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Cited by 112Noncommutative and Quantum Gravity TheoriesRelativity and Gravitational TheoryBlack Holes and Theoretical Physics

Volume and quantizations

Open access
Lewandowski, Jerzy·1997·Classical and Quantum Gravity
DOI 10.1088/0264-9381/14/1/010
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Abstract

The differential structure of the Ashtekar - Isham quantum configuration space of canonical gravity allows the expression of the operators, representing various geometrical objects, by compact analytic formulae. In this paper such a formula is presented for the Rovelli - Smolin volume operator. This operator is compared with the quantum volume defined by Ashtekar and Lewandowski and a difference is indicated.

AI snapshot

Written from the paper’s full text.

TL;DR

The paper presents a compact analytic formula for the Rovelli-Smolin volume operator in the Ashtekar-Isham graph-projective framework and compares it with the Ashtekar-Lewandowski volume operator. It shows that the two operators differ in how they treat graph intersections, with consequences for lattice volume operators and eigenvalue comparisons.

Key findings

  • The Rovelli-Smolin volume operator is expressed analytically as a sum over vertices of triple products of edge-based operators, with an overall factor \(1/(8\cdot 3!)\) and orientation-dependent coefficients; the expression is independent of the chosen graph for cylindrical functions.
  • The graph-projective volume operator depends on diffeomorphic details of intersections—for example, it vanishes on planar graphs—whereas the Rovelli-Smolin operator treats every triple of edges at a vertex equally and is preserved by all homeomorphisms.
  • At vertices with at most three edges the two operators coincide; at four-valent vertices they satisfy \(q_v = \kappa(v) q^{RS}_v\), where \(\kappa(v)\) encodes intersection characteristics. This explains why some lattice eigenvalue relations hold for a single vertex but generally fail when contributions from multiple vertices are included.
  • Loll’s first lattice volume is neither the Rovelli-Smolin nor the graph-projective operator; her modified lattice volume matches the graph-projective one after a correction, and her annihilation result for vertices with no more than three intersecting edges extends to the other volume operators.

Methods

  • Mathematical derivation within canonical quantum gravity using Ashtekar variables, cylindrical functions on \(A/G\), piecewise analytic embedded graphs, Haar measure, and elementary spin-like operators \(J\).
  • The regularizations are compared by rewriting the Rovelli-Smolin and graph-projective volume operators in a common graph-projective framework and using the Gauss constraint at three- and four-valent vertices.
  • No empirical sample or statistical analysis is used; the results are analytic operator identities and comparisons.

Limitations

  • The paper states that the physical Hilbert space is not yet known, so the volume operator’s action on physical states is determined only by duality from the kinematic Hilbert space; the volume itself is not an observable.
  • The paper explicitly defers discussion of the origins and status of the differences between regularizations to a forthcoming Ashtekar-Lewandowski paper.
  • Evident from the text: the analysis is purely formal and theoretical, with no numerical or experimental validation, and the comparison is restricted to the kinematic level and specific graph valences.

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

Integrity signals

Registry

Crossref · 10.1088/0264-9381/14/1/010

Cites

16 of 16 references verified0 citation errors · 0 not checkable

Statistics

Nothing to recompute.

Claims the paper makes

0 of 44 cited sources checked · 0 supported · 0 flagged

  • Findingnone given

    The operator q^RS_v gives equal treatment to every triple of edges meeting at a vertex, whether or not those edges are tangential.

    “the Rovelli-Smolin operator, qRS v , extracted by our decomposition, regards each triple of edges at an intersection equally, irrespective of whether they are tangential or not”
    Quote in the paperSubstantial claim
  • Findingreasoning

    For a four-valent vertex, every triple product ε_{ijk} J_{eI}^i J_{eJ}^j J_{eK}^k equals the others up to a sign.

    “That is, all the triple products, J ve I i J ve J j J ve K k ij k , are equal modulo a ±1 factor.”
    Quote in the paperSubstantial claim
  • Findingnone given

    If a vertex has three or fewer incident edges, equations (5) and (6) agree with equations (8) and (9).

    “When there are at the most three edges, say e 1 , e 2 and e 3 , at a vertex v the formulae (5) and ( 6) coincide with those of ( 8) and (9).”
    Quote in the paperSubstantial claim
  • Findingnone given

    When the graph is planar, every q_v vanishes identically.

    “For a planar graph for instance, each qv is identically zero.”
    Quote in the paperSubstantial claim
  • Findingreasoning

    Every homeomorphism leaves the Rovelli-Smolin volume vol^RS invariant.

    “Thus, the Rovelli-Smolin volume, vol RS , is preserved by all the homeomorphisms!”
    Quote in the paperSubstantial claim
  • Classified as probably not a substantial claim. 5
  • May combine two claims. 6
  • The quote only partly states this claim. 1

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

    Findingcitation

    If read literally, the definition in [16] would produce products of J operators at distinct vertices, and such products would fail gauge invariance.

    “taken literally as it stands in [16], that definition would give products of operators J vei J v e j ... at different points v = v which would not be gauge invariant”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [16] Ashtekar A, 1995

      Not checked in the free run.

  • The quote only partly states this claim.

    Findingreasoning

    The same annihilation result holds for all the other volume operators [14,16,18] when each lattice vertex has no more than three intersecting edges.

    “Remarkably, this result continues to be true for any of the remaining volume operators [14,16,18] which can be easily seen by using our formula for the Rovelli-Smolin volume.”
    Quote in the paperSubstantial claim0 of 3 sources checked
    • cites [14] source

      Not checked in the free run.

    • cites [16] Ashtekar A, 1995

      Not checked in the free run.

    • cites [18] Ashtekar A

      Not checked in the free run.

  • May combine two claims.

    Premisecitation

    The initial lattice operator from [13] uses equations (5) and (6), but its sum in (6) runs only over outgoing edges and therefore lacks the 1/8 factor.

    “The first operator published in [13] is given by equations ( 5) and ( 6), but with the sum in (6) ranging only over the outgoing edges (with respect to an orientation of the lattice) and, in consequence, without the factor 1 8 .”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [13] source

      Not checked in the free run.

  • Classified as probably not a substantial claim.

    Premisecitation

    The lattice operator has been treated as if it were a restriction of the Rovelli-Smolin volume.

    “The misunderstanding which we indicate here is that the lattice operator appears to be considered a restriction of the Rovelli-Smolin volume”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [18] Ashtekar A

      Not checked in the free run.

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

    Premisecitation

    The results of reference [18] were compared with those of references [13,16], and agreement was reported.

    “the results derived in [18] were compared therein with those of [13,16] and an agreement was found.”
    Quote in the paperSubstantial claim0 of 3 sources checked
    • cites [18] Ashtekar A

      Not checked in the free run.

    • cites [13] source

      Not checked in the free run.

    • cites [16] Ashtekar A, 1995

      Not checked in the free run.

  • May combine two claims.

    Interpretationcitation

    Loll's first lattice volume operator matches neither the Rovelli-Smolin volume operator nor the graph projective framework volume operator.

    “Our point is that the first lattice volume, that of [13], corresponds neither to the Rovelli-Smolin nor to the graph projective framework volume operator.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [13] source

      Not checked in the free run.

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

    Interpretationnone given

    The volume operator's action on physical quantum states is fixed by duality from its action on the kinematic Hilbert space L2(A/G).

    “In any case, the action of the volume operator, which we are considering below, on physical quantum states will be determined-by the duality-by its action in the kinematic Hilbert space, L 2 (A/G).”
    Quote in the paperNot a substantial claim
    • Rests on an assumption: The eventual Hilbert space of physical quantum states is not yet known.

  • May combine two claims.

    Premisenone given

    On any fixed graph γ, the operators J^a_{veI} admit an algebraic interpretation as standard spin operators, with the given commutation relations and with each edge representing a separate degree of freedom subject to a constraint.

    “For every fixed graph γ the operators J ve I i can be thought of, algebraically, as the familiar spin operators, [J ve I i , J v e J j ] = iδ vv δ I J ij k J ve I k , edges labeling different degrees of freedom with a constraint”
    Quote in the paperSubstantial claim

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

    Premisecitation

    States analogous to those that diagonalize the spin operators were introduced in the L2(A/G) context and are known as spin-network functions.

    “The analogue of states used to diagonalize the spin operators was also introduced in the context of the L 2 (A/G); they are called the spin-network functions [9,21,22].”
    Quote in the paperNot a substantial claim0 of 3 sources checked
    • cites [9] source

      Not checked in the free run.

    • cites [21] Baez J, 1994, Lett. Math. Phys. source

      Not checked in the free run.

    • cites [22] Baez J, 1994

      Not checked in the free run.

  • May combine two claims.

    Interpretationnone given

    For a cylindrical function, summing over all vertices v and over all triples of distinct J operators at each v yields a finite number of nonzero terms that match the earlier sums, giving explicitly graph-independent expressions.

    “To have explicitly graph independent expressions, one can sum with respect to all the points v of in (5) and(8) with respect to all the triples of different J operators at each v; given a cylindrical function, the resulting sums would consist of a finite number of nonvanishing terms coinciding with the sums above.”
    Quote in the paperSubstantial claim
10 of 10

  • The paper relies on this without support.

    The volume functional for a region, or for the whole space, is not an observable.

    “In fact, the volume functional assigned to a region in or even to itself is not an observable.”

  • The paper relies on this without support.

    The eventual Hilbert space of physical quantum states is not yet known.

    “At this point we should recall that the eventual Hilbert space of physical quantum states is not yet known.”

  • The paper relies on this without support.

    For every edge e and its two endpoints v and v', the relation J_ve^j J_ve^j = J_ve'^j J_ve'^j holds.

    “which holds for every edge e and its two ends v, v .”

  • The paper relies on this without support.

    For a fixed graph γ, any operator that is a finite linear combination of products of spin operators J_ve^I_k with constant coefficients has a discrete spectrum.

    “For a fixed graph γ , any operator which can be expressed as a finite linear combination of products of the 'spin operators', J ve I k , with constant coefficients has a discrete spectrum that is well known to every physicist.”
4 of 4

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?

  • Spin foams also arise naturally in lattice gauge theory, where the problems of canonical quantum gravity are absent.

    “It is reassuring therefore that spin foams also arise naturally in lattice gauge theory, where these problems are absent.”

    From Spin foam models (1998) · cites this paper as [32]

  • A path integral in lattice gauge theory can be represented as a sum over spin foams living in the lattice.

    “The basic idea, due to Iwasaki [32] and Reisenberger [48], is that a path integral in lattice gauge theory can be represented as a sum over spin foams living in the lattice.”

    From Spin foam models (1998) · cites this paper as [32]

  • The discreteness of the spectra of area and volume is a central result in loop quantum gravity, first obtained by Rovelli and Smolin in [20], and later confirmed and clarified by a number of authors [21-25].

    “The discreteness of the spectra of area and volume -and the 'quantized' structure of space that these spectra suggests- is a central result in loop quantum gravity, first obtained by Rovelli and Smolin in [20], and later confirmed and clarified by a number of authors [21-25].”

    From “Sum over surfaces” form of loop quantum gravity (1997) · cites this paper as [25]

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.

  • Citation [16] does not match the attributed lattice work

    Minor

    The discussion presents [16] as a source for a Loll lattice volume operator, whereas reference [16] is attributed to Ashtekar.

  • Citation [17] names different authors

    Minor

    The text identifies [17] as a forthcoming paper by Ashtekar and Lewandowski, but the reference list assigns [17] to Loll.

  • Citation [18] names different authors

    Minor

    The text credits [18] to De Pietri and Rovelli, while the reference list assigns [18] to Ashtekar and Lewandowski.

3 of 3

Cites

Flagged first, then by citations

Research works printed but not linked: 2

  1. Ashtekar A, 1995, Phys. Rev., 53, 2865
  2. Rovelli C, 1995, Phys. Rev., 53, 5743

Cited by

Sorted by citations

Authors
Lewandowski, J.
Affiliations
University of Warsaw (PL)
Corresponding author
Lewandowski, Jerzy
ORCID
Lewandowski, Jerzy
Venue
Classical and Quantum Gravity
Publisher
IOP Publishing
Published
1997-01-01
Volume
14
Issue
1
Pages
71-76
ISSN
  • 0264-9381 (ISSN-L)
  • 1361-6382
DOI
10.1088/0264-9381/14/1/010
OpenAlex
W2021621923
Semantic Scholar
4789e18ee110d4ddcaa57b723f44765c21deb9d6
Cited by
112
References
16
Citations by year
2026: 1 · 2024: 3 · 2023: 2 · 2021: 2 · 2020: 3 · 2019: 2 · 2018: 2 · 2017: 3 · 2016: 1 · 2015: 4 · 2014: 7 · 2013: 2 · 2012: 6
FWCI
12.87
Type
journal article
Language
en
Open access
Locations
  • Classical and Quantum Gravity (publishedVersion)
  • arXiv (Cornell University) (submittedVersion)
  • CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
  • arXiv (Cornell 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 › Physics and Astronomy › Nuclear and High Energy Physics › Black Holes and Theoretical Physics
Keywords
volume operator, canonical quantum gravity, quantum geometry

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