pith. sign in

arxiv: 2606.28788 · v1 · pith:AEEUBRW4new · submitted 2026-06-27 · 🌀 gr-qc · hep-th

Non-perturbative, background independent canonical quantum gravity in Fock representations

Thomas Thiemann This is my paper

Pith reviewed 2026-06-30 09:04 UTC · model grok-4.3

classification 🌀 gr-qc hep-th
keywords canonical quantum gravityFock representationsbackground independenceloop quantum gravityWheeler-DeWitt constraintmatter fieldsHilbert space separabilityconstraint quantisation
0
0 comments X

The pith

Background independent Fock representations exist for non-perturbative quantum gravity when matter fields accompany geometry.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that Fock-type representations can be realized in a background-independent manner inside the canonical, non-perturbative formulation of quantum gravity. Suitable matter fields must be present; their excitations become entangled with geometry excitations already at the level of the Fock vacuum. The construction proceeds by direct application of the constraint quantisation method. The resulting Hilbert space is separable, which permits the Hamiltonian constraint to be treated as a densely defined quadratic form.

Core claim

There exist background independent representations of Fock type within the manifestly non-perturbative, canonical approach to quantum gravity. Mandatory for their existence is the presence of suitable matter fields next to the geometry field. In particular, the excitations of the corresponding Fock vacuum necessarily entangle matter and geometry. The Fock quantum gravity Hilbert space, in contrast to the loop quantum gravity Hilbert space, is separable, allowing the Hamiltonian constraint to be implemented as a densely defined quadratic form.

What carries the argument

Constraint quantisation method applied to the canonical variables of gravity plus matter, producing background-independent Fock representations whose vacuum entangles the two sectors.

If this is right

  • The Fock quantum gravity Hilbert space is separable.
  • The Hamiltonian constraint can be implemented as a densely defined quadratic form.
  • Excitations of the Fock vacuum necessarily entangle matter and geometry.
  • The Fock incarnation differs from the loop quantum gravity incarnation primarily through separability of the Hilbert space.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The separability of the Hilbert space may allow standard techniques from quantum field theory to be applied more directly to the constraint operators.
  • The requirement that matter fields be present suggests that pure-gravity Fock representations may not exist within this construction.
  • The entanglement between matter and geometry at the vacuum level could alter the interpretation of semiclassical states compared with decoupled treatments.

Load-bearing premise

The constraint quantisation method can be applied directly to produce these Fock representations while preserving background independence.

What would settle it

An explicit demonstration that the Fock states obtained from the constraint quantisation procedure are not background independent or that no choice of matter fields yields a consistent representation.

read the original abstract

It is commonly believed that a quantum field theory of General Relativity requires a non-perturbative formulation. In addition, the background independence of classical General Relativity supplies a physical selection criterion for suitable Hilbert space representations of the corresponding quantum field theory. In this contribution we show that there exist background independent representations of Fock type within the manifestly non-perturbative, canonical approach to quantum gravity. Mandatory for their existence is the presence of suitable matter fields next to the geometry field. In particular, the excitations of the corresponding Fock vacuum necessarily entangles matter and geometry. In this article we use the constraint quantisation method. We compare the resulting Fock incarnation of background independent, non-perturbative canonical quantum gravity with the well known Loop quantum gravity incarnation. One of the most important differences is that the Fock quantum gravity (FQG) Hilbert space, in contrast to the Loop quantum gravity (LQG) Hilbert space, is separable. This has many advantages when attempting to implement the Hamiltonian constraint, also known as Wheeler-DeWitt constraint, as a densely defined quadratic form.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript claims that background-independent Fock-type representations exist within the manifestly non-perturbative canonical approach to quantum gravity, provided suitable matter fields are included alongside the geometry; these representations are obtained via constraint quantization, yield a Fock vacuum that necessarily entangles matter and geometry, and produce a separable Hilbert space (in contrast to the non-separable LQG space) that facilitates implementing the Hamiltonian (Wheeler-DeWitt) constraint as a densely defined quadratic form.

Significance. If the claimed construction is valid, the result would supply a separable Hilbert space for background-independent quantum gravity that avoids the technical obstacles LQG encounters when promoting the Hamiltonian constraint to an operator, while preserving the non-perturbative character of the theory. The separability and the entanglement between matter and geometry are presented as concrete advantages over existing approaches.

major comments (2)
  1. [Abstract] Abstract, paragraph 3: the assertion that 'the constraint quantisation method' directly produces a Fock representation whose vacuum and operators remain invariant under the diffeomorphism constraint without selecting a background metric is the load-bearing step; no explicit construction of the positive-frequency modes, the inner product, or the resulting creation/annihilation operators is supplied, leaving open the possibility that background dependence enters through the definition of the Fock vacuum.
  2. [Abstract] Abstract, final paragraph: the claim that the FQG Hilbert space is separable and thereby advantageous for the Hamiltonian constraint is presented as a key distinction from LQG, yet without a concrete definition of the Fock space or verification that the constraint operators are densely defined quadratic forms, the advantage cannot be assessed.
minor comments (1)
  1. [Abstract] The abstract refers to 'suitable matter fields' without specifying which fields or their coupling; a brief indication of the minimal matter content required would improve clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting these important points regarding the presentation of our results. We provide point-by-point responses below.

read point-by-point responses

  1. Referee: [Abstract] Abstract, paragraph 3: the assertion that 'the constraint quantisation method' directly produces a Fock representation whose vacuum and operators remain invariant under the diffeomorphism constraint without selecting a background metric is the load-bearing step; no explicit construction of the positive-frequency modes, the inner product, or the resulting creation/annihilation operators is supplied, leaving open the possibility that background dependence enters through the definition of the Fock vacuum.

    Authors: The constraint quantization procedure is the mechanism that enforces diffeomorphism invariance at the level of the representation without introducing a background metric; the resulting Fock vacuum is defined such that its excitations entangle matter and geometry precisely because the modes are selected to satisfy the constraints. We agree that the abstract does not contain the explicit construction of the positive-frequency modes, inner product, or creation/annihilation operators. In the revised manuscript we will add a concise outline of these elements, drawn from the constraint quantization steps already used in the body of the paper, to make the background independence explicit. revision: yes

  2. Referee: [Abstract] Abstract, final paragraph: the claim that the FQG Hilbert space is separable and thereby advantageous for the Hamiltonian constraint is presented as a key distinction from LQG, yet without a concrete definition of the Fock space or verification that the constraint operators are densely defined quadratic forms, the advantage cannot be assessed.

    Authors: Separability follows directly from the Fock construction over the entangled vacuum once suitable matter fields are included; this is the structural difference from the non-separable LQG space. We concur that the abstract states the advantage without supplying the concrete Fock-space definition or the verification that the Hamiltonian constraint acts as a densely defined quadratic form. The revised manuscript will include an explicit definition of the Fock space together with the argument establishing that the constraint operators are densely defined quadratic forms. revision: yes

Circularity Check

0 steps flagged

Derivation chain self-contained; no circular reductions identified

full rationale

The abstract asserts existence of background-independent Fock representations via constraint quantisation in the presence of matter, with the resulting Hilbert space being separable. No equations, fitted parameters, or self-citations are exhibited that reduce any claimed result to its own inputs by construction. The central claim is presented as a direct consequence of applying the quantisation method, without visible renaming, smuggling of ansatze, or load-bearing self-citations that would force the outcome. This is the normal case of a self-contained existence argument.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no information on free parameters, axioms, or invented entities.

pith-pipeline@v0.9.1-grok · 5713 in / 981 out tokens · 32413 ms · 2026-06-30T09:04:45.157638+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

54 extracted references · 28 canonical work pages · 19 internal anchors

  1. [1]

    Goroff, A

    M.H. Goroff, A. Sagnotti, Phys. Lett.B160(1985) 81; Nucl. Phys.B266(1986) 709

    1985

  2. [2]

    Percacci

    R. Percacci. An introduction to covariant quantum gravity and asymptotic safety. World Scientific, Singapore, 2017. M. Reuter, F. Saueressig. Quantum gravity and the functional renormalization group. Cambridge mono- graphs on mathematical physics, Cambridge, 2019

    2017

  3. [3]

    S. Surya. The causal set approach to quantum gravity. Living reviews in relativity22(2019) 1. arXiv:1903.11544 [gr-qc] F. Dowker. Introduction to causal sets and their phenomenology. Gen. Rel. Grav.45(2013) 9, 1651-1667

    work page arXiv 2019

  4. [4]

    R. Loll. Quantum gravity from causal dynamical triangulations: a review. Class. Quantum Grav.37 013002. e-Print: 1905.08669 [hep-th] J. Ambjorn, A. Goerlich, J. Jurkiewicz, R. Loll. Causal dynamical triangulations and the search for a theory of quantum gravity. In: Proceedings of MG13. R. T. Jantzen, K. Rosquist, R. Ruffini. 120-137. World scientific, Sin...

    work page arXiv 1905

  5. [5]

    Quantum Gravity

    C. Rovelli, “Quantum Gravity”, Cambridge University Press, Cambridge, 2004. T. Thiemann, “Modern Canonical Quantum General Relativity”, Cambridge University Press, Cam- bridge, 2007 J. Pullin, R. Gambini, “A first course in Loop Quantum Gravity”, Oxford University Press, New York, 2011 C. Rovelli, F. Vidotto, “Covariant Loop Quantum Gravity”, Cambridge Un...

    2004

  6. [6]

    H. Nstase. introduction to the AdS/CFT correspondence. Cambridge university Press, Cambridge, 2015

    2015

  7. [7]

    Rev.73(1948) 1092; Rev

    P.A.M Dirac, Phys. Rev.73(1948) 1092; Rev. Mod. Phys.21(1949) 392

    1948

  8. [8]

    The Coordinate Group Symmetries of General Relativity

    P. G. Bergmann, A. Komar, “The Coordinate Group Symmetries of General Relativity”, Int. J. Theor. Phys.5(1972) 15 P. G. Bergmann, A. Komar. “The Phase Space Formulation of General Relativity and Approaches Towards its Canonical Quantization”, Gen. Rel. Grav.,1(1981) 227-254

    1972

  9. [9]

    R. L. Arnowitt, S. Deser, C. W. Misner. Canonical variables for general relativity. Phys. Rev.117 (1960) 1595-1602

    1960

  10. [10]

    B. S. DeWitt, Phys. Rev.160(1967) 1113; Phys. Rev.162(1967) 1195; Phys. Rev.162(1967) 1239

    1967

  11. [11]

    Geometrodynamics

    J. A. Wheeler, “Geometrodynamics”, Academic Press, New York, 1962

    1962

  12. [12]

    New Variables for Classical and Quantum Gravity

    A. Ashtekar, “New Variables for Classical and Quantum Gravity” Phys. Rev. Lett.57(1986) 2244-2247

    1986

  13. [13]

    A real polynomial formulation of general relativity in terms of connections

    J. F. G. Barbero, “A real polynomial formulation of general relativity in terms of connections”, Phys. Rev.D49(1994) 6935-6938

    1994

  14. [14]

    Operator Algebras and Quantum Statistical Mechanics

    O. Bratteli, D. W. Robinson, “Operator Algebras and Quantum Statistical Mechanics”, vol. 1,2, Springer Verlag, Berlin, 1997

    1997

  15. [15]

    Local Quantum Physics

    R. Haag, “Local Quantum Physics”, Springer Verlag, Berlin, 1984

    1984

  16. [16]

    Thiemann

    T. Thiemann. Observations on representations of the spatial diffeomorphism group and algebra in all dimensions Phys. Rev.D110(2024) 12, 124022. e-Print: 2405.01201 [gr-qc]

    work page arXiv 2024

  17. [17]

    Thiemann

    T. Thiemann. Non-perturbative quantum gravity in Fock representations Phys. Rev. D110(2024) 12, 124023. e-Print: 2405.01212 [gr-qc]

    work page arXiv 2024

  18. [18]

    Geometrodynamics Regained

    S. A. Hojman, K. Kuchar, C. Teitelboim, “Geometrodynamics Regained”, Annals Phys.96(1976) 88-135 38

    1976

  19. [19]

    R. P. Woodard. Ostrogradsky’s theorem on Hamiltonian instability. Scholarpedia 10 (2015) 8, 32243. e-Print: 1506.02210 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  20. [20]

    Giesel, T

    K. Giesel, T. Thiemann. Hamiltonian theory: dynamics. In: Handbook of Quantum Gravity. C. Bambi, L. Modesto, I. Shapiro (eds.), Springer Verlag, Berlin, 2023

    2023

  21. [21]

    Scalar Material Reference Systems and Loop Quantum Gravity

    K. Giesel, T. Thiemann, “Scalar Material Reference Systems and Loop Quantum Gravity”, Class. Quant. Grav.32(2015) 135015, [arXiv:1206.3807]

    work page arXiv 2015

  22. [22]

    Thiemann

    T. Thiemann. Canonical Quantum Gravity, Constructive QFT, and Renormalisation. Front. in Phys. 8(2020) 548232, Front.in Phys.0(2020) 457. e-Print: 2003.13622 [gr-qc]

    work page arXiv 2020

  23. [23]

    Thiemann Non-degenerate metrics, hypersurface deformation algebra, non-anomalous representa- tions and density weights in quantum gravity

    T. Thiemann Non-degenerate metrics, hypersurface deformation algebra, non-anomalous representa- tions and density weights in quantum gravity. Gen. Rel. Grav.56(2024) 10, 122. e-Print: 2207.08299 [gr-qc]

    work page arXiv 2024

  24. [24]

    Thiemann

    T. Thiemann. Quantum gravity in the triangular gauge. e-Print: 2305.06724 [gr-qc] T. Lang, S. Schander. Quantum Geometrodynamics Revived I. Classical Constraint Algebra. e-Print: 2305.10097 [gr-qc] II. Hilbert Space of Positive Definite Metrics. 2305.09650 [gr-qc]

    work page arXiv

  25. [25]

    Quantum Spin Dynamics VIII. The Master Constraint

    T. Thiemann, “Quantum spin dynamics. VIII. The Master constraint”, Class. Quant. Grav.23(2006) 2249-2266, [gr-qc/0510011]

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  26. [26]

    D. A. Dubin, M. A. Hennings, T B Smith. Mathematical Aspects of Weyl Quantization and Phase. World Scientific, Singapore, 2000

    2000

  27. [27]

    G. I. Sharygin. Lectures on Deformation Quantisation. Peking University Series in Mathematics: Vol- ume 8. World Scientific, Singapore, 2025

    2025

  28. [28]

    Introduction to representations of the canonical commutation and anticommutation relations

    J. Derezinski. Introduction to Representations of the Canonical Commutation and Anticommutation Relations. Lecture Notes Physics 695 (2006) 63-143. eprint: [math-ph/0511030]

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  29. [29]

    Thiemann

    T. Thiemann. Non-perturbative, background independent Fock representations for canonical quantum gravity. Companion paper - Jurek memorial article

  30. [30]

    Generalised Functions

    I. M. Gel’fand and N. Ya. Vilenkin, “Generalised Functions”, vol. 4: Applications of Harmonic Analysis, Academic Press, New York, London, 1964

    1964

  31. [31]

    On the Generality of Refined Algebraic Quantization

    D. Marolf, D. Giulini. On the generality of refined algebraic quantization. Class. Quant. Grav.16(1999) 2479-2488; e-Print: gr-qc/9812024 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 1999

  32. [32]

    The domain of dependence

    R. Geroch, “The domain of dependence”, Journ. Math. Phys.11(1970), 437 - 509

    1970

  33. [33]

    On smooth Cauchy hypersurfaces and Geroch's splitting theorem

    A. N. Bernal, M. Sanchez, “On Smooth Cauchy hypersurfaces and Geroch’s splitting theorem” Com- mun. Math. Phys.243(2003) 461-470, [gr-qc/0306108]

    work page internal anchor Pith review Pith/arXiv arXiv 2003

  34. [34]

    Towards Loop Quantum Supergravity (LQSG)

    N. Bodendorfer, A. Thurn, T. Thiemann. Towards Loop Quantum Supergravity (LQSG). Phys. Lett. B 711(2012) 205-211. e-Print: 1106.1103 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  35. [35]

    Representations of the Weyl Algebra in Quantum Geometry

    C. Fleischhack, “Representations of the Weyl algebra in quantum geometry”, Commun. Math. Phys. 285(2009) 67-140, [math-ph/0407006] J. Lewandowski, A. Okolow, H. Sahlmann, T. Thiemann, “Uniqueness of diffeomorphism invariant states on holonomy-flux algebras” Commun. Math. Phys.267(2006) 703-733, [gr-qc/0504147]

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  36. [36]

    Representations of the holonomy algebras of gravity and non-Abelian gauge theories

    A. Ashtekar, C.J. Isham, “Representations of the Holonomy Algebras of Gravity and Non-Abelean Gauge Theories”, Class. Quantum Grav.9(1992) 1433, [hep-th/9202053]

    work page internal anchor Pith review Pith/arXiv arXiv 1992

  37. [37]

    Spin Networks and Quantum Gravity

    C. Rovelli, L. Smolin. Spin networks and quantum gravity. Phys.Rev.D 52 (1995) 5743-5759. e-Print: gr-qc/9505006 [gr-qc] J. C. Baez. Spin network states in gauge theory. Adv. Math.117(1996) 253-272. e-Print: gr-qc/9411007 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 1995

  38. [38]

    Narnhofer, W.E

    H. Narnhofer, W.E. Thirring. Covariant QED without indefinite metric. Rev. Math. Phys.4(1992) spec01, 197-211

    1992

  39. [39]

    Projective Techniques and Functional Integration

    A. Ashtekar, J. Lewandowski, “Representation theory of analytic HolonomyC ⋆ algebras”, in “Knots and Quantum Gravity”, J. Baez (ed.), Oxford University Press, Oxford 1994 A. Ashtekar, J. Lewandowski, “Projective Techniques and Functional Integration for Gauge Theories”, J. Math. Phys.36, 2170 (1995), [gr-qc/9411046]

    work page internal anchor Pith review Pith/arXiv arXiv 1994

  40. [40]

    Complexifier Coherent States for Quantum General Relativity

    T. Thiemann, “Complexifier coherent states for canonical quantum general relativity”, Class. Quant. Grav.23(2006) 2063-2118, [gr-qc/0206037] 39

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  41. [41]

    Kinematical Hilbert Spaces for Fermionic and Higgs Quantum Field Theories

    T. Thiemann. Kinematical Hilbert spaces for Fermionic and Higgs quantum field theories. Class. Quant. Grav.15(1998) 1487-1512; e-Print: gr-qc/9705021 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  42. [42]

    Quantization of diffeomorphism invariant theories of connections with local degrees of freedom

    A. Ashtekar, J. Lewandowski, D. Marolf, J. Mourao, T.Thiemann. Quantization of diffeomorphism invariant theories of connections with local degrees of freedom J. Math. Phys.36(1995) 6456-6493. e-Print: gr-qc/9504018 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 1995

  43. [43]

    A length operator for canonical quantum gravity

    T. Thiemann. A Length operator for canonical quantum gravity. J. Math. Phys.39(1998) 3372-3392. e-Print: gr-qc/9606092 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  44. [44]

    Discreteness of area and volume in quantum gravity

    C. Rovelli and L. Smolin. Discreteness of volume and area in quantum gravity. Nucl. Phys.B442(1995), 593-622; Erratum: Nucl. Phys.B456(1995) 753, [gr-qc/9411005] A. Ashtekar and J. Lewandowski. Quantum theory of geometry I: Area Operators. Class. Quant. Grav. 14(1997) A55-A82, [gr-qc/9602046]

    work page internal anchor Pith review Pith/arXiv arXiv 1995

  45. [45]

    Quantum Theory of Geometry II: Volume operators

    A. Ashtekar and J. Lewandowski. Quantum theory of geometry II: Volume operators. Adv. Theo. Math. Phys.1(1997) 388-429, [gr-qc/9711031]

    work page internal anchor Pith review Pith/arXiv arXiv 1997

  46. [46]

    Anomaly-free formulation of non-perturbative, four-dimensional Lorentzian quantum gravity

    T. Thiemann, “Anomaly-free Formulation of non-perturbative, four-dimensional Lorentzian Quantum Gravity”, Physics LettersB380(1996) 257-264, [gr-qc/9606088] T. Thiemann, “Quantum Spin Dynamics (QSD)”, Class. Quantum Grav.15(1998) 839-73, [gr- qc/9606089] T. Thiemann, “Quantum Spin Dynamics (QSD) : II. The Kernel of the Wheeler-DeWitt Constraint Operator”,...

    work page internal anchor Pith review Pith/arXiv arXiv 1996

  47. [47]

    On the consistency of the constraint algebra in spin network quantum gravity

    R. Gambini, J. Lewandowski, D. Marolf, J. Pullin On the consistency of the constraint algebra in spin network quantum gravity. Int. J. Mod. Phys.D 7(1998) 97-109; e-Print: gr-qc/9710018 [gr-qc] M. Varadarajan. Euclidean LQG Dynamics: An Electric Shift in Perspective. Class. Quant. Grav.38 (2021) 13, 135020. e-Print: 2101.03115 [gr-qc] M. Varadarajan. Anom...

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  48. [48]

    Reality conditions inducing transforms for quantum gauge field theory and quantum gravity

    T. Thiemann. Reality conditions inducing transforms for quantum gauge field theory and quantum gravity. Class. Quant. Grav.13(1996) 1383-1404. e-Print: gr-qc/9511057 [gr-qc] M. Varadarajan. From Euclidean to Lorentzian Loop Quantum Gravity via a Positive Complexifier Class. Quant. Grav.36(2019) 1, 015016. e-Print: 1808.00673 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 1996

  49. [49]

    N. Rosen. Bimetric theory of gravitation. In: Topics in Theoretical and Experimental Gravitation Physics, V. Sabbata, J. Weber (eds.). Springer Verlag, Berlin, 1977

    1977

  50. [50]

    Quantum Physics

    J. Glimm and A. Jaffe, “Quantum Physics”, Springer Verlag, New York, 1987

    1987

  51. [51]

    Methods of modern mathematical physics

    M. Reed, B. Simon, “Methods of modern mathematical physics”, vol. I-IV, Academic Press, 1980

    1980

  52. [52]

    K. Kuchar. Dirac Constraint Quantization of a Parametrized Field Theory by Anomaly - Free Operator Representations of Space-time Diffeomorphisms. Phys. Rev.D 39(1989) 2263-2280. K. Kuchar. Parametrized Scalar Field on R X S(1): Dynamical Pictures, Space-time Diffeomorphisms, and Conformal Isometries. Phys. Rev.D 39(1989) 1579-1593

    1989

  53. [53]

    Y. D. Luke. Integrals of Bessel functions. McGraw Hill, USA, 1962

    1962

  54. [54]

    T. L. M. Guedes, G. A. Mena Marugan, M. Mueller, F. Vidotto. Computing the Graph-Changing Dynamics of Loop Quantum Gravity. Universe 11 (2025) 12, 387. e-Print: 2412.20257 [gr-qc] T. L. M. Guedes, G. A. Mena Marugan, M. Mueller, F. Vidotto. Taming Thiemann’s Hamiltonian constraint in canonical loop quantum gravity: Reversibility, eigenstates, and graph-ch...

    work page arXiv 2025