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Cited by 297Noncommutative and Quantum Gravity TheoriesAdvanced Operator Algebra ResearchRelativity and Gravitational Theory

Uniqueness of Diffeomorphism Invariant States on Holonomy–Flux Algebras

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Lewandowski, Jerzy, Okołów, Andrzej, Sahlmann, Hanno, Thiemann, Thomas·2006·Communications in Mathematical Physics
DOI 10.1007/s00220-006-0100-7
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Written from the paper’s full text.

TL;DR

The paper defines an abstract quantum holonomy–flux ∗-algebra for connection theories and proves that, under Yang–Mills gauge and spatial diffeomorphism invariance, there is exactly one invariant state on it. This state is the known Ashtekar–Lewandowski-type state used in loop quantum gravity, so the invariant GNS representation is unique.

Key findings

  • There exists exactly one Yang–Mills gauge invariant and diffeomorphism invariant state on the quantum holonomy–flux ∗-algebra \(\mathcal A\) (Theorem 4.2).
  • The known state \(\omega_0\), defined by setting \(\omega_0(a\cdot \hat Y)=0\) for flux generators and integrating cylindrical functions with the Haar measure, is invariant and is the unique such state.
  • In any GNS representation from an invariant state, every flux vector field vanishes as a vector: \([\hat X_{S,f}]=0\) for every face \(S\) and smearing vector field \(f\) (Lemma 4.3).
  • The result is formulated for arbitrary dimension \(D\ge 2\), arbitrary compact connected structure group \(G\), and arbitrary bundle \(P\), using a semianalytic category rather than analytic diffeomorphisms.

Methods

  • The authors define a classical Ashtekar–Corichi–Zapata holonomy–flux Lie algebra from cylindrical functions and flux derivations, then construct an abstract quantum ∗-algebra \(\mathcal A\) as a quotient of the enveloping algebra by relations encoding linearity, commutators, and the \(\mathrm{Cyl}\)-module structure.
  • They introduce semianalytic manifolds, edges, faces, and diffeomorphisms to obtain a symmetry group containing local diffeomorphisms while preserving finite intersections between edges and faces.
  • The uniqueness proof uses GNS representations of invariant states, a partition-of-unity decomposition of smearing vector fields, local semianalytic diffeomorphisms, and reduction to the known uniqueness of the invariant measure on the space of generalized connections.

Limitations

  • The paper states that whether the result holds for smaller symmetry groups, or even only for analytic diffeomorphisms, is an open question.
  • The proof relies on compact support of the smearing functions \(f\) in the flux variables; the authors show that dropping compact support can produce a different invariant state in the example \(\Sigma=\mathbb R^2\), \(G=U(1)\), though compact support makes \(\int_S df=0\) and removes the contradiction.
  • The treatment assumes a common dense domain for all flux operators and cylindrical functions; the authors note Fleischhack’s later result avoids this but needs an additional assumption on bundle automorphisms.

Model-written summary. Check it against the paper before relying on it.

Integrity signals

Registry

Crossref · 10.1007/s00220-006-0100-7

Cites

26 of 26 references verified0 citation errors · 0 not checkable

Statistics

Nothing to recompute.

Claims the paper makes

0 of 23 cited sources checked · 0 supported · 0 flagged

  • Findingown data
    § Abstract · p. 1

    The only state on the holonomy-flux algebra invariant under the group of diffeomorphisms described in the paper is the one used in loop quantum gravity.

    “We show that there is only one cyclic representation invariant under spatial diffeomorphisms.”
    • Also stated in: § 1 Introduction · p. 6 · boosted
      “Our analysis will show that this is not the case: the only state that is invariant under the group of diffeomorphisms described above is the one used in LQG.”
    • Also stated in: § 1 Introduction · p. 3 · boosted
      “We will show that in the case of diffeomorphism invariant theories 1 , upon restricting to diffeomorphism invariant representations, this issue will not arise: we find a unique cyclic representation”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The manifold Σ is equipped with a semianalytic structure.
    • The authors state this as an assumption: The structure group G of the principal fiber bundle is compact and connected.
    • The authors state this as an assumption: The principal fiber bundle is semianalytic.
    • The authors state this as an assumption: The differentiability class C^m of the manifold is fixed with m ≥ 1.
  • Findingreasoning
    § 4 The uniqueness theorem · p. 20

    There exists exactly one Yang-Mills gauge invariant and diffeomorphism invariant state on the quantum holonomy-flux ∗-algebra A.

    “Theorem 4.2. There exists exactly one Yang-Mills gauge invariant and diffeomorphism invariant state on the quantum holonomy-flux ∗ -algebra A .”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The smearing functions f used in the flux variables E_{S,f} have compact support on S.
    • The authors state this as an assumption: The treatment assumes a common dense domain for all flux operators and cylindrical functions.
  • Findingreasoning
    § A.1 semianalytic functions in R n · p. 31

    If φ: U→U' is a semianalytic bijection between open subsets of R^n, and for every x0∈U there is a semianalytic partition compatible with φ such that each restriction extends to an analytic injective function with analytic inverse, then φ^{-1} is semianalytic.

    “Then, φ -1 is semianalytic.”
    Quote in the paperNot a substantial claim
  • Findingreasoning
    § 2.7 Symmetries of A · p. 15

    The pullback Ad∗ ̃φ preserves the space of cylindrical functions.

    “The pullback Ad ∗ ˜ ϕ preserves the space of the cylindrical functions.”
    Quote in the paperNot a substantial claim
  • Findingnone given
    § A.1 semianalytic functions in R n · p. 33

    If P(U,h) is a semi-semianalytic partition of an open U⊂R^n, then every x∈U has a neighborhood that admits a semianalytic partition finer than P(Ũ,h).

    “Suppose P ( U , h ) is a semi-semianalytic partition of an open U ⊂ R n . Then, every x ∈ U has a neighborhood ˜ U which admits a semianalytic partition finer than P ( ˜ U , h ) .”
    Quote in the paperNot a substantial claim
  • May combine two claims. 17
  • Classified as probably not a substantial claim. 30

  • May combine two claims.

    Findingreasoning
    § 3 States, GNS. · p. 17

    If a functional ω on Cyl satisfies the equalities and inequality (55) for every Ψ, Ψ' and α, then ω is continuous with respect to the norm and determines a unique extension to a state on the C*-algebra Cyl.

    “Lemma 3.2. 8 Suppose that ω : Cyl → C satisfies for every Ψ , Ψ ′ ∈ Cyl and α ∈ C the following equalities and inequality,”
    • Also stated in: § 3 States, GNS. · p. 18 · plain
      “Therefore, ω is continuous with respect to the norm ‖ · ‖ and determines a unique extension to a state defined on the C ∗ -algebra Cyl .”
    Quote in the paperSubstantial claim

  • Classified as probably not a substantial claim. May combine two claims.

    Findingnone given
    § 2.3 The Ashtekar-Isham quantum configuration space A · p. 9

    Every cylindrical function extends naturally to the space of generalized connections, and the result is a unique function independent of the compatible graph and function used.

    “(the result defines a unique function on A , independent of choice of the compatible γ and ψ .)”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based). May combine two claims.

    Findingnone given
    § 2.4 Generalized vector fields tangent to A · p. 11

    The definition of the flux vector field via the flow is equivalent to the Poisson-bracket definition (3) and is its manifestly trivialization-independent version.

    “This definition is equivalent to (3) and it is its manifestly trivialization independent version.”
    Quote in the paperNot a substantial claim

  • Classified as probably not a substantial claim. May combine two claims.

    Findingreasoning
    § 2.2 The cylindrical functions · p. 8

    The cylindrical functions form a subalgebra of the algebra of complex-valued functions on the space of connections, and this subalgebra admits an involution and a norm.

    “It is easy to see, that all the cylindrical functions set up a subalgebra of the algebra of all the complex valued functions defined on A ; we denote it by Cyl. In a natural way it admits definition of an involution and a norm”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Findingreasoning
    § 2.6 The elements of A · p. 14

    Every element of A is a finite linear combination of elements of the form given in (35).

    “Every element of the algebra A is a finite linear combination of elements of the form”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S4 (based). May combine two claims.

    Findingnone given
    § 4 The uniqueness theorem · p. 22

    The product (·|·) is bilinear and symmetric.

    “- ( i ) it is bilinear and symmetric,”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingreasoning
    § 2.7 Symmetries of A · p. 14

    The bundle automorphism group induces an isomorphism of the ACZ classical Lie algebra A_class and a ∗-isomorphism of the quantum ∗-algebra A.

    “In this way, the bundle automorphism group induces an isomorphism of the ACZ classical Lie algebra A class , and finally a ∗ -isomorphism of the quantum ∗ -algebra A .”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Findingcitation
    § A.1 semianalytic functions in R n · p. 34

    For every semianalytic partition P(U,h) of an open U⊂R^n, every point x∈U has a neighborhood that admits an analytic partition finer than P(Ũ,h).

    “For every semianalytic partition P ( U , h ) of an open U ⊂ R n , every point x ∈ U has a neighborhood ˜ U which admits an analytic partition finer than P ( ˜ U , h ) .”
    Quote in the paperNot a substantial claim0 of 2 sources checked
    • cites [2] Łojasiewicz S. (1964) Triangulation of semi-analytic sets. Ann. Scuola. Norm. Sup. Pisa 18, 449–474

      Not checked in the free run.

    • cites [3] Bierstone E., Milman P.D. (1988) Semianalytic and Subanalytic sets. Publ. Maths. IHES 67, 5–42 source

      Not checked in the free run.

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Findingreasoning
    § 2.7 Symmetries of A · p. 15

    The map ̃φ is smooth.

    “It will be relevant later that the map is smooth which can easily be seen by fixing any trivialization of the fibers of P in question.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingnone given
    § 2.4 Generalized vector fields tangent to A · p. 11

    The pullback θ(t)* preserves the space of cylindrical functions, and for every cylindrical function its derivative is a well-defined element of that space.

    “Importantly, the pullback θ ( t ) ∗ preserves Cyl and for every cylindrical function Ψ, the derivative”
    • Also stated in: § 2.4 Generalized vector fields tangent to A · p. 11 · plain
      “is a well defined element of Cyl.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The manifold Σ is equipped with a semianalytic structure.

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Findingnone given
    § 2.4 Generalized vector fields tangent to A · p. 12

    Every element Y of the complexified space Γ(TA)(C) is a derivation in Cyl.

    “Note, that every Y ∈ Γ( T A ) ( C ) is a derivation in Cyl, that is”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Findingreasoning
    § A.1 semianalytic functions in R n · p. 30

    If f': U'→R^m and φ: U→U' are semianalytic functions on open subsets U⊂R^n and U'⊂R^n', then the composition f'∘φ: U→R^m is semianalytic.

    “Let U ⊂ R n and U ′ ⊂ R n ′ be open subsets. Suppose the functions f ′ : U ′ → R m and φ : U → U ′ are semianalytic. Then, the composition function f ′ ◦ φ : U → R m is semianalytic.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingreasoning
    § 4 The uniqueness theorem · p. 19

    The state ω0 is Yang-Mills gauge and diffeomorphism invariant.

    “The state ω 0 is Yang-Mills gauge and diffeomorphism invariant.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Findingnone given
    § 2.4 Generalized vector fields tangent to A · p. 11

    The action of the flux vector field X_{S,f} on cylindrical functions is linear in the smearing vector field f.

    “An important observation is that the action of the X S,f on the cylindrical functions is linear in the vector field f , i.e. X S,f 1 + f 2 = X S,f 1 + X S,f 2 .”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

    Findingnone given
    § 2.4 Generalized vector fields tangent to A · p. 11

    Given an orientation of the face S, the resulting flow is independent of the choice of the neighborhoods U, U-, U+.

    “Given an orientation of S , the resulting flow is independent of the choice of the neighborhoods U , U -, U + .”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Findingreasoning
    § 2.4 Generalized vector fields tangent to A · p. 11

    The flow used to define the flux vector field is semianalytic; in a local trivialization it corresponds to an element of the Lie algebra of G and is expressed by the usual exponential map.

    “It is easy to show, that the flow is semianalytic: in a local trivialization of P S , the vector field f corresponds to an element of the Lie algebra of G and the flow can be expressed by the usual exponential map.”
    Quote in the paperNot a substantial claim
    • The authors state this as an assumption: The manifold Σ is equipped with a semianalytic structure.

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Findingnone given
    § A.1 semianalytic functions in R n · p. 28

    If f: U→R^n is semianalytic and V⊂U is open, then the restriction f|V is semianalytic.

    “Clearly, if f : U → R n is semianalytic, and V ⊂ U is open, then the restriction function f |V is semianalytic.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

    Findingnone given
    § A.1 semianalytic functions in R n · p. 29

    Every analytic map φ: U→U' between open subsets U⊂R^n and U'⊂R^n' pullbacks all semianalytic functions defined on U' into semianalytic functions defined on U.

    “It is obvious, that every analytic map φ : U → U ′ between two open subsets U ⊂ R n and U ′ ⊂ R n ′ pullbacks all the semianalytic functions defined on U ′ into semianalytic functions defined on U .”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

    Findingnone given
    § A.1 semianalytic functions in R n · p. 29

    If f: U→R is a semianalytic function that does not vanish on an open set U, then 1/f is also semianalytic.

    “It is obvious, that if f : U → R is a semianalytic function and it does not vanish on an open set U , then 1 f is also semianalytic.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

    Findingnone given
    § A.1 semianalytic functions in R n · p. 31

    In general, the inverse of an invertible semianalytic function is not necessarily semianalytic.

    “In general, the inverse of an invertible semianalytic function is not necessarily semianalytic.”
    Quote in the paperNot a substantial claim

  • Classified as probably not a substantial claim.

    Interpretationnone given
    § Abstract · p. 1

    The uniqueness result is relevant for any theory invariant under spatial diffeomorphisms or being part of a diffeomorphism invariant theory.

    “The uniqueness result is relevant for any such theory invariant under spatial diffeomorphisms or being a part of a diffeomorphism invariant theory.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Interpretationnone given
    § Abstract · p. 1

    The precise definition of the abstract *-algebra of basic kinematical observables given in the paper is valid for any theory in which the configuration variable is a connection with a compact structure group.

    “The precise definition of the abstract ∗ -algebra of the basic kinematical observables we give could be used for any theory in which the configuration variable is a connection with a compact structure group.”
    • Also stated in: § 1 Introduction · p. 2 · plain
      “Our definitions are valid for arbitrary dimension D ≥ 2 of the base manifold Σ, arbitrary compact structure group G , and arbitrary bundle P .”
    Quote in the paperNot a substantial claim
    • The authors state this as an assumption: The structure group G of the principal fiber bundle is compact and connected.

  • Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

    Interpretationnone given
    § 1 Introduction · p. 5

    The representation corresponding to that state can be immediately generalized to arbitrary dimension and arbitrary compact gauge group.

    “Moreover, this representation can be immediately generalized to arbitrary dimension and arbitrary compact gauge group.”
    Quote in the paperNot a substantial claim
    • The authors state this as an assumption: The structure group G of the principal fiber bundle is compact and connected.

  • Probably not a substantial claim: may not pass S4 (based).

    Premisenone given
    § 2.1 Semianalytic structures · p. 6

    A necessary condition for the Poisson bracket between the variables to be finite is that every edge intersects every face in at most finitely many isolated intersection points plus finitely many connected segments.

    “A necessary condition for the Poisson bracket between the variables to be finite is that every edge intersects every face in at most finite number of isolated intersection points plus a finite number of connected segments (i.e. edges in themselves).”
    Quote in the paperNot a substantial claim
    • The authors state this as an assumption: The manifold Σ is equipped with a semianalytic structure.

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Premisecitation
    § 2.2 The cylindrical functions · p. 8

    Every cylindrical function is compatible with many sets of edges, and without loss of generality the set of edges may be assumed to be an embedded graph.

    “Every cylindrical function is compatible with many sets of edges. Without lack of generality, we may assume that γ is an embedded graph, that is, if two edges e I = e J intersect, then the intersection is contained in the boundary of each of them [22].”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [22] Ashtekar A., Lewandowski J., (1994) Representation theory of analytic holonomy algebras. In: Baez J.C. (eds) Knots and Quantum Gravity. Oxford, Oxford University Press

      Not checked in the free run.

  • May combine two claims.

    Interpretationreasoning
    § 5 Closing remarks · p. 26

    Under the assumptions made in this paper, the counterexample state ω equals ω0, so there is no contradiction.

    “Hence under the assumptions made in this paper, ω = ω 0 , and there is no contradiction.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The smearing functions f used in the flux variables E_{S,f} have compact support on S.

  • Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based). May combine two claims.

    Premisereasoning
    § 2.1 Semianalytic structures · p. 6

    A semianalytic structure is weaker than an analytic one, so it can be determined on the manifold by choosing an arbitrary analytic structure.

    “A semianalytic structure is weaker than an analytic one, therefore it can be determined on Σ for example by choosing an arbitrary analytic structure.”
    Quote in the paperNot a substantial claim
    • The authors state this as an assumption: The manifold Σ is equipped with a semianalytic structure.

  • Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

    Interpretationnone given
    § 5 Closing remarks · p. 25

    The uniqueness result holds for any enlargement of the symmetry group that contains the diffeomorphisms considered above.

    “Certainly the result holds for any enlargement of the symmetry group that contains the diffeomorphisms considered above.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S4 (based).

    Premisenone given
    § 5 Closing remarks · p. 26

    The semianalytic diffeomorphisms form a subgroup of the C^m diffeomorphism group, the group of symmetries induced by the action of the diffeomorphisms of Σ in the classical phase-space and preserving the classical Ashtekar-Corichi-Zapata algebra.

    “The corresponding diffeomorphisms form a subgroup of the C m diffeomorphism group, the group of symmetries induced by the action of the diffeomorphisms of Σ in the classical phase-space and preserving our classical Ashtekar-Corichi-Zapata algebra.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Premisecitation
    § 4 The uniqueness theorem · p. 20

    The state ω0 is well known and extensively used in loop quantization.

    “The state ω 0 is well known, and is extensively used in the loop quantization [22, 6].”
    Quote in the paperNot a substantial claim0 of 2 sources checked
    • cites [22] Ashtekar A., Lewandowski J., (1994) Representation theory of analytic holonomy algebras. In: Baez J.C. (eds) Knots and Quantum Gravity. Oxford, Oxford University Press

      Not checked in the free run.

    • cites [6] Ashtekar A., Lewandowski J. (2004) Background independent quantum gravity: A status report. Class. Quant. Grav. 21, R53 source

      Not checked in the free run.

  • May combine two claims.

    Premisereasoning
    § 1 Introduction · p. 4

    The analyticity requirement breaks the local character of the non-analytic diffeomorphism group, and the analytic diffeomorphism group usually considered in this context does not have that local character.

    “The analyticity requirement, however, breaks the local character of the non-analytic diffeomorphisms group.”
    • Also stated in: § 2.1 Semianalytic structures · p. 6 · plain
      “This is not the case, for example, for the analytic diffeomorphism group that has usually been considered in this context.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation
    § 5 Closing remarks · p. 27

    There are several similar, non-equivalent extensions of the analytic diffeomorphisms recently introduced in the literature.

    “There are several similar, non-equivalent extension of the analytic diffeomorphisms recently introduced in literature.”
    Quote in the paperNot a substantial claim0 of 2 sources checked
    • cites [29] Fleischhack, C.: Representations of the Weyl algebra in quantum geometry. http://arxiv.org/list/ math-ph/0407006, 2004

      Not checked in the free run.

    • cites [11] Fairbairn W. Rovelli C. (2004) Separable Hilbert space in loop quantum gravity, J. Math. Phys. 45, 2802–2814 source

      Not checked in the free run.

  • Probably not a substantial claim: may not pass S4 (based).

    Premisenone given
    § 1 Introduction · p. 4

    The product (3) is well defined provided that the intersection between the path α and the surface S contains finitely many isolated points.

    “The product (3) is well defined provided that the intersection between the path α with the surface S contains finitely many isolated points.”
    Quote in the paperNot a substantial claim

  • Classified as probably not a substantial claim. May combine two claims.

    Premisecitation
    § 1 Introduction · p. 4

    Functions depending on the connection only through Wilson loop functionals form the algebra of cylindrical functions, and every flux variable acts as a derivation on this algebra defined by the Poisson bracket.

    “The functions Ψ : A → C depending on A via the Wilson loop functionals only form the algebra of cylindrical functions and every flux variable E S,f acts as a derivation X S,f on this algebra, defined by the Poisson bracket 2”
    Quote in the paperSubstantial claim0 of 2 sources checked
    • cites [9] Ashtekar A., Lewandowski J. (1997) Quantum theory of geometry I: Area operators. Class. Quant. Grav. 14, A55–A82 source

      Not checked in the free run.

    • cites [10] Ashtekar A., Corichi A., Zapata J.A. (1998) Quantum theory of geometry III: Non-commutativity of Riemannian structures. Class. Quant. Grav. 15, 2955–2972 source

      Not checked in the free run.

    • The authors state this as an assumption: The structure group G of the principal fiber bundle is compact and connected.

  • May combine two claims.

    Premisecitation
    § 1 Introduction · p. 4

    The Poisson bracket in (3) preserves the space of cylindrical functions, and the Dirac delta is absorbed completely by the integrations in the definitions of holonomy and flux.

    “The Poisson bracket in (3) preserves the space of cylindrical functions and the Dirac delta is absorbed completely by the integrations involved in the definitions of the holonomy and flux. This fact was pointed out for the first time in [1, 8].”
    Quote in the paperNot a substantial claim0 of 2 sources checked
    • cites [1] Ashtekar A., (1991) Lectures on non-perturbative canonical gravity. Notes prepared in collaboration with R. S. Tate. Singapore, World Scientific source

      Not checked in the free run.

    • cites [8] Rovelli C. (1991) Ashtekar formulation of general relativity and loop space non-perturbative quantum gravity: a report. Class. Quant. Grav. 8, 1613–1675 source

      Not checked in the free run.

    • The authors state this as an assumption: The structure group G of the principal fiber bundle is compact and connected.

  • Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

    Interpretationnone given
    § 5 Closing remarks · p. 26

    All results that use smearing functions with noncompact support (such as the definition of the volume operator) can be recovered by taking appropriate limits once the state is fixed to be ω0.

    “all results that use smearing functions with noncompact support (such as the definition of the volume operator) can be recovered by taking appropriate limits once the state is fixed to be ω 0 .”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Interpretationcitation
    § 5 Closing remarks · p. 27

    The Rovelli-Fairbairn generalized diffeomorphisms are defined to be smooth everywhere except a finite set of points, and therefore they would not be useful in our case.

    “The Rovelli-Fairbairn generalized diffeomorphisms, however, are defined to be smooth everywhere except a finite set of points, therefore they would not be useful in our case.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [11] Fairbairn W. Rovelli C. (2004) Separable Hilbert space in loop quantum gravity, J. Math. Phys. 45, 2802–2814 source

      Not checked in the free run.

37 of 37

  • The authors state this as an assumption.

    The manifold Σ is equipped with a semianalytic structure.

    “Next, we assume that the manifold Σ is equipped with a semianalytic structure.”

  • The authors state this as an assumption.

    The structure group G of the principal fiber bundle is compact and connected.

    “The structure group of P is denoted by G , and it is assumed to be compact and connected.”

  • The authors state this as an assumption.

    The principal fiber bundle is semianalytic.

    “We are assuming the bundle is semianalytic.”

  • The authors state this as an assumption.

    The differentiability class C^m of the manifold is fixed with m ≥ 1.

    “The differentiability class C m is fixed, m ≥ 1.”

  • The authors state this as an assumption.

    The smearing functions f used in the flux variables E_{S,f} have compact support on S.

    “the uniqueness theorem fails if a rather innocent looking assumption - that of compact support on S for the smearing functions f used in the flux variables E S,f - is removed”
    “for smearing functions f with compact support in S , ∫ S df =0.”

  • The authors state this as an assumption.

    The treatment assumes a common dense domain for all flux operators and cylindrical functions.

    “It does not make the assumption of a common dense domain for all flux operators and cylindrical functions that is implicit in our treatment.”
6 of 6

Claims other papers make about this one

Each is checked against this paper’s own text: does it say what the citing paper says it does?

  • The operator representation of the basic unconstrained phase-space variables in loop quantum gravity has been proven unique.

    “The representation of the basic unconstrained phase-space variables as suitable operators in used in LQG — which has the additional key property of containing a special state that is diffeomorphism invariant — has been shown to be unique [245].”

    From The Spin-Foam Approach to Quantum Gravity (2013) · cites this paper as [245]

  • The holonomy-flux algebra admits a unique quantization in a Hilbert space with diffeomorphism-invariant states.

    “It has been shown that the associated (Poisson) holonomy-flux algebra admits a unique quantization in a Hilbert space with diffeomorphism-invariant states [176, 245].”

    From The Spin-Foam Approach to Quantum Gravity (2013) · cites this paper as [245]

Self-check

Percentages, sums and reported differences re-computed from the paper’s own numbers.

Each contradiction between two passages listed here was confirmed by an independent check.

  • Composition rule gives every composed edge the same beginning and end

    Minor

    Under the stated condition, the beginning of e is already the end of e ′. The stated endpoint rule therefore gives e ◦ e ′ coincident endpoints, whereas the definition requires an edge to have a two-point boundary.

  • Claimed scope includes disconnected groups, but the definition excludes them

    Minor

    The introduction says the definitions apply to an arbitrary compact structure group. The definition of the bundle used to construct the algebra additionally requires that group to be connected.

2 of 2

Cites

Flagged first, then by citations

Research works printed but not linked: 5

  1. Łojasiewicz S. (1964) Triangulation of semi-analytic sets. Ann. Scuola. Norm. Sup. Pisa 18, 449–474
  2. Schmüdgen, K.: Unbounded Operator Algebras and Representation Theory. In: Operator Theory: Advances and Applications. Vol. 37, Basel: Birkhäuser, 1990
  3. Sahlmann, H.: Some comments on the representation theory of the algebra underlying loop quantum gravity. http://arxiv.org/list/ gr-qc/0207111, 2002 · arXiv
  4. Sahlmann, H.: When do measures on the space of connections support the triad operators of loop quantum gravity? http://arxiv.org/list/ gr-qc/0207112, 2002 · arXiv
  5. Reed M., Simon B. (1980) Methods of Modern Mathematical Physics Vol.1: Functional Analysis. New York, Academic Press

Cited by

Sorted by citations

Authors
Lewandowski, J., Okołów, A., Sahlmann, H., & Thiemann, T.
Affiliations
  • Pennsylvania State University (US)
  • Louisiana State University (US)
  • University of Waterloo (CA)
Corresponding author
Thiemann, Thomas
ORCID
Venue
Communications in Mathematical Physics
Publisher
Springer Science and Business Media LLC
Published
2006-08-22
Volume
267
Issue
3
Pages
703-733
ISSN
  • 0010-3616 (ISSN-L)
  • 1432-0916
DOI
10.1007/s00220-006-0100-7
OpenAlex
W2014240016
Cited by
297
References
26
Citations by year
2026: 4 · 2025: 20 · 2024: 10 · 2023: 9 · 2022: 6 · 2021: 10 · 2020: 7 · 2019: 11 · 2018: 11 · 2017: 11 · 2016: 19 · 2015: 7 · 2014: 17 · 2013: 22 · 2012: 16
FWCI
13.73
Type
journal article
Language
en
Open access
Licence
www.springer.com/tdm (tdm)
Locations
  • Communications in Mathematical Physics (publishedVersion)
  • arXiv (Cornell University) (acceptedVersion)
  • Max Planck Institute for Plasma Physics (submittedVersion)
  • Max Planck Digital Library (submittedVersion, other-oa)
  • MPG.PuRe (Max Planck Society) (submittedVersion, other-oa)
  • arXiv (Cornell University)
Funders
  • National Science Foundation
  • Pennsylvania State University
  • Institut Périmètre de physique théorique
  • Natural Sciences and Engineering Research Council of Canada
Topics
  • Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Noncommutative and Quantum Gravity Theories
  • Physical Sciences › Mathematics › Mathematical Physics › Advanced Operator Algebra Research
  • Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Relativity and Gravitational Theory
Keywords
loop quantum gravity, diffeomorphism invariance, cyclic representations, uniqueness of representation, holonomy, flux, Dirac quantization, constraint quantization

About these checks

Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.

Claims read by deepseek-v4.1-flash under definition version 3 (extract-v3) on 2026-10-06.

Classified by jev-1.13-20260917 (gold-guideline-v0.2) on 2026-10-06.

Statistics and arithmetic are recomputed, not judged.

Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.

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