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Cited by 96Noncommutative and Quantum Gravity TheoriesRelativity and Gravitational TheoryAdvanced Operator Algebra Research

Are the spectra of geometrical operators in loop quantum gravity really discrete?

Open access
Dittrich, Bianca, Thiemann, Thomas·2009·Journal of Mathematical Physics
DOI 10.1063/1.3054277
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Abstract

One of the celebrated results of loop quantum gravity (LQG) is the discreteness of the spectrum of geometrical operators such as length, area, and volume operators. This is an indication that the Planck scale geometry in LQG is discontinuous rather than smooth. However, there is no rigorous proof thereof at present. Because the aforementioned operators are not gauge invariant, they do not commute with the quantum constraints. The relational formalism in the incarnation of Rovelli’s partial and complete observables provides a possible mechanism for turning a non-gauge-invariant operator into a gauge invariant one. In this paper we investigate whether the spectrum of such a physical, that is, gauge invariant, observable can be predicted from the spectrum of the corresponding gauge variant observables. We will not do this in full LQG but rather consider much simpler examples where field theoretical complications are absent. We find, even in those simpler cases, that kinematical discreteness of the spectrum does not necessarily survive at the gauge invariant level. Whether or not this happens depends crucially on how the gauge invariant completion is performed. This indicates that “fundamental discreteness at the Planck scale in LQG” is far from established. To prove it, one must provide the detailed construction of gauge invariant versions of geometrical operators.

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Crossref · 10.1063/1.3054277

Cites

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Claims the paper makes

0 of 9 cited sources checked · 0 supported · 0 flagged

  • Findingreasoning

    For the momentum constraint C_c = p1, the quantized constraint has continuous spectrum.

    “We will work with the momentum constraint C c = p 1 , whose quantization C ˆc =-iប‫ץ‬ 1 has a continuous spectrum.”
    Quote in the paperSubstantial claim
  • Findingreasoning

    In the (d,d,d,d) example with T=x1^2+p1^2 and f=x2^2+p2^2, the complete observable F(τ)=x1^2+x2^2+p1^2+p2^2 - τ has discrete spectrum.

    “and find F͑͒ = x 1 2 + x 2 2 + p 1 2 + p 2 2 -, ͑3.6͒ which is the sum of two harmonic oscillator Hamiltonians minus a constant .The corresponding quantum operator has a discrete spectrum.”
    Quote in the paperSubstantial claim
  • Findingreasoning

    In the two-constraint example with C1=p1, C2=p3, T1=x1^2, T2=x3^2, the complete observable F(λ1,λ2) has a discrete spectrum.

    “Hence, we have an example with two constraints and two clock variables with a continuous spectrum and a partial observable f and a complete observable F͑ 1 , 2 ͒ with a discrete spectrum.”
    Quote in the paperSubstantial claim
  • Findingown data

    The paper's examples realize every possible combination of discrete or continuous spectra for the constraint, clock, partial observable, and complete observable.

    “In Sec.III we will show with the help of concrete examples that every possible combination of continuous and discrete spectra for the constraint, the partial observable, the clock, and the complete observable can be realized.”
    Quote in the paperSubstantial claim
  • Findingown data

    In the Appendix B toy model, a kinematically discrete spectrum becomes a gauge-invariant continuous one, contrary to arguments in Ref. 2.

    “We find that in contrast to the arguments spelled out there, the kinematically discrete spectrum switches to a gauge invariant continuous one.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites Ref. 2

      Not checked in the free run.

  • May combine two claims. 15
  • Classified as probably not a substantial claim. 9

  • May combine two claims.

    Findingnone given

    For the rotation constraint, the gauge-invariant functions r^2 = x1^2+x2^2 and h = x1^2+x2^2+p1^2+p2^2 have continuous and discrete spectra, respectively.

    “The gauge invariant functions we will encounter in our examples of complete observables are given by r 2 ª x 1 2 + x 2 2 and h ª x 1 2 + x 2 2 + p 1 2 + p 2 2 .The corresponding quantizations have a continuous spectrum and a discrete spectrum, respectively.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning

    In the one-constraint model with clock T = x1^2 + γ x2^2, the complete observable F has a discrete spectrum for 0 ≤ γ < 1 and a continuous spectrum for γ ≥ 1.

    “with 0 Յ ␥ Ͻ 1 or ␥ Ն 1 for a discrete spectrum of the complete observable or a continuous spectrum of the complete observable, respectively.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning

    For the relativistic particle, the partial observable f = x0 p1 - x1 p0 has discrete spectrum, while the corresponding complete observable F(τ=0) = -x0 p1 + x1 p0 has continuous spectrum.

    “With the relativistic particle we already have an example where the spectrum of the quantization of the partial observable f = x 0 p 1x 1 p 0 is discrete ͓since it generates rotations in the ͑x 0 , x 1 ͒ plane͔ and the spectrum of the quantization of the corresponding complete observable F͑ =0͒ = -x 0 p 1x 1 p 0 is continuous ͑since it generates boosts͒.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning

    For the momentum constraint, the Dirac observable algebra is spanned by x2 and p2; physical wave functions are independent of x1, and the physical Hilbert space is L2(R, dx2).

    “The Dirac observable algebra for this constraint is spanned by x 2 and p 2 , i.e., it is given by phase space functions which do not depend on x 1 or p 1 .Physical wave functions are wave functions which do not depend on x 1 .”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingown data

    In simple finite-dimensional examples, the spectrum of a complete observable is not determined by the spectra of the partial observable and clock; kinematical discreteness need not persist after gauge-invariant completion.

    “We find, even in those simpler cases, that kinematical discreteness of the spectrum does not necessarily survive at the gauge invariant level.”
    • Also stated in: § I. INTRODUCTION · boosted
      “Contrary to this expectation, we will show that in this note, without further assumptions, it is impossible to make any predictions about the spectra of the complete observables, even if the spectra of the partial observables and the clocks are known.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingnone given

    The constraint Cˆd has a pure point spectrum that includes zero, so there are solutions to Cˆd ψ_phys = 0 in the kinematical Hilbert space.

    “The constraint has a pure point spectrum and, moreover, the point zero is included in this spectrum so that we have solutions phys to the constraint equation C ˆd phys = 0 that are elements of the kinematical Hilbert space.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingown data

    The cylinder example satisfies both properties missing in earlier models (compact configuration space and f,T Poisson commuting) and still the switch occurs.

    “Thus, we now explicitly display an example where both of the above-mentioned properties that are missing in our models are satisfied and still the switch occurs.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingnone given

    The representation of the Dirac observable algebra (A1) is unitary and irreducible.

    “The representation ͑A4͒ of the Dirac observable algebra ͑A1͒ is unitary and irreducible.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning

    The flux operator Eˆ has a discrete spectrum taking integer values with eigenfunctions T_n = h^n.

    “The spectrum of E ˆis discrete and takes integer values with eigenfunctions given by T n = h n ͑“spin network functions”͒.”
    Quote in the paperSubstantial claim

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

    Interpretationcitation

    A complete observable may fail to be well defined: the surface T=τ can intersect a gauge orbit more than once, yielding multiple solutions and a multiple-valued complete observable.

    “We want to remark that the complete observable might not always be well defined.For instance, the surface T = might meet a gauge orbit several times which leads to multiple solutions of the equation T͑t͒= ! and hence might lead to multiple valued complete observables as is discussed in the first reference in Ref. 8.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites first reference in Ref. 8

      Not checked in the free run.

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

    Premisecitation

    In loop quantum gravity, spatial geometrical operators such as area, volume, and length have been constructed on the kinematical Hilbert space and possess discrete spectra.

    “One of the celebrated results of loop quantum gravity ͑LQG͒ is the discreteness of the spectrum of geometrical operators such as length, area, and volume operators.”
    • Also stated in: § I. INTRODUCTION · plain
      “One of the successes of LQG is the rigorous construction of ͑spatial͒ geometrical operators, such as the area, the volume, and the length operator, 2,3 on the so-called kinematical Hilbert space of LQG. 4 Moreover, it turns out that these geometrical operators have a discrete spectrum.”
    Quote in the paperSubstantial claim0 of 2 sources checked
    • cites [2] source

      Not checked in the free run.

    • cites [3]

      Not checked in the free run.

  • Classified as probably not a substantial claim.

    Interpretationreasoning

    To prove fundamental discreteness at the Planck scale in loop quantum gravity, one must provide a detailed construction of gauge-invariant versions of geometrical operators.

    “To prove it, one must provide the detailed construction of gauge invariant versions of geometrical operators.”
    Quote in the paperSubstantial claim

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

    Interpretationnone given

    The examples are simple and omit complications expected for geometrical operators in LQG.

    “The examples considered here are very simple and do not include further complications one would expect for the geometrical operators in LQG.”
    Quote in the paperNot a substantial claim

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

    Interpretationnone given

    In more complicated field-theory examples, the physical Hilbert space may not be unique and observable spectra could depend on the choice of Hilbert space.

    “In more complicated ͑field theory͒ examples, even the physical Hilbert space might not be unique and the spectra of observables could, in principle, depend on the choice of the Hilbert space.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Premisecitation

    Under a good gauge fixing T=τ, the space of complete observables with clock T is symplectomorphic to the reduced phase space equipped with the Dirac bracket, and the map sends the complete observable associated with f to the gauge restriction of f.

    “The space of all complete observables with this choice of clock variable T can be mapped via a symplectomorphism onto the reduced phase space obtained with the gauge fixing T = and equipped with the Dirac bracket ͕•,•͖ D .”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites Refs.8 and 9

      Not checked in the free run.

    • The authors state this as an assumption: The equation T=τ defines a good gauge fixing: the surface T=τ meets each gauge orbit exactly once.

  • May combine two claims.

    Premisereasoning

    The geometrical operators constructed so far are gauge dependent, not invariant under spacetime diffeomorphisms, so their discreteness is only a kinematical result.

    “However, so far the discreteness of their spectrum is a result at the kinematical level only.By “kinematical level” we mean that the geometrical operators defined so far are gauge dependent, i.e., they are not invariant under space-time diffeomorphisms.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning

    Whether the orbits generated by f are compact cannot generally be decided from local considerations alone; it can change under projection to the reduced phase space, and global properties of the reduced phase space and its embedding matter.

    “Now to determine whether the orbits generated by the vector field f are compact or not cannot, in general, be concluded from local considerations only.It might change under the projectionfoot_2 onto the reduced phase space.Also, the global properties of the reduced phase space and how the reduced phase space is embedded into the full phase space are important.”
    Quote in the paperSubstantial claim

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

    Interpretationreasoning

    Physical observables might have very unexpected properties.

    “As a more general lesson these examples show that physical observables might have very unexpected properties.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S2 (checkable).

    Interpretationnone given

    It is hard to get complete control over the reduced phase space of general relativity.

    “However, it is very hard to get complete control over the reduced phase space of general relativity.”
    Quote in the paperNot a substantial claim

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

    Premisecitation

    In Ref. 17, there is an example based on sl(2,R) where a continuous kinematical spectrum switches to a discrete gauge invariant one.

    “In Ref. 17 we find an example based on sl͑2,R͒ where a continuous kinematical spectrum switches to a discrete gauge invariant one.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites Ref. 17

      Not checked in the free run.

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

    Premisecitation

    The Bohr-Sommerfeld quantization rule is a heuristic method connecting energy gaps to properties of phase space orbits generated by the Hamiltonian.

    “For instance, the Bohr-Sommerfeld quantization rule ͑Ref.14͒ applicable to systems with periodic motion is a heuristic method which connects the energy gaps to properties of the phase space orbits as generated by the Hamiltonian.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites Ref.14

      Not checked in the free run.

  • May combine two claims.

    Premisecitation

    In Ref. 17, the configuration space is noncompact and the switch happens in the wrong direction.

    “However, there the configuration space is still noncompact and, furthermore, the switch happens in the wrong direction.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites Ref. 17

      Not checked in the free run.

22 of 22

  • The authors state this as an assumption.

    The parameter τ is chosen so that the complete observable is defined at least on an open set of phase space; in the examples it can be analytically continued to the whole phase space.

    “Here we will assume that the parameter is always chosen such that one can define the complete observable at least on an open set of the phase space.In our examples this complete observable can always be continued analytically to a gauge invariant function defined on the whole phase space.”

  • The authors state this as an assumption.

    The partial and complete observables have either entirely discrete or entirely continuous spectra, and the spectral type of the complete observable does not depend on the parameter τ.

    “We will assume that the partial and complete observables have either an entirely discrete ͑i.e., pure point͒ spectrum or an entirely continuous ͑i.e., absolutely continuous͒ spectrum and that the spectral type of the complete observable does not depend on the particular value of the parameter .”

  • The authors state this as an assumption.

    Dirac observables constructed from matter fields can be quantized as self-adjoint operators on a physical Hilbert space that is yet to be constructed.

    “assuming that such Dirac observables can be quantized as self-adjoint operators on a yet to be constructed physical Hilbert space”

  • The authors state this as an assumption.

    The equation T=τ defines a good gauge fixing: the surface T=τ meets each gauge orbit exactly once.

    “Assume that the equation T = defines a good gauge fixing, i.e., that the surface T = meets each gauge orbit once and only once.”
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?

  • At the level of gauge-invariant observables, some discrete features might vanish.

    “on the other, at the level of gauge-invariant observables, some discrete aspects could disappear [154]”

    From The Spin-Foam Approach to Quantum Gravity (2013) · cites this paper as [154]

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.

  • Physical inner product omits the wrong coordinate

    Minor

    Physical wave functions are independent of x1, so constructing an inner product on the stated physical Hilbert space requires omitting integration over x1, not x2.

1 of 1

Cites

Flagged first, then by citations

Research works printed but not linked: 14

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  3. 2007, Introduction to Modern Canonical Quantum General Relativity
  4. 1998, Adv. Theor. Math. Phys., 1, 388
  5. C. Fleischhack , “Representations of the Weyl algebra in quantum geometry,” e-print arXiv:math-ph/0407006v2; · arXiv
  6. 2006, Gen. Relativ. Gravit., 23, 6155
  7. 2007, Class. Quantum Grav., 24, 4543, 10.1088/0264-9381/24/4/001 · DOI
  8. B. Dittrich and J.Tambornino e-print arXiv:gr-qc/0610060; · arXiv
  9. B. Dittrich and J. Tamborninoe-print arXiv:gr-qc/0702093. · arXiv
  10. Ashtekar, 1991, Conceptional Problems in Quantum Gravity, 126
  11. 1961, Quantum Mechanics
  12. R. Brunetti and K.Fredenhagen , “When does a detector click?,” e-print arXiv:quant-ph/0103144; · arXiv
  13. e-print arXiv:quant-ph/0207048; · arXiv
  14. Dobrev, 2006, Quantum Theory and Symmetries IV

Cited by

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Related

Authors
Dittrich, B., & Thiemann, T.
Affiliations
Perimeter Institute (CA)
Venue
Journal of Mathematical Physics
Publisher
AIP Publishing
Published
2009-01-01
Volume
50
Issue
1
ISSN
  • 0022-2488 (ISSN-L)
  • 1089-7658
DOI
10.1063/1.3054277
OpenAlex
W2167002495
Cited by
96
References
43
Citations by year
2026: 3 · 2025: 16 · 2023: 2 · 2022: 18 · 2021: 6 · 2020: 3 · 2019: 1 · 2018: 2 · 2017: 5 · 2016: 6 · 2015: 3 · 2014: 7 · 2013: 5 · 2012: 9
FWCI
4.36
Type
journal article
Language
en
Open access
Locations
  • Journal of Mathematical Physics (publishedVersion)
  • arXiv (Cornell University) (submittedVersion)
  • Max Planck Institute for Plasma Physics (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 › Mathematics › Mathematical Physics › Advanced Operator Algebra Research
Keywords
loop quantum gravity, area operator, volume operator, gauge invariance, Planck scale discreteness

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