lemma 3.1, the existence of a smooth connection realizing an arbitrary prescribed $n$-tuple of holonomies around those loops. That lemma does double duty: it converts the apparently weak pointwise bound on a candidate spectrum element into the full sup-norm bound needed for continuity, and it identifies the quotient $[A/G]/sim$ with the compact space $SU(2)^n/Ad$, which is what makes cylindrical functions and cylindrical measures available in a non-linear setting where no linear duality exists.

Extended reading notes

Core claim

The paper's central result (Theorem 3.5) is a characterization of the Gel'fand spectrum $A/G$ of the Ashtekar–Isham holonomy $C^*-$algebra: every point $A$ of the spectrum defines a homomorphism $H$ from the hoop group $HG$ to $SU(2)$ with $A(hoop) = (1/2) Tr H(hoop)$, and conversely every homomorphism from $HG$ to $SU(2)$ defines a point of the spectrum, the correspondence being one-to-one modulo conjugation by a fixed element of $SU(2)$. The authors emphasize that the homomorphism need not be continuous — no topology on the hoop group enters — so the characterization is purely algebraic and usable in practice. A companion result (Theorem 4.4) equips the spectrum with an 'induced Haar measure': for each finitely generated subgroup of the hoop group generated by $n$ strongly independent hoops, one projects $A/G$ onto $SU(2)^n/Ad$ and integrates there with the normalized Haar measure; the authors show the resulting integrals are compatible across different choices of subgroup, so a genuine regular measure exists, and it is strictly positive and diffeomorphism invariant because no extra structure on the manifold was introduced. The generating functional of the resulting representation vanishes on smoothly embedded loops and is nonzero only on retraced or otherwise generalized loops, so it is a generalized knot invariant rather than a standard one.

Load-bearing premise

The whole argument rests on the loops being piecewise analytic, since only then does any finite set of loops decompose into finitely many strongly independent loops, and the authors themselves note in Appendix A that for non-Abelian groups it is an open question whether the results survive if loops are only piecewise smooth.

Editorial extensions

If this is right

Every cyclic representation of the holonomy $C^*-$algebra is of the form $L^2(A/G, d\mu)$ for a regular measure, with Wilson-loop operators acting by multiplication, and the induced Haar measure gives a faithful, diffeomorphism-invariant example.
The loop transform becomes a rigorous map from $L^2(A/G, d\mu)$ to functions on the hoop group, with the Wilson-loop action on hoop states taking the simple form $T_{\tilde\gamma}\,\psi(\tilde\alpha) = \tfrac12(\psi(\tilde\alpha\cdot\tilde\gamma) + \psi(\tilde\alpha\cdot\tilde\gamma^{-1}))$.
Diffeomorphism-invariant measures on the completed connection space correspond one-to-one with generalized knot and link invariants satisfying the positivity conditions of the GNS construction.
The completion $A/G$ contains generalized connections supported on a single point, on a 2-dimensional submanifold, or depending only on the tangent directions of a loop at an intersection point, so the completion is strictly larger than the classical configuration space.
The framework extends to $U(N)$ and $SU(N)$ gauge groups on $n$-manifolds with piecewise analytic loops, and the resulting algebra turns out to be independent of which principal bundle one starts from.

Reading between the lines

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

Editorial: the algebraic characterization makes the spectrum computable in a practical sense — checking whether a candidate generalized connection is a point of $A/G$ reduces to checking it is a group homomorphism, which is a finite algebraic test on any finite set of hoops rather than an analytic continuity argument.
Editorial: because the measure is built from Haar measure alone, the construction suggests a family rather than a single object; replacing the Haar measure by other adjoint-invariant measures on $SU(2)^n$ should produce further diffeomorphism-invariant measures, subject to the same compatibility conditions.
Editorial: the vanishing of the generating functional on smoothly embedded loops means the induced Haar measure does not see ordinary knot classes; any knot information in this representation must come from generalized loops with kinks and retracings, which is a sharp constraint on how knot invariants can arise as measures here.
Editorial: the paper's own remark that a Plancherel-type theorem relating integrals over $A/G$ and over the hoop group is still missing marks the natural next target — without it the loop representation has no intrinsic inner product.

Original abstract

Integral calculus on the space of gauge equivalent connections is developed. Loops, knots, links and graphs feature prominently in this description. The framework is well--suited for quantization of diffeomorphism invariant theories of connections. The general setting is provided by the abelian C* algebra of functions on the quotient space of connections generated by Wilson loops (i.e., by the traces of holonomies of connections around closed loops). The representation theory of this algebra leads to an interesting and powerful ``duality'' between gauge--equivalence classes of connections and certain equivalence classes of closed loops. In particular, regular measures on (a suitable completion of) connections/gauges are in 1--1 correspondence with certain functions of loops and diffeomorphism invariant measures correspond to (generalized) knot and link invariants. By carrying out a non--linear extension of the theory of cylindrical measures on topological vector spaces, a faithful, diffeomorphism invariant measure is introduced. This measure can be used to define the Hilbert space of quantum states in theories of connections. The Wilson--loop functionals then serve as the configuration operators in the quantum 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, 6,828 input tokens, $0.0003 reported.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.07 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.04 · compares with prior results 0.06 · mentions code or data availability 0.02.

The abstract, sentence by sentence

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

Integral calculus on the space of gauge equivalent connections is developed. Loops, knots, links and graphs feature prominently in this description. The framework is well--suited for quantization of diffeomorphism invariant theories of connections. The general setting is provided by the abelian C* algebra of functions on the quotient space of connections generated by Wilson loops (i.e., by the traces of holonomies of connections around closed loops). The representation theory of this algebra leads to an interesting and powerful ``duality'' between gauge--equivalence classes of connections and certain equivalence classes of closed loops. In particular, regular measures on (a suitable completion of) connections/gauges are in 1--1 correspondence with certain functions of loops and diffeomorphism invariant measures correspond to (generalized) knot and link invariants. By carrying out a non--linear extension of the theory of cylindrical measures on topological vector spaces, a faithful, diffeomorphism invariant measure is introduced. This measure can be used to define the Hilbert space of quantum states in theories of connections. The Wilson--loop functionals then serve as the configuration operators in the quantum 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	Integral calculus on the space of gauge equivalent connections is developed.	contribution 0.91	0.82	0.06	asserted 1.00
s1	Loops, knots, links and graphs feature prominently in this description.	background 0.64	0.32	0.03	asserted 1.00
s2	The framework is well--suited for quantization of diffeomorphism invariant theories of connections.	interpretation 0.94	0.41	0.10	asserted 0.94
s3	The general setting is provided by the abelian C* algebra of functions on the quotient space of connections generated by Wilson loops (i.e., by the traces of holonomies of connections around closed loops).	method 0.43	0.43	0.11	asserted 1.00
s4	The representation theory of this algebra leads to an interesting and powerful ``duality'' between gauge--equivalence classes of connections and certain equivalence classes of closed loops.	result 0.78	0.78	0.52	asserted 1.00
s5	In particular, regular measures on (a suitable completion of) connections/gauges are in 1--1 correspondence with certain functions of loops and diffeomorphism invariant measures correspond to (generalized) knot and link invariants.	result 0.89	0.74	0.45	asserted 1.00
s6	By carrying out a non--linear extension of the theory of cylindrical measures on topological vector spaces, a faithful, diffeomorphism invariant measure is introduced.	method 0.52	0.90	0.13	asserted 1.00
s7	This measure can be used to define the Hilbert space of quantum states in theories of connections.	interpretation 0.65	0.76	0.11	asserted 0.91
s8	The Wilson--loop functionals then serve as the configuration operators in the quantum theory.	interpretation 0.54	0.69	0.13	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.79 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v2

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]

A. Ashtekar, Non–perturbative canonical quantum gravity (Notes pre- pared in collaboration with R.S. Tate), World Scientiﬁc, Singapore, (1991)

View on Pith 1991
[2]

A. Ashtekar and C. Isham, Class. Quantum Grav. 9, 1433–85 (1992)

View on Pith 1992
[3]

R. Gambini and A. Trias, Nucl. Phys. B278, 436–448 (1986)

View on Pith 1986
[4]

C. Rovelli and L. Smolin, Nucl. Phys. B331, 80–152 (1992)

View on Pith 1992
[5]

A. Rendall, Class. Quantum Grav. 10, (1993)

View on Pith 1993
[6]

J. Baez, In: Proceedings of the conference on quantum topology (to ap- pear)

View on Pith
[7]

J. Lewandowski, Class. Quantum Grav. 10, 879–904 (1993)

View on Pith 1993
[8]

P.K. Mitter and C. Viallet, Commun. Math. Phys. 79, 43–58 (1981)

View on Pith 1981
[9]

C. Di Bortolo, R. Gambini and J. Griego, preprint IFFI/9301; gr – qc/9303010

View on Pith
[10]

R. Giles, Phys. Rev. D24, 2160–8 (1981)

View on Pith 1981
[11]

J.W. Barrett, Int. J. Theor. Phys. 30, 1171–1215 (1991)

View on Pith 1991
[12]

A.N. Kolmogorov, Foundations of the Theory of Probability Chelsea, New York (1956) Holonomy C∗ Algebras 41

View on Pith 1956
[13]

Y. Choquet–Bruhat, C. DeWitt–Morette and M. Dillard–Bleick, Anal- ysis, Manifolds and Physics , North–Holland, Amsterdam (1982), pp 573– 587

View on Pith 1982
[14]

M. Creutz, Quarks, Gluons and Lattices , Cambridge University Press, Cambridge, 1988, Chapter 8

View on Pith 1988
[15]

A. Trautman, Chech. J. Phys. B 29 , 107–111 (1979)

View on Pith 1979
[16]

C. Procesi, Adv. Math.19, 306-381 (1976)

View on Pith 1976

Forward citations

Cited by 22 Pith papers

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

Constraining quantum-gravity predictions for evolving dark energy
astro-ph.CO 2026-09 unverdicted novelty 7.0 of 10

Group field theory quantum gravity predicts dark energy equations of state that are constrained by BAO and supernova data, with the logarithmic branch resembling a cosmological constant and oscillatory solutions allow...
Quantum geometry from higher gauge theory
gr-qc 2019-08 conditional novelty 7.0 of 10

The BFCG-Yetter model and the KBF state sum produce the same single-4-simplex amplitude when evaluated on boundary states called G-networks.
The loop gravity string
gr-qc 2016-11 conditional novelty 7.0 of 10

Boundary degrees of freedom of canonical gravity localize at punctures and carry a Virasoro algebra with central charge 3 per puncture, so the boundary is a conformal field theory.
Quantum gravity kinematics from extended TQFTs
hep-th 2016-04 accept novelty 7.0 of 10

A root-of-unity quantum-deformed representation of the 2+1 dimensional holonomy-flux algebra is built on a Turaev-Viro vacuum, with holonomies and fluxes realized as ribbon operators acting on Drinfeld-centre excitations.
New Variables for Classical and Quantum Gravity in all Dimensions I. Hamiltonian Analysis
gr-qc 2011-05 conditional novelty 7.0 of 10

A new canonical connection formulation of general relativity in D+1 dimensions (D>1) with structure group SO(D+1) or SO(1,D), whose symplectic reduction by Gauss and simplicity constraints recovers the ADM phase space.

Show 17 more citing papers

Uniqueness of diffeomorphism invariant states on holonomy-flux algebras
gr-qc 2005-04 accept novelty 7.0 of 10

Under semianalytic local diffeomorphism invariance, the holonomy-flux *-algebra admits exactly one Yang-Mills gauge invariant and diffeomorphism invariant state, the Ashtekar-Lewandowski state.
Coarse-grained models for loop quantum gravity and their renormalization
hep-th 2026-08 unverdicted novelty 6.0 of 10

The paper constructs coarse-grained Hilbert spaces and effective Hamiltonians for loop quantum gravity and writes down non-perturbative renormalization flow equations for them.
The one-loop effective action from the coherent state path integral of loop quantum gravity
gr-qc 2025-02 conditional novelty 6.0 of 10

The one-loop effective action from the LQG coherent state path integral is computed and found to be UV-finite, with the dynamical area j0 > 0 at the quantum level.
Fock representation of gravitational boundary modes and the discreteness of the area spectrum
gr-qc 2017-06 conditional novelty 6.0 of 10

Quantizing boundary spinors on a null surface in a Fock representation turns cross-sectional area into the difference of two number operators, giving a discrete spectrum in multiples of 8πβG.
On entanglement entropy in non-Abelian lattice gauge theory and 3D quantum gravity
hep-th 2016-09 conditional novelty 6.0 of 10

A fusion-basis definition of entanglement entropy for lattice gauge theories yields the non-Abelian magnetic-centre choice and vanishes on the flat BF vacuum.
Emergent Friedmann dynamics with a quantum bounce from quantum gravity condensates
gr-qc 2016-02 conditional novelty 6.0 of 10

Effective Friedmann equations derived from group field theory condensates are non-singular: a conserved charge keeps the volume from reaching zero, producing a bounce in the Planck regime.
A new vacuum for Loop Quantum Gravity
gr-qc 2014-01 conditional novelty 6.0 of 10

A flat-connection (BF) vacuum dual to the Ashtekar-Lewandowski vacuum is constructed for 2+1 dimensional loop quantum gravity, together with a modified holonomy-flux algebra that is claimed to be consistent under Pach...
New Variables for Classical and Quantum Gravity in all Dimensions III. Quantum Theory
gr-qc 2011-05 conditional novelty 6.0 of 10

The constraints and geometric operators of a new SO(D+1) connection formulation of General Relativity can be quantised with loop quantum gravity methods in all dimensions D+1 >= 3.
Non-commutative flux representation for loop quantum gravity
hep-th 2010-04 conditional novelty 6.0 of 10

A non-commutative Fourier transform is shown to define a well-defined flux representation of the full loop quantum gravity state space, with fluxes acting by star-multiplication and holonomies by translation.
The length operator in Loop Quantum Gravity
gr-qc 2008-06 conditional novelty 6.0 of 10

A new length operator in Loop Quantum Gravity, built from a two-hand flux operator and a regularized inverse volume, has a discrete spectrum and reproduces the classical edge length of a tetrahedron in the semiclassic...
Algebraic Quantum Gravity (AQG) IV. Reduced Phase Space Quantisation of Loop Quantum Gravity
gr-qc 2007-11 conditional novelty 6.0 of 10

By using pressure-free dust as a relational clock, the authors turn the kinematical Hilbert space of loop quantum gravity into a physical Hilbert space and construct a candidate physical Hamiltonian operator.
Quantum-reduced loop gravity: New perspectives on the kinematics and dynamics
gr-qc 2024-12 conditional novelty 5.0 of 10

The paper reformulates quantum-reduced loop gravity with a master constraint operator and finds that its single-node Hamiltonian matches the Bianchi I loop quantum cosmology Hamiltonian.
New Variables for Classical and Quantum Gravity in all Dimensions II. Lagrangian Analysis
gr-qc 2011-05 conditional novelty 5.0 of 10

Applying gauge unfixing to the higher-dimensional Palatini action reproduces the first-class connection formulation of general relativity previously obtained from an extension of the ADM phase space.
Quantum Gravity, Hydrodynamics and Emergent Cosmology: A Collection of Perspectives
gr-qc 2024-11 unverdicted novelty 4.0 of 10

The paper argues that an effective 'hydrodynamics on superspace' framework, realized in TGFT condensate cosmology, can unify quantum gravity and cosmology.
An elementary introduction to loop quantum gravity
gr-qc 2016-07 accept novelty 2.0 of 10

Five lecture notes introduce loop quantum gravity from motivation through kinematics to dynamics, and give a candid inventory of the field's unresolved problems.
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.
Introduction to Loop Quantum Cosmology
gr-qc 2011-09 accept novelty 2.0 of 10

A structured survey of loop quantum cosmology reporting that minisuperspace models replace the big bang with a bounce, while midisuperspace and effective-dynamics work remains incomplete.

How later papers used it

56 later papers cite it

Evidence: substantial. 56 of the citing papers share no author with it, and 7 of those carry a confident label from Pith's citation classifier.

Pith read 21 of the 97 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

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

Uses its method or tool 1 of 6 papers with this label

Adopted a labeling introduced in this paper.

Constrained projective quantum states for the degenerate Plebanski gravity (2017) · 0.97 · paraphrased: the licence does not allow quoting

How the literature uses it

56 arXiv papers cite this in 97 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 · 71 sentences without a confident label

Uses its method or tool 6 papers

Our formulation provides access to the quantisation techniques developed in [ref] and allows for a rigorous construction of a quantum theory.

New Variables for Classical and Quantum Gravity in all Dimensions IV. Matter Coupling · Introduction · 0.87
These triples are selected as in expression [ref], and each cylindrical function, defined in the same manner as in CLQG [ref] , depends on the connection only through a linear holonomy.

Relativistic classical theory II. Holonomy-flux representation of gravitational degrees of freedom · Summary · 0.90
A labeling we are going to use was introduced in [ref] .

Constrained projective quantum states for the degenerate Plebanski gravity · Finite sets of configurational d.o.f. · 0.97

Compares with it 1 paper

In particular, the obtained mean-field vacuum open ups the possibility to compare it with the Ashtekar-Lewandowski vacuum [ref] and the Geiller-Dittrich vacuum [ref] studied in LQG and spin foam gravity.

Quantum Gravity, Hydrodynamics and Emergent Cosmology: A Collection of Perspectives · Three Ways to Mean-Field Group Field Theory by Johannes Thürigen · 0.93

Cites it as background 15 papers

Since several categories (for example: analytic [ref] , semianalytic [ref] , stratified [ref] , smooth [ref] ) have been used in LQG, we have chosen to base the analysis on a number of assumptions about the ability of diffeomorphisms to realize certain maps between graphs in a local fashion.

A Fock space structure for the diffeomorphism invariant Hilbert space of loop quantum gravity and its applications · Introduction · 0.88
The Hilbert space used in LQG was obtained in [ref] by breaking the $SL(2,)$ symmetry to the $SU(2)$ one (which amounts to breaking the Lorentz symmetry of GR to that of three-dimensional rotations) and in [ref] Lanéry and Thiemann constructed their projective quantum states also for LQG with the $SU(2)$ symmetry.

A modification of the projective construction of quantum states for field theories · Partial solutions · 0.81
We call the group of based loops associated to $$ for the hoop group (holonomy loops, see [ref] ), denoted $_$.

On Spectral Triples in Quantum Gravity I · Simplicial complexes and their loop algebras · 0.81

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]

A. Ashtekar, Non–perturbative canonical quantum gravity (Notes pre- pared in collaboration with R.S. Tate), World Scientiﬁc, Singapore, (1991)

View on Pith 1991

[2] [2]

A. Ashtekar and C. Isham, Class. Quantum Grav. 9, 1433–85 (1992)

View on Pith 1992

[3] [3]

R. Gambini and A. Trias, Nucl. Phys. B278, 436–448 (1986)

View on Pith 1986

[4] [4]

C. Rovelli and L. Smolin, Nucl. Phys. B331, 80–152 (1992)

View on Pith 1992

[5] [5]

A. Rendall, Class. Quantum Grav. 10, (1993)

View on Pith 1993

[6] [6]

J. Baez, In: Proceedings of the conference on quantum topology (to ap- pear)

View on Pith

[7] [7]

J. Lewandowski, Class. Quantum Grav. 10, 879–904 (1993)

View on Pith 1993

[8] [8]

P.K. Mitter and C. Viallet, Commun. Math. Phys. 79, 43–58 (1981)

View on Pith 1981

[9] [9]

C. Di Bortolo, R. Gambini and J. Griego, preprint IFFI/9301; gr – qc/9303010

View on Pith

[10] [10]

R. Giles, Phys. Rev. D24, 2160–8 (1981)

View on Pith 1981

[11] [11]

J.W. Barrett, Int. J. Theor. Phys. 30, 1171–1215 (1991)

View on Pith 1991

[12] [12]

A.N. Kolmogorov, Foundations of the Theory of Probability Chelsea, New York (1956) Holonomy C∗ Algebras 41

View on Pith 1956

[13] [13]

Y. Choquet–Bruhat, C. DeWitt–Morette and M. Dillard–Bleick, Anal- ysis, Manifolds and Physics , North–Holland, Amsterdam (1982), pp 573– 587

View on Pith 1982

[14] [14]

M. Creutz, Quarks, Gluons and Lattices , Cambridge University Press, Cambridge, 1988, Chapter 8

View on Pith 1988

[15] [15]

A. Trautman, Chech. J. Phys. B 29 , 107–111 (1979)

View on Pith 1979

[16] [16]

C. Procesi, Adv. Math.19, 306-381 (1976)

View on Pith 1976