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Cited by 303Noncommutative and Quantum Gravity TheoriesRelativity and Gravitational TheoryAdvanced Operator Algebra Research

Differential geometry on the space of connections via graphs and projective limits

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Ashtekar, Abhay, Lewandowski, Jerzy·1995·Journal of Geometry and Physics
DOI 10.1016/0393-0440(95)00028-g
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TL;DR

The paper develops differential geometry on the completion \(A/G\) of the space of gauge-equivalent connections by realizing it as a projective limit of compact Hausdorff analytic manifolds labelled by graphs. It introduces forms, exterior derivatives, volume forms, vector fields, Lie brackets, divergence, Laplacians, heat kernels, and heat-kernel measures without requiring a background metric, making the framework suitable for diffeomorphism-invariant theories such as quantum general relativity.

Key findings

  • \(A/G\) is obtained as the projective limit of a projective family of compact, Hausdorff, analytic manifolds labelled by graphs, and is naturally isomorphic to the Gel’fand spectrum of the holonomy \(C^\star\)-algebra \(HA\); it is independent of the initial choice of principal bundle \(B(M,G)\).
  • Differential-geometric notions can be induced on \(A/G\) through consistent families on the projective family: \(C^n\) differential forms, exterior derivatives, volume forms, vector fields, Lie brackets, divergence with respect to a volume form, and momentum operators that are essentially self-adjoint on \(L^2(A,\mu)\) when compatible with the volume form.
  • The normalized Haar volume form on \(G\) yields a natural volume form \(\mu_o\) on \(A\); its pushforward gives the induced Haar measure on \(A/G\), invariant under all diffeomorphisms of \(M\). Heat-kernel measures are constructed from an edge-metric \(l\), and the resulting measure is faithful iff \(l(e)>0\) for every non-zero edge \(e\).
  • \(A/G\) admits a natural degenerate contravariant metric \(k_o\), giving a Laplace-like operator and, for \(G=SU(2)\) and oriented 3-dimensional \(M\), a natural third-order operator related to the volume observable in quantum gravity.

Methods

  • Mathematical/theoretical construction: fix an \(n\)-dimensional analytic manifold \(M\) and a smooth principal bundle \(B(M,G)\) with compact connected Lie group \(G\); use the set of analytic graphs in \(M\) as labels and build projective families for \(A\), \(G\), and \(A/G\).
  • Analysis: define cylindrical functions, forms, volume forms, vector fields, and operators as consistent families on the projective family; prove propositions/theorems on consistency, essential self-adjointness, \(G\)-invariance, and diffeomorphism invariance.
  • No empirical data, numerical experiments, or quantitative outcomes are reported.

Limitations

  • The paper states that the quotient spaces \(A_\gamma/G_\gamma\) are not compact analytic manifolds because the quotient construction introduces kinks and boundaries; therefore differential geometry is developed “upstairs” on \(A\) using \(G\)-invariant structures.
  • For non-compact structure groups, the paper states that the \(A/G\) construction via the holonomy \(C^\star\)-algebra does not go through, the projective limit is no longer compact, and self-consistent families generally provide only cylindrical measures rather than regular Borel measures.
  • Evident from the text: the work is formal/projective rather than numerical; several extensions are deferred, including possible replacement of \(SL(2,\mathbb{C})\) by its quantum version \(SL(2)_q\), detailed derivation of local area and Hamiltonian operators, and further work on weakened consistency conditions for vector fields.

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

Integrity signals

Registry

Crossref · 10.1016/0393-0440(95)00028-g

Cites

14 of 14 references verified0 citation errors · 0 not checkable

Statistics

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Claims the paper makes

0 of 70 cited sources checked · 0 supported · 0 flagged

  • Findingnone given
    § Proposition 1 : · p. 17

    The map (26) defines a one-to-one correspondence between the projective limit A and the space P(B) of generalized parallel transports in B.

    “Proposition 2 : The map (26) defines a one-to-one correspondence between the projective limit A and the space P ( B ) of generalized parallel transports in B .”
    Quote in the paperNot a substantial claim
    • The authors state this as an assumption: M is an n-dimensional analytic manifold.
    • The authors state this as an assumption: B(M,G) is a smooth principal fiber bundle with structure group G.
  • Findingreasoning
    § Proposition 7 : · p. 28

    If X is compatible with μ, then for every f,g ∈ Cyl^1(A), the divergence of X with respect to μ satisfies the product rule.

    “i) Let X be a vector field and µ , a smooth volume form on A such that X is compatible with µ . Then, for every f, g ∈ Cyl 1 ( A ) ,”
    Quote in the paperSubstantial claim
  • Findingcitation
    § 3.4 The quotient A / G and the projective family for A / G . · p. 21

    A/G is naturally isomorphic to the Gel'fand spectrum of the holonomy C*-algebra HA.

    “Finally, by combining the results of [5] and [8], one can show that the space A / G is naturally isomorphic to the Gel'fand spectrum of the holonomy C ⋆ -algebra HA introduced in section 1.”
    Quote in the paperSubstantial claim0 of 2 sources checked
    • cites [5] [5] J. Baez, 'Generalized Measures in gauge theory', Lett. Math. Phys. 31 , 213-223 (1994). -

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

    • cites [8] [8] A. Ashtekar and J. Lewandowski, 'Projective techniques and functional integration for gauge theories', preprint CGPG-94/10-6, for the special JMP issue on functional integration, edited by C. DeWitt-Morette. -

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • Findingreasoning
    § 5.1 Forms and volume forms · p. 33

    If ν is a volume form on A, then averaging over G yields a G-invariant volume form on A such that for every G-invariant f ∈ C0(A), the integral of f with respect to the averaged form equals the integral of f with respect to ν.

    “Proposition 8 : Suppose ν = ( ν γ ) γ ∈ L is a volume form on A . Then, if R ¯ g denotes the action of ¯ g ∈ G on A , and d ¯ g denotes the Haar measure on G ,”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The structure group G is compact.
  • Findingnone given
    § 6.1 A Laplace operator. · p. 43

    The strategy of using the product metric on Aγ to define a Laplacian fails because the resulting family of operators (∆'_γ) fails to be self consistent.

    “Unfortunately, this strategy fails: the resulting family of operators (∆ ′ γ ) γ ∈ L fails to be self consistent.”
    Quote in the paperSubstantial claim
  • May combine two claims. 32
  • The quote only partly states this claim. 7
  • Classified as probably not a substantial claim. 20

  • May combine two claims. The quote only partly states this claim.

    Findingreasoning
    § Proposition 7 : · p. 28

    If Y is another vector field compatible with μ, then the commutator [X,Y] is compatible with μ and its divergence satisfies the usual formula.

    “ii) Suppose that Y is another vector field on A which is compatible with µ . Then, the commutator [ X,Y ] also is compatible with µ , and”
    Quote in the paperSubstantial claim

  • May combine two claims. The quote only partly states this claim.

    Findingreasoning
    § Theorem 2 : · p. 32

    If Y is another smooth vector field compatible with μ, then [X,Y] is compatible with μ and the corresponding momentum operators satisfy the usual commutator formula.

    “(ii) Let Y be another smooth vector field on the projective limit, also compatible with the measure µ . Then, the vector field [ X,Y ] also is compatible with µ and”
    Quote in the paperSubstantial claim

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

    Findingnone given
    § Proposition 7 : · p. 29

    The second term in definition (54) of the momentum operators is necessary to ensure that it is a symmetric operator.

    “As one might expect from this analogy, the second term in the definition (54) of the momentum operators is necessary to ensure that it is a symmetric operator.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingnone given
    § Proposition 1 : · p. 15

    The map Λγ is bijective and the analytic manifold structure it defines on Aγ does not depend on the initial choice of points in the fibers over vertices or orientations of edges.

    “The map Λ γ of (17) is bijective and the analytic manifold structure defined on A γ by Λ γ does not depend on the initial choice of points in the fibers of B over the vertices of γ and orientation of the edges made in its definition;”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: M is an n-dimensional analytic manifold.
    • The authors state this as an assumption: B(M,G) is a smooth principal fiber bundle with structure group G.
    • The authors state this as an assumption: G is a compact and connected Lie group.

  • The quote only partly states this claim.

    Findingreasoning
    § Theorem 2 : · p. 32

    If X is compatible with μ, then the operator (P(X), Cyl^1(A)) is essentially self-adjoint on L^2(A,μ).

    “(i)Let X = ( X γ ) γ ≥ γ 0 and µ = ( µ γ ) γ ∈ L be a smooth vector field and volume form on the projective limit A . Suppose X is compatible with µ ; then, the operator ( P ( Y ) , Cyl 1 ( A )) of (54) is essentially self-adjoint on L 2 ( A , µ ) ;”
    Quote in the paperSubstantial claim

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

    Findingnone given
    § 1 Introduction · p. 6

    There is a third order differential operator representing the volume observable in quantum gravity.

    “Another example is a third order differential operator representing the 'volume observable' in quantum gravity.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingreasoning
    § 5.2 Vector fields on A / G · p. 34

    If X is a C^{n+1} G-invariant vector field on A, then X is compatible with the Haar measure μ_o on A and div X ∈ Cyl^n(A) is G-invariant.

    “Theorem 3 : Let X = ( X γ ) γ ≥ γ 0 be a C n +1 vector field on A . If X is G invariant then it is compatible with the Haar measure µ o on A and div X ∈ Cyl n ( A ) is G invariant.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning
    § 1 Introduction · p. 6

    A/G admits a natural degenerate contravariant metric and a Laplace-like operator whose construction does not use any background structure on M, so its action respects diffeomorphism invariance.

    “In section 7, we point out that A / G admits a natural (degenerate) contravariant metric and use it to introduce a Laplace-like operator. Since this construction does not use any background structure on M , the action of the operator respects diffeomorphism invariance.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: M is an n-dimensional analytic manifold.

  • May combine two claims.

    Findingcitation
    § Proposition 3 : · p. 19

    The consistent family of measures on Gγ defines a regular Borel probability measure ̄μ_o on G, which is the Haar measure on G, with total volume finite (unit).

    “Hence, it defines a regular, Borel Probability measure ¯ µ o on G . This is the just Haar measure on G . Thus, by enlarging the group G to G , one can obtain a compact group of generalized gauge transformations whose total volume is fi nite (actually, unit). This observation was first made by Baez [6].”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [6] [6] J. Baez, 'Diffeomorphism invariant generalized measures on the space of connections modulo gauge transformations', hep-th/9305045, to appear in the Proceedings of the conference on quantum topology, D. Yetter (ed) (World Scientific, Singapore, 1994). -

      Not checked in the free run.

    • The authors state this as an assumption: G is a compact and connected Lie group.

  • May combine two claims.

    Findingreasoning
    § 5.3 Characterization of vector fields on A / G · p. 39

    If both vertices of an edge lie outside γ0, the corresponding vector fields on G must be both right and left invariant; since the gauge group is semi-simple, they must vanish.

    “Since both vertices of these edges lie outside γ 0 , the corresponding vector fields on G have to be both right and left invariant. Since we have assumed that the gauge group is semi-simple, this implies that they must vanish.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The gauge group G is semi-simple.

  • The quote only partly states this claim.

    Findingreasoning
    § 5.2 Vector fields on A / G · p. 38

    If X is a G-invariant vector field on A, then the operator with domain Cyl^1(A/G) is essentially self-adjoint on L^2(A/G, μ_o).

    “Theorem 4 Let X be a G invariant vector field on A . The operator”
    Quote in the paperSubstantial claim

  • May combine two claims. The quote only partly states this claim.

    Findingnone given
    § 5.2 Vector fields on A / G · p. 38

    If Y is another G-invariant vector field on A/G, then [X,Y] is also a G-invariant vector field on A/G and the corresponding operators satisfy the usual commutator formula on Cyl^2(A/G).

    “Suppose Y is another G invariant vector field on A / G ; then, [ X,Y ] is also a G invariant vector field on A / G and”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingreasoning
    § 7.3 A natural Laplacian · p. 59

    If γ' is constructed by adding an extra edge e'' to γ, then the projection kills every term containing R_e''i, and the remaining terms coincide with those of ∆_γ.

    “On the other hand, if γ ′ is constructed by adding an extra edge, say e ′′ , to γ then the projection kills every term containing R e ′′ i . The remaining terms coincide with those of ∆ γ .”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning
    § 7.3 A natural Laplacian · p. 61

    The family (q_γ, C^3(A_γ))_{γ∈L} of operators is self consistent. The operator (q, Cyl_3(A)) is natural, G invariant and defines a natural operator on Cyl_3(A/G).

    “Proposition 13 : The family ( q γ , C 3 ( A γ )) γ ∈ L of operators is self consistent. The operator ( q, Cyl 3 ( A )) is natural, G invariant and defines a natural operator on Cyl 3 ( A / G ) .”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingcitation
    § 6.3 Non-compact structure groups. · p. 51

    The projective limit is well-defined and the notion of self-consistent families of measures (μ_γ) is still valid for non-compact G.

    “Nonetheless, the projective limit is well-defined and the notion of self-consistent families of measures ( µ γ ) γ ∈ L is still valid.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning
    § Theorem 5 : · p. 45

    The operator ∆(l) is G-invariant and thus defines an essentially self-adjoint operator (∆(l), Cyl^2(A/G)) in L^2(A/G, μ'_o).

    “( iii ) the operator ∆ ( l ) is G invariant and thus defines an essentially selfadjoint operator (∆ ( l ) , Cyl 2 ( A / G )) in L 2 ( A / G , µ ′ o ) , where µ ′ o is the pushforward of µ o to A / G .”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning
    § 6.2 The heat equation and the associated volume forms. · p. 48

    There exists a linear map ρ_t⋆: C0(A) → C0(A) defined by (100), and it is continuous with respect to the sup-norm.

    “(i)there exists a linear map ρ t ⋆ : C 0 ( A ) → C 0 ( A ) defined by”
    • Also stated in: § 6.2 The heat equation and the associated volume forms. · p. 48 · plain
      “which is continuous with respect to the sup-norm;”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The structure group G is compact.

  • May combine two claims.

    Findingreasoning
    § 6.3 Non-compact structure groups. · p. 51

    The members A_γ of this family have the manifold structure of G^E and are therefore no longer compact.

    “As in section 3, the members A γ of this family have the manifold structure of G E (where E is the number of edges in γ ) and are therefore no longer compact.”
    Quote in the paperSubstantial claim

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

    Findingreasoning
    § 7.1 Co-metric tensors. · p. 56

    If kγ is non-degenerate, ∆(k) is the Laplacian it defines.

    “If k γ is non-degenerate, ∆ ( k ) is the Laplacian it defines.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingreasoning
    § 7.2 A natural co-metric tensor. · p. 57

    The family k_o = (kγ) is a natural co-metric tensor on A; k_o is G invariant, and defines a natural co-metric on A/G.

    “Proposition 11 : The family k o = ( k γ ) γ ∈ L is a natural co-metric tensor on A ; k o is G invariant, and defines a natural co-metric on A / G .”
    Quote in the paperSubstantial claim

  • Classified as probably not a substantial claim.

    Findingnone given
    § 7.2 A natural co-metric tensor. · p. 58

    k_o is diffeomorphism invariant.

    “The diffeomorphism invariance of k o is obvious.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning
    § 6.1 A Laplace operator. · p. 44

    The weighted Laplacian ∆_{γ,(l)} is well-defined and insensitive to the ambiguity in the choice of (g_e1,...,g_eE) that label the points of Aγ.

    “However, two such isomorphisms are in essence related by an element of G γ and since ∆ is left and right invariant on G , ∆ γ, ( l ) is well-defined; it is insensitive to the ambiguity in the choice of ( g e 1 , ..., g e E ) that label the points of A γ .”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingcitation
    § 8 Discussion · p. 61

    Neither the construction of A/G nor the introduction of a large family of measures on it requires a metric on M.

    “Neither the construction of A / G nor the introduction of a large family of measures on it requires a metric on M .”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The algebra of Wilson loop variables should be faithfully represented in the quantum theory.

  • The quote only partly states this claim.

    Findingreasoning
    § Theorem 5 : · p. 45

    To be consistent with a projection of the third kind, the numbers l(e) must satisfy l(e)=l(e1)+l(e2) whenever e=e2∘e1.

    “However, to be consistent with a projection of the third kind, the numbers l ( e ) have to satisfy the following necessary and sufficient condition. Let f be a function on G . Whenever we divide an edge e into e = e 2 ◦ e 1 , then”
    Quote in the paperSubstantial claim

  • The quote only partly states this claim.

    Findingreasoning
    § 8 Discussion · p. 65

    Although A-bar is very big, it is small enough to admit a rich geometry.

    “The answer turned out to be in the negative; although A is very big, it is small enough to admit a rich geometry.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationcitation
    § 1 Introduction · p. 5

    Many of the Diff(M)-invariant measures on A/G are faithful, indicating that all of A/G would be relevant to quantum gravity.

    “Many of these are faithful indicating that all of A / G would be relevant to quantum gravity.”
    Quote in the paperSubstantial claim0 of 5 sources checked
    • cites [4] [4] A. Ashtekar and J. Lewandowski, 'Representation theory of analytic holonomy C ⋆ algebras', in Knots and quantum gravity , J. Baez (ed), (Oxford University Press, Oxford 1994). -

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

    • cites [5] [5] J. Baez, 'Generalized Measures in gauge theory', Lett. Math. Phys. 31 , 213-223 (1994). -

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

    • cites [6] [6] J. Baez, 'Diffeomorphism invariant generalized measures on the space of connections modulo gauge transformations', hep-th/9305045, to appear in the Proceedings of the conference on quantum topology, D. Yetter (ed) (World Scientific, Singapore, 1994). - source

      Not checked in the free run.

    • cites [12] [12] A. Ashtekar, J. Lewandowski, D. Marolf, J. Mour˜ ao and T. Thiemann, 'Quantization of theories of connections with local degrees of freedom', J. Math. Phys. (in press) -

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

    • cites [8] [8] A. Ashtekar and J. Lewandowski, 'Projective techniques and functional integration for gauge theories', preprint CGPG-94/10-6, for the special JMP issue on functional integration, edited by C. DeWitt-Morette. -

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

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

    Interpretationnone given
    § 1 Introduction · p. 7

    The general spirit of this approach to geometry on A/G is the same as that of non-commutative geometry and quantum groups: geometrical notions can be developed by exploiting the properties of the algebra of functions.

    “A striking aspect of this approach to geometry on A / G is that its general spirit is the same as that of non-commutative geometry and quantum groups: even though there is no underlying differentiable manifold, geometrical notions can be developed by exploiting the properties of the algebra of functions.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationreasoning
    § 1 Introduction · p. 6

    A/G is large enough to serve as a universal home for measures and, at the same time, small enough to be mathematically interesting and physically useful.

    “While it is large enough to serve as a 'universal home' for measures, A / G is, at the same time, small enough to be mathematically interesting and physically useful. This is the main message of this paper.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Premisecitation
    § 2 Projective Techniques: general framework · p. 11

    For a compact Hausdorff projective family, regular Borel probability measures on the projective limit correspond bijectively to consistent families of regular Borel probability measures on the members, and faithfulness is preserved.

    “Using this notion, we can now characterize measures on X [8]: Theorem 1 : Let ( X γ , p γγ ′ ) γγ ′ ∈ L be a compact, Hausdorff projective family and X be its projective limit;”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [8] [8] A. Ashtekar and J. Lewandowski, 'Projective techniques and functional integration for gauge theories', preprint CGPG-94/10-6, for the special JMP issue on functional integration, edited by C. DeWitt-Morette. -

      Not checked in the free run.

  • May combine two claims.

    Interpretationcitation
    § 3.4 The quotient A / G and the projective family for A / G . · p. 21

    A/G is independent of the choice of the bundle B and is tied only to the underlying manifold M.

    “Thus, although it is not obvious from our construction, A / G is independent of the choice of the bundle B we made in the beginning; it is tied only to the underlying manifold M . (See [3, 4, 5, 7] for the bundle independent definitions.)”
    Quote in the paperSubstantial claim0 of 4 sources checked
    • cites [3] [3] A. Ashtekar and C.J. Isham, 'Representations of the holonomy algebras for gravity and non-Abelian gauge theories', Class. & Quan. Grav. 9 , 1433-1467 (1992). -

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

    • cites [4] [4] A. Ashtekar and J. Lewandowski, 'Representation theory of analytic holonomy C ⋆ algebras', in Knots and quantum gravity , J. Baez (ed), (Oxford University Press, Oxford 1994). -

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

    • cites [5] [5] J. Baez, 'Generalized Measures in gauge theory', Lett. Math. Phys. 31 , 213-223 (1994). -

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

    • cites [7] [7] D. Marolf and J. M. Mour˜ ao, 'On the support of the AshtekarLewandowski measure', Commun. Math. Phys. (in press). -

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • Probably not a substantial claim: may not pass S2 (checkable), S3 (load-bearing).

    Interpretationreasoning
    § 8 Discussion · p. 62

    The results of this paper can be used to give a rigorous meaning to various operators that have played an important role in heuristic treatments.

    “Specifically, it appears that results of this paper can be used to give a rigorous meaning to various operators that have played an important role in various heuristic treatments [10, 11].”
    Quote in the paperNot a substantial claim0 of 2 sources checked
    • cites [10] [10] A. Ashtekar, Non-Perturbative Canonical Gravity , Lectures notes prepared in collaboration with R.S. Tate (World Scientific, Singapore, 1991) -

      Not checked in the free run.

    • cites [11] [11] C. Rovelli and L Smolin, 'Loop representation of quantum general relativity', Nucl. Phys. B331 , 80-152 (1990); 'Discreteness of Area and Volume in Quantum Gravity, gr-qc/9411005 -

      Not checked in the free run.

  • May combine two claims.

    Interpretationreasoning
    § 7.1 Co-metric tensors. · p. 56

    Assuming the necessary differentiability, k(ω) has a well-defined divergence with respect to the Haar volume form μ_o on A, and div_μo k(ω) = [div kγ(ωγ)]∼.

    “Hence, assuming the necessary differentiability, k ( ω ) has a well defined divergence with respect to the Haar volume form µ o on A , div µ o k ( ω ) = [div k γ ( ω γ )] ∼ .”
    Quote in the paperSubstantial claim

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

    Premisecitation
    § 7.3 A natural Laplacian · p. 59

    The family and the transformations ρ_t* coincide with those introduced in [20].

    “The family and the transformations ρ t⋆ coincide with those introduced in [20].”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [20] [20] A. Ashtekar, J. Lewandowski, D. Marolf, J. Mour˜ ao and T. Thiemann, 'Coherent State Transforms for Spaces of Connections' J. Funct. Analysis (in press) gr-qc/9412014. -

      Not checked in the free run.

  • May combine two claims.

    Interpretationreasoning
    § 6.2 The heat equation and the associated volume forms. · p. 50

    On A/G, there is a natural origin [A0], so the freedom of the choice of A0 can be eliminated.

    “Finally, on A / G , there is a natural origin [ ¯ A 0 ] whence the freedom is the choice of ¯ A 0 can be eliminated.”
    Quote in the paperNot a substantial claim
    • The authors state this as an assumption: The structure group G is compact.

  • Classified as probably not a substantial claim.

    Premisecitation
    § 6.2 The heat equation and the associated volume forms. · p. 47

    For each t, the heat kernel ρ_t on G is a positive, smooth function on G.

    “As is well-known [19], for each t , ρ t is a positive, smooth function on G .”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [19] [19] E. M. Stein, Topics in analysis related to the Littlewood-Paley theory (Princeton University Press, Princeton, 1970) page 38. -

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • May combine two claims.

    Premisereasoning
    § 6.3 Non-compact structure groups. · p. 51

    For non-compact G, the construction of the space A/G given in [3] does not go through, since the Wilson loop functions are unbounded and one cannot endow them with the structure of a C⋆-algebra in a straightforward fashion.

    “To begin with, the construction of the space A / G given in [3] does not go through, since the Wilson loop functions are now unboun[216zded and one can not endow them with the structure of a C ⋆ -algebra in a straightforward fashion.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [3] [3] A. Ashtekar and C.J. Isham, 'Representations of the holonomy algebras for gravity and non-Abelian gauge theories', Class. & Quan. Grav. 9 , 1433-1467 (1992). -

      Not checked in the free run.

    • Rests on an assumption: For non-compact G, the Wilson loop functions are unbounded.

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

    Interpretationnone given
    § 8 Discussion · p. 61

    This success may be primarily due to the fact that in 2 dimensions, the Yang-Mills theory is invariant under all volume preserving diffeomorphisms.

    “This success may, however, be primarily due to the fact that in 2 dimensions, the Yang-Mills theory is invariant under all volume preserving diffeomorphisms.”
    Quote in the paperNot a substantial claim
    • The authors state this as an assumption: The algebra of Wilson loop variables should be faithfully represented in the quantum theory.

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

    Premisecitation
    § 8 Discussion · p. 63

    A/G first arose as the Gel'fand spectrum of an Abelian C* algebra HA constructed from Wilson loop functions on the space A/G of smooth connections modulo gauge transformations.

    “As mentioned in the Introduction, A / G fi rst arose as the Gel'fand spectrum of an Abelian C ⋆ algebra HA constructed from the so-called Wilson loop functions on the space A / G of smooth connections modulo gauge transformations [3].”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [3] [3] A. Ashtekar and C.J. Isham, 'Representations of the holonomy algebras for gravity and non-Abelian gauge theories', Class. & Quan. Grav. 9 , 1433-1467 (1992). -

      Not checked in the free run.

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

    Premisecitation
    § 8 Discussion · p. 61

    The framework has been successful in dealing with Yang-Mills theory in 2 space-time dimensions [18] without a significant new input.

    “The framework has been successful in dealing with Yang-Mills theory in 2 space-time dimensions [18] without a significant new input.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [18] [18] A. Ashtekar, J. Lewandowski, D. Marolf, J. Mour˜ ao and T. Thiemann, 'A manifestly gauge invariant approach to quantum theories of gauge fields', in Geometry of constrained dynamical systems , J. Charap (ed) (Cambridge University Press, Cambridge, 1994); 'Constructive quantum gauge field theory in two space-time dimensions' (preprint). -

      Not checked in the free run.

    • The authors state this as an assumption: The algebra of Wilson loop variables should be faithfully represented in the quantum theory.

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

    Premisecitation
    § 8 Discussion · p. 61

    The framework was introduced in [3, 4] and extended in [5,8] to a theory of diffeomorphism invariant measures.

    “Such a framework was introduced in [3, 4] and extended in [5,8] to a theory of diffeomorphism invariant measures.”
    Quote in the paperNot a substantial claim0 of 4 sources checked
    • cites [3] [3] A. Ashtekar and C.J. Isham, 'Representations of the holonomy algebras for gravity and non-Abelian gauge theories', Class. & Quan. Grav. 9 , 1433-1467 (1992). -

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

    • cites [4] [4] A. Ashtekar and J. Lewandowski, 'Representation theory of analytic holonomy C ⋆ algebras', in Knots and quantum gravity , J. Baez (ed), (Oxford University Press, Oxford 1994). -

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

    • cites [5] [5] J. Baez, 'Generalized Measures in gauge theory', Lett. Math. Phys. 31 , 213-223 (1994). -

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

    • cites [8] [8] A. Ashtekar and J. Lewandowski, 'Projective techniques and functional integration for gauge theories', preprint CGPG-94/10-6, for the special JMP issue on functional integration, edited by C. DeWitt-Morette. -

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • May combine two claims.

    Interpretationreasoning
    § 6.2 The heat equation and the associated volume forms. · p. 50

    According to Theorem 6, the heat kernel ρ_t naturally projects to the quotient A/G, and hence there is a well-defined Gaussian measure (105) on A/G.

    “According to Theorem (6), the heat kernel ρ t naturally projects to this quotient and hence we have a well-defined Gaussian measure (105) on A / G .”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The structure group G is compact.

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

    Premisecitation
    § 7.3 A natural Laplacian · p. 61

    The operator q is closely related to the total volume (of M) operator of Riemannian quantum gravity [11, 12].

    “This operator is closely related to the total volume (of M ) operator of Riemannian quantum gravity [11, 12].”
    Quote in the paperNot a substantial claim0 of 2 sources checked
    • cites [11] [11] C. Rovelli and L Smolin, 'Loop representation of quantum general relativity', Nucl. Phys. B331 , 80-152 (1990); 'Discreteness of Area and Volume in Quantum Gravity, gr-qc/9411005 -

      Not checked in the free run.

    • cites [12] [12] A. Ashtekar, J. Lewandowski, D. Marolf, J. Mour˜ ao and T. Thiemann, 'Quantization of theories of connections with local degrees of freedom', J. Math. Phys. (in press) -

      Not checked in the free run.

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

    Interpretationreasoning
    § 8 Discussion · p. 65

    The three ways of constructing A/G are complementary, and together they show that A/G has a surprisingly rich structure.

    “The three ways of constructing A / G are complementary, and together they show that A / G has a surprisingly rich structure.”
    Quote in the paperSubstantial claim

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

    Premisecitation
    § 8 Discussion · p. 64

    Independently, graphs were introduced to analyze the so-called strip-operators as a consistent family of vector fields.

    “Independently, the graphs were introduced to analyze the so-called 'strip-operators' (which serve as gauge invariant momentum operators) as a consistent family of vector fields in [21].”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [21] [21] J. Lewandowski, 'Topological measure and graph-differential geometry on the quotient space of connections' Int. J. Mod. Phys. D3 , 207-210 (1994). -

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • May combine two claims.

    Interpretationreasoning
    § 8 Discussion · p. 65

    For diffeomorphism invariant theories, there is no easy substitute for the extended spaces A-bar and A/G-bar; one has to learn to deal directly with quantum configuration spaces.

    “To summarize, for diffeomorphism invariant theories, there is no easy substitute for the extended spaces A and A / G ; one has to learn to deal directly with quantum configuration spaces.”
    Quote in the paperSubstantial claim

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

    Premisecitation
    § 8 Discussion · p. 63

    The label set of the Marolf-Mourao family is the set of all finitely generated subgroups of the hoop group.

    “The label set of the family they used is, however, different from the one we used in this paper: it is the set of all finitely generated subgroups of the hoop group.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [7] [7] D. Marolf and J. M. Mour˜ ao, 'On the support of the AshtekarLewandowski measure', Commun. Math. Phys. (in press). -

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

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

    Premisecitation
    § 8 Discussion · p. 64

    Baez discovered powerful theorems that simplify obtaining measures and, using them, obtained a large class of diffeomorphism invariant measures on A/G.

    “His main result was two-folds: he discovered powerful theorems that simplify the task of obtaining measures and, using them, obtained a large class of diffeomorphism invariant measures on A / G .”
    Quote in the paperNot a substantial claim0 of 2 sources checked
    • cites [5] [5] J. Baez, 'Generalized Measures in gauge theory', Lett. Math. Phys. 31 , 213-223 (1994). -

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

    • cites [6] [6] J. Baez, 'Diffeomorphism invariant generalized measures on the space of connections modulo gauge transformations', hep-th/9305045, to appear in the Proceedings of the conference on quantum topology, D. Yetter (ed) (World Scientific, Singapore, 1994). - source

      Not checked in the free run.

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

    Premisenone given
    § 8 Discussion · p. 65

    There are three characterizations of A/G: one as a Gel'fand spectrum of an Abelian C* algebra and two as projective limits.

    “Fortunately, this task is made manageable because there are three characterizations of A / G : One as a Gel'fand spectrum of an Abelian C ⋆ -algebra and two as projective limits.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Premisecitation
    § 8 Discussion · p. 65

    Marolf and Mourao have shown that the support of the natural measure mu0 is concentrated precisely on A-bar minus A, and the space A is contained in a set of zero mu0-measure.

    “First, as Marolf and Mour˜ ao [7] have shown, the support of the natural measure µ o is concentrated precisely on A -A ; the space A is contained in a set of zero µ o -measure.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [7] [7] D. Marolf and J. M. Mour˜ ao, 'On the support of the AshtekarLewandowski measure', Commun. Math. Phys. (in press). -

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

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

    Premisecitation
    § 8 Discussion · p. 63

    Marolf and Mourao introduced a projective family of compact Hausdorff spaces and showed that its projective limit is naturally isomorphic to A/G.

    “Marolf and Mour˜ ao [7] obtained another characterization of A / G : using various techniques employed in [4], they introduced a projective family of compact Hausdorff spaces and showed that its projective limit is naturally isomorphic to A / G .”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [7] [7] D. Marolf and J. M. Mour˜ ao, 'On the support of the AshtekarLewandowski measure', Commun. Math. Phys. (in press). -

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

  • Classified as probably not a substantial claim.

    Premisecitation
    § 8 Discussion · p. 64

    The shift from hoops to graphs was suggested and explored by Baez.

    “The shift from hoops to graphs was suggested and explored by Baez [5, 6].”
    Quote in the paperSubstantial claim0 of 2 sources checked
    • cites [5] [5] J. Baez, 'Generalized Measures in gauge theory', Lett. Math. Phys. 31 , 213-223 (1994). -

      Couldn’t be matched to a paper in Crossref, so it can’t be checked.

    • cites [6] [6] J. Baez, 'Diffeomorphism invariant generalized measures on the space of connections modulo gauge transformations', hep-th/9305045, to appear in the Proceedings of the conference on quantum topology, D. Yetter (ed) (World Scientific, Singapore, 1994). - source

      Not checked in the free run.

  • May combine two claims.

    Interpretationnone given
    § 8 Discussion · p. 63

    At a sufficiently general level, the two projective families are equivalent, with subgroups of the hoop group recovered in the second picture as the fundamental groups of graphs.

    “At a sufficiently general level, the two families are equivalent, the subgroups of the hoop group being recovered in the second picture as the fundamental groups of graphs.”
    Quote in the paperNot a substantial claim
52 of 52

  • The authors state this as an assumption.

    M is an n-dimensional analytic manifold.

    “Fix an n-dimensional, analytic manifold M and a smooth principal fiber bundle B ( M,G ) with the structure group G which we will assume to be a compact and connected Lie group.”

  • The authors state this as an assumption.

    B(M,G) is a smooth principal fiber bundle with structure group G.

    “Fix an n-dimensional, analytic manifold M and a smooth principal fiber bundle B ( M,G ) with the structure group G which we will assume to be a compact and connected Lie group.”

  • The authors state this as an assumption.

    G is a compact and connected Lie group.

    “Fix an n-dimensional, analytic manifold M and a smooth principal fiber bundle B ( M,G ) with the structure group G which we will assume to be a compact and connected Lie group.”

  • The paper relies on this without support.

    A does not have a natural manifold structure.

    “The problem is that A does not have a natural manifold structure.”

  • The authors state this as an assumption.

    The gauge group G is semi-simple.

    “in the case when G is assumed to be semi-simple.”

  • The authors state this as an assumption.

    The structure group G is compact.

    “since G is compact”
    “We now wish to relax the assumption that the structure group be compact and let G be any connected Lie group.”

  • The paper relies on this without support.

    The heat kernel ρ_t on G is Ad(G) invariant.

    “(Note that, since the heat kernel ρ t on G is Ad( G ) invariant, the above formula is independent of the choice of coordinates.)”

  • The paper relies on this without support.

    The heat kernel on G is nonvanishing.

    “The proof follows from Theorem 6, and from the nonvanishing of the heat kernel on G .”

  • The paper relies on this without support.

    For non-compact G, the Wilson loop functions are unbounded.

    “since the Wilson loop functions are now unboun[216zded”

  • The authors state this as an assumption.

    The algebra of Wilson loop variables should be faithfully represented in the quantum theory.

    “Here, the basic assumption is that the algebra of Wilson loop variables should be faithfully represented in the quantum theory and it leads one directly to the completion A / G .”
10 of 10

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?

  • Over the past three years, the non-perturbative canonical approach to quantum gravity based on connections has been put on a firm mathematical footing through the development of a new functional calculus on the space of gauge equivalent connections.

    “Over the past three years, this approach has been put on a firm mathematical footing through the development of a new functional calculus on the space of gauge equivalent connections [4-11].”

    From Quantum theory of geometry: I. Area operators (1997) · cites this paper as [10]

    Split into 2 atoms — 0 confirmed, 2 partially supported, 0 disputed, 0 misattributed, 0 unverifiable. Expand the claim to see per-atom verdicts.

    Experimental — accuracy not yet measured

  • The measure ̃μ_o on A/G is selected by the reality conditions; the requirement that the corresponding operators on the quantum Hilbert space be self-adjoint selects ̃μ_o.

    “Physically, the measure ˜ µ o is selected by the so-called 'reality conditions'. More precisely, the classical phase space admits an (over)complete set of naturally defined configuration and momentum variables which are real, and the requirement that the corresponding operators on the quantum Hilbert space be self-adjoint selects for us the measure ˜ µ o [10].”

    From Quantum theory of geometry: I. Area operators (1997) · cites this paper as [10]

    The Introduction (p. 4) says that finding suitable measures is tied to making momentum operators essentially self-adjoint. Theorem 2 (p. 32) establishes self-adjointness for operators compatible with a measure, but the paper does not say that reality conditions or self-adjointness uniquely select the induced Haar measure μ̃₀. (LLM, conf=0.88)

    Experimental — accuracy not yet measured

  • The new functional calculus does not use any background fields and is therefore well-suited to a fully nonperturbative treatment.

    “This calculus does not use any background fields (such as a metric) and is therefore well-suited to a fully nonperturbative treatment.”

    From Quantum theory of geometry: I. Area operators (1997) · cites this paper as [10]

  • On H_{γ,π}, the action of gauge transformations group G is trivial and there is an orthogonal decomposition (4.23) of H̃_o; this recovers result on spin network states obtained in [9,10].

    “On H γ,⃗ π , the action of the gauge transformations group G is trivial and we have the following orthogonal decomposition of the Hilbert space of gauge invariant cylindrical functions,”

    From Quantum theory of geometry: I. Area operators (1997) · cites this paper as [10]

    Section 3.3 (p. 17) describes gauge transformations acting on graph-level connections, and the Discussion (p. 63) describes cylindrical functions on the gauge quotient. Neither passage establishes a trivial action on H_{γ,π}, the claimed orthogonal decomposition, or a spin-network result; searches for those terms found no decisive passage in our copy. (LLM, conf=0.72)

    Experimental — accuracy not yet measured

  • Every π_I is necessarily non-trivial whereas ρ_α may be trivial; in gauge invariant context [9,10], ρ_α are all trivial whence items iii and v are unnecessary.

    “It should be emphasized that every π I is necessarily non-trivial whereas ρ α may be trivial (i.e., 1-dimensional). In the gauge invariant context [9,10], ρ α are all trivial whence the items iii) and v) are unnecessary.”

    From Quantum theory of geometry: I. Area operators (1997) · cites this paper as [10]

    The cited paper discusses gauge-invariant functions in the Introduction (p. 4) and gauge-invariant vector fields in §5.3 (pp. 39–42), but those passages do not address whether ρ_α are trivial or whether items iii) and v) are unnecessary. Searches for the representation notation, trivial representations, and the claimed gauge-invariant simplification found no passage that settles the attributed point. (LLM, conf=0.72)

    Experimental — accuracy not yet measured

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.

  • Theorem 2 names the operator for Y rather than its stated vector field X

    Minor

    Theorem 2(i) assumes a compatible vector field X but states its conclusion for P(Y). The preceding definition assigns P(X) to X, while Y is introduced only in part (ii) of the theorem.

  • Cylindrical functions are alternately identified with and only dense in continuous functions

    Minor

    Section 2 says cylindrical continuous functions require completion to obtain all continuous functions. Section 4.1 instead says the two algebras are naturally isomorphic.

  • V is defined as the number of edges in a vertex-indexed gauge group

    Minor

    The construction gives one gauge-group factor per vertex, but the subsequent description of that product uses V as the number of edges. Graphs need not have equal numbers of edges and vertices.

  • The third projective family is both said to consist of manifolds and not to

    Minor

    Section 3 says every member of all three projective families is an analytic manifold. Section 3.4 says this is not true of the quotient spaces that constitute the third family.

4 of 4

Cites

Flagged first, then by citations

Research works printed but not linked: 3

  1. Ashtekar, 1987, New Hamiltonian formulation of general relativity, Phys. Rev., D36, 1587
  2. A. Ashtekar, J. Lewandowski, D. Marolf, J. Mourão and T. Thiemann, Quantization of theories of connections with local degrees of freedom, J. Math. Phys., in press.
  3. D. Marolf and J.M. Mourão, On the support of the Ashtekar-Lewandowski measure, Commun. Math. Phys., in press.

Cited by

Sorted by citations

Authors
Ashtekar, A., & Lewandowski, J.
Affiliations
  • Pennsylvania State University (US)
  • University of Warsaw (PL)
Corresponding author
Ashtekar, Abhay
ORCID
Venue
Journal of Geometry and Physics
Publisher
Elsevier BV
Published
1995-11
Volume
17
Issue
3
Pages
191-230
ISSN
0393-0440 (ISSN-L)
DOI
10.1016/0393-0440(95)00028-g
OpenAlex
W2037266697
Cited by
303
References
14
Citations by year
2026: 4 · 2025: 2 · 2024: 4 · 2023: 2 · 2022: 4 · 2021: 5 · 2020: 3 · 2019: 5 · 2018: 8 · 2017: 10 · 2016: 14 · 2015: 18 · 2014: 11 · 2013: 21 · 2012: 8
FWCI
14.02
Type
journal article
Language
en
Open access
Licence
Locations
  • Journal of Geometry and Physics (publishedVersion)
  • arXiv (Cornell University) (submittedVersion)
  • CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
  • arXiv (Cornell University)
Funders
  • National Science Foundation (93-96246, PHY91-07007)
  • Isaac Newton Institute for Mathematical Sciences
  • Pennsylvania State University
Topics
  • Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Noncommutative and Quantum Gravity Theories
  • Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Relativity and Gravitational Theory
  • Physical Sciences › Mathematics › Mathematical Physics › Advanced Operator Algebra Research
Keywords
inverse limits, gauge theory, quantum general relativity, diffeomorphism invariance, differential forms, exterior derivative, volume forms, vector fields, Lie brackets, divergence, Laplacian, heat kernel, heat kernel measure

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