- Physical Review Journals
All JournalsPhysics Magazine

Physical Review D

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Structure of the continuum limit of spin foams

Matteo Bruno*

Eugenia Colafranceschi†

Fabio M. Mele‡

Carlo Rovelli§

  • *Contact author: bruno@cpt.univ-mrs.fr
  • †Contact author: ecolafra@ucm.es
  • ‡Contact author: fmele1@lsu.edu
  • §Contact author: crovelli@uwo.ca
PDF

Phys. Rev. D 114, 066005 – Published 8 September, 2026

DOI: https://doi.org/10.1103/7493-9nb7

Abstract

The spin foam approach to quantum gravity aims at providing a covariant path-integral formulation of canonical loop quantum gravity. Since spin foam amplitudes are defined through discretizations of spacetime, understanding the continuum limit of the theory remains a central open problem. In this work, we investigate the structural aspects of this limit in a model-independent manner. We introduce an axiomatic framework for spin foam amplitudes inspired by Atiyah’s formulation of topological quantum field theories (TQFTs). In this setting, Hilbert spaces and amplitudes are assigned to combinatorial and topological data associated with triangulated manifolds. By equipping the set of triangulations with suitable orders, this framework provides a precise notion of continuum limit and allows us to analyse its properties independently of any specific model. We systematically investigate how the specifics of the limit procedure allow to go beyond TQFT in the continuum. Under natural assumptions on the convergence of spin foam amplitudes, we establish a no-go result: sufficiently strong notions of convergence necessarily lead to a topological theory. Motivated by this obstruction, we weaken the notion of convergence and consider the continuum limit of spin foam amplitudes in a distributional sense, in the spirit of refined algebraic quantization. Under this assumption, the amplitude associated with the cylinder defines a rigging map, yielding a canonical construction of the physical Hilbert space. The resulting continuum amplitudes act as well-defined distributions on this space of physical states, characterizing this formulation of the gravitational path integral as physical in a precise sense.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (100)

  1. C. Rovelli and F. Vidotto, Covariant Loop Quantum Gravity: An Elementary Introduction to Quantum Gravity and Spinfoam Theory (Cambridge University Press, Cambridge, England, 2014).
  2. J. C. Baez, An introduction to spin foam models of BF theory and quantum gravity, Lect. Notes Phys. 543, 25 (2000).
  3. A. Perez, The spin-foam approach to quantum gravity, Living Rev. Relativity 16, 3 (2013).
  4. E. Bianchi, Spinfoam gravity, in Loop Quantum Gravity: The First 30 Years (WSP, Singapore, 2017), pp. 97–124.
  5. J. Engle and S. Speziale, Spin foams: Foundations, in Handbook of Quantum Gravity (Springer, Singapore, 2023), pp. 1–40.
  6. M. P. Reisenberger and C. Rovelli, “Sum over surfaces” form of loop quantum gravity, Phys. Rev. D 56, 3490–3508 (1997).
  7. M. Reisenberger and C. Rovelli, Spin foams as Feynman diagrams, in 25th Johns Hopkins Workshop on Current Problems in Particle Theory: 2001: A Relativistic Spacetime Odyssey. Experiments and Theoretical Viewpoints on General Relativity and Quantum Gravity (2000), pp. 431–448.
  8. B. Dittrich, Diffeomorphism symmetry in quantum gravity models, Adv. Sci. Lett. 2, 151 (2009).
  9. B. Bahr, B. Dittrich, and S. He, Coarse graining free theories with gauge symmetries: The linearized case, New J. Phys. 13, 045009 (2011).
  10. B. Bahr, B. Dittrich, and S. Steinhaus, Perfect discretization of reparametrization invariant path integrals, Phys. Rev. D 83, 105026 (2011).
  11. C. Rovelli and M. Smerlak, In quantum gravity, summing is refining, Classical Quantum Gravity 29, 055004 (2012).
  12. B. Dittrich, From the discrete to the continuous: Towards a cylindrically consistent dynamics, New J. Phys. 14, 123004 (2012).
  13. B. Dittrich and S. Steinhaus, Time evolution as refining, coarse graining and entangling, New J. Phys. 16, 123041 (2014).
  14. D. Oriti, J. P. Ryan, and J. Thürigen, Group field theories for all loop quantum gravity, New J. Phys. 17, 023042 (2015).
  15. B. Dittrich, The continuum limit of loop quantum gravity: A framework for solving the theory, in Loop Quantum Gravity: The First 30 Years (World Scientific, Singapore, 2017), pp. 153–179.
  16. C. Rovelli, Discretizing parametrized systems: The magic of Dittrich invariance, Phys. Rev. D 106, 104062 (2022).
  17. S. K. Asante, B. Dittrich, and S. Steinhaus, Spin Foams, Refinement Limit, and Renormalization, in Handbook of Quantum Gravity (Springer, Singapore, 2023), pp. 1–37.
  18. M. Han, On the summation and triangulation independence of Lorentzian spinfoam amplitudes for all LQG, Phys. Rev. D 113, 084034 (2026).
  19. M. Han, Ultraviolet fixed point in covariant loop quantum gravity, Phys. Rev. D 114, 044040 (2026).
  20. M. F. Atiyah, Topological quantum field theory, Publ. Math. IHES 68, 175 (1988).
  21. E. Witten, Topological quantum field theory, Commun. Math. Phys. 117, 353 (1988).
  22. E. Witten, 2+1 dimensional gravity as an exactly soluble system, Nucl. Phys. B311, 46 (1988).
  23. E. Witten, Quantum field theory and the Jones polynomial, Commun. Math. Phys. 121, 351 (1989).
  24. C. Rovelli, The basis of the Ponzano-Regge-Turaev-Viro-Ooguri model is the loop representation basis, Phys. Rev. D 48, 2702 (1993).
  25. L. Crane, Topological field theory as the key to quantum gravity, in Knots and Quantum Gravity (Oxford University Press, Oxford, England, 1994), pp. 121–132.
  26. J. W. Barrett, Quantum gravity as topological quantum field theory, J. Math. Phys. (N.Y.) 36, 6161 (1995).
  27. B. S. DeWitt, Quantum theory of gravity. I. The canonical theory, Phys. Rev. 160, 1113 (1967).
  28. D. Giulini and D. Marolf, A uniqueness theorem for constraint quantization, Classical Quantum Gravity 16, 2489 (1999).
  29. D. Giulini and D. Marolf, On the generality of refined algebraic quantization, Classical Quantum Gravity 16, 2479 (1999).
  30. T. Thiemann, Modern Canonical Quantum General Relativity, Cambridge Monographs on Mathematical Physics (Cambridge University Press, Cambridge, England, 2007).
  31. M. Atiyah, The Geometry and Physics of Knots, Lezioni Lincee (Cambridge University Press, Cambridge, England, 1990).
  32. J. C. Baez and J. Dolan, Higher-dimensional algebra and topological quantum field theory, J. Math. Phys. (N.Y.) 36, 6073 (1995).
  33. B. Bakalov and A. A. Kirillov, Lectures on Tensor Categories and Modular Functors (American Mathematical Soc., Providence, 2001).
  34. L. S. Abrams, Two-dimensional topological quantum field theories and Frobenius algebras, J. Knot Theory Ramif. 5, 569 (1996).
  35. J. Kock, Frobenius Algebras and 2d Topological Quantum Field Theories (Cambridge University Press, Cambridge, England, 2004).
  36. N. Reshetikhin and V. G. Turaev, Invariants of three manifolds via link polynomials and quantum groups, Invent. Math. 103, 547 (1991).
  37. V. G. Turaev, Quantum Invariants of Knots and 3-Manifolds (de Gruyter Studies in Mathematics 18, de Gruyter & Co., 1994).
  38. M. Bärenz and J. Barrett, Dichromatic state sum models for four-manifolds from pivotal functors, Commun. Math. Phys. 360, 663 (2018).
  39. C. Meusburger, V. Mulevicius, and F. Torzewska, Categorical 4-manifold invariants from trisection diagrams, arXiv:2511.19384.
  40. C. Rovelli, Simple model for quantum general relativity from loop quantum gravity, J. Phys. Conf. Ser. 314, 012006 (2011).
  41. C. Rovelli, Zakopane lectures on loop gravity, Proc. Sci. QGQGS2011 (2011) 003.
  42. J. C. Baez, Spin foam models, Classical Quantum Gravity 15, 1827 (1998).
  43. R. Oeckl, A “general boundary” formulation for quantum mechanics and quantum gravity, Phys. Lett. B 575, 318 (2003).
  44. T. Jacobson, Thermodynamics of spacetime: The Einstein equation of state, Phys. Rev. Lett. 75, 1260 (1995).
  45. L. Smolin, General relativity as the equation of state of spin foam, Classical Quantum Gravity 31, 195007 (2014).
  46. L. Smolin, Thermodynamics of quantum spacetime histories, Phys. Rev. D 96, 104042 (2017).
  47. E. Bianchi, Entropy of non-extremal black holes from loop gravity, arXiv:1204.5122.
  48. C. Rovelli, Quantum Gravity, Cambridge Monographs on Mathematical Physics (Cambridge University Press, Cambridge, England, 2004).
  49. W. Kamiński, M. Kisielowski, and J. Lewandowski, Spin-foams for all loop quantum gravity, Classical Quantum Gravity 27, 095006 (2010).
  50. M. P. Reisenberger, World sheet formulations of gauge theories and gravity, in Proceedings of the Seventh Marcel Grossmann Meeting on General Relativity (World Scientific, Singapore, 1996).
  51. J. Engle, E. Livine, R. Pereira, and C. Rovelli, LQG vertex with finite Immirzi parameter, Nucl. Phys. B799, 136 (2008).
  52. L. Freidel and K. Krasnov, A new spin foam model for 4D gravity, Classical Quantum Gravity 25, 125018 (2008).
  53. H. Ooguri, Topological lattice models in four dimensions, Mod. Phys. Lett. A 07, 2799 (1992).
  54. T. Regge, General relativity without coordinates, Nuovo Cimento (1955–1965) 19, 558 (1961).
  55. G. Ponzano and T. Regge, Semiclassical limit of racah coefficients, Spectroscopic and Group Theoretical Methods in Physics, edited by F. Block (John Wiley and Sons, Inc., New York, 1968).
  56. F. Conrady and L. Freidel, Path integral representation of spin foam models of 4d gravity, Classical Quantum Gravity 25, 245010 (2008).
  57. E. Bianchi, E. Magliaro, and C. Perini, Spinfoams in the holomorphic representation, Phys. Rev. D 82, 124031 (2010).
  58. E. Magliaro and C. Perini, Regge gravity from spinfoams, Int. J. Mod. Phys. D 22, 1 (2013).
  59. E. Magliaro and C. Perini, Curvature in spinfoams, Classical Quantum Gravity 28, 145028 (2011).
  60. M. Han and M. Zhang, Asymptotics of spinfoam amplitude on simplicial manifold: Lorentzian theory, Classical Quantum Gravity 30, 165012 (2013).
  61. M. Han, Z. Huang, H. Liu, and D. Qu, Complex critical points and curved geometries in four-dimensional Lorentzian spinfoam quantum gravity, Phys. Rev. D 106, 044005 (2022).
  62. J. F. P. Hudson, Piecewise Linear Topology (W. A. Benjamin, New York, 1969).
  63. J. H. C. Whitehead, On C1-complexes, Ann. Math. 41, 809 (1940).
  64. F. Hellmann, State sums and geometry, arXiv:1102.1688.
  65. V. Turaev and O. Viro, State sum invariants of 3-manifolds and quantum 6j-symbols, Topology 31, 865 (1992).
  66. J. W. Barrett and I. Naish-Guzman, The Ponzano-Regge model, Classical Quantum Gravity 26, 155014 (2009).
  67. H. Ooguri and N. Sasakura, Discrete and continuum approaches to three-dimensional quantum gravity, Mod. Phys. Lett. A 06, 3591 (1991).
  68. H. Ooguri, Partition functions and topology-changing amplitudes in the 3D lattice gravity of Ponzano and Regge, Nucl. Phys. B382, 276 (1992).
  69. B. Dittrich and M. Geiller, Flux formulation of loop quantum gravity: Classical framework, Classical Quantum Gravity 32, 135016 (2015).
  70. C. Rovelli, The Projector on physical states in loop quantum gravity, Phys. Rev. D 59, 104015 (1999).
  71. K. Noui and A. Perez, Three-dimensional loop quantum gravity: Physical scalar product and spin-foam models, Classical Quantum Gravity 22, 1739 (2005).
  72. J. J. Halliwell and J. B. Hartle, Wave functions constructed from an invariant sum over histories satisfy constraints, Phys. Rev. D 43, 1170 (1991).
  73. J. B. Hartle and S. W. Hawking, Wave function of the Universe, Phys. Rev. D 28, 2960 (1983).
  74. E. Alesci, T. Thiemann, and A. Zipfel, Linking covariant and canonical loop quantum gravity: New solutions to the Euclidean scalar constraint, Phys. Rev. D 86, 024017 (2012).
  75. T. Thiemann and A. Zipfel, Linking covariant and canonical LQG II: Spin foam projector, Classical Quantum Gravity 31, 125008 (2014).
  76. T. Thiemann, Quantum spin dynamics (QSD), Classical Quantum Gravity 15, 839 (1998).
  77. T. Thiemann, Quantum spin dynamics (QSD): II. The kernel of the Wheeler—DeWitt constraint operator, Classical Quantum Gravity 15, 875 (1998).
  78. T. Thiemann, Quantum spin dynamics (QSD): III. Quantum constraint algebra and physical scalar product in quantum general relativity, Classical Quantum Gravity 15, 1207 (1998).
  79. W. Donnelly and L. Freidel, Local subsystems in gauge theory and gravity, J. High Energy Phys. 09 (2016) 102.
  80. A. J. Speranza, Local phase space and edge modes for diffeomorphism-invariant theories, J. High Energy Phys. 02 (2018) 021.
  81. L. Freidel, M. Geiller, and D. Pranzetti, Edge modes of gravity. Part I. Corner potentials and charges, J. High Energy Phys. 11 (2020) 026.
  82. L. Freidel, M. Geiller, and D. Pranzetti, Edge modes of gravity. Part II. Corner metric and Lorentz charges, J. High Energy Phys. 11 (2020) 027.
  83. L. Freidel, M. Geiller, and D. Pranzetti, Edge modes of gravity. Part III. Corner simplicity constraints, J. High Energy Phys. 01 (2021) 100.
  84. L. Ciambelli, R. G. Leigh, and P.-C. Pai, Embeddings and integrable charges for extended corner symmetry, Phys. Rev. Lett. 128, 171302 (2022).
  85. S. Carrozza, S. Eccles, and P. A. Hoehn, Edge modes as dynamical frames: Charges from post-selection in generally covariant theories, SciPost Phys. 17, 048 (2024).
  86. L. Freidel, A. Perez, and D. Pranzetti, Loop gravity string, Phys. Rev. D 95, 106002 (2017).
  87. L. Freidel, F. Girelli, and B. Shoshany, 2+1D loop quantum gravity on the edge, Phys. Rev. D 99, 046003 (2019).
  88. L. Freidel, E. R. Livine, and D. Pranzetti, Gravitational edge modes: From Kac–Moody charges to Poincaré networks, Classical Quantum Gravity 36, 195014 (2019).
  89. L. Freidel, M. Geiller, and W. Wieland, Corner Symmetry and Quantum Geometry, in Handbook of Quantum Gravity, edited by C. Bambi, L. Modesto, and I. Shapiro (Springer, Singapore, 2024), pp. 1–36.
  90. E. Colafranceschi, X. Dong, D. Marolf, and Z. Wang, Algebras and Hilbert spaces from gravitational path integrals. Understanding Ryu-Takayanagi/HRT as entropy without AdS/CFT, J. High Energy Phys. 10 (2024) 063.
  91. R. Oeckl and J. O. Almada, Compositional quantum field theory: An axiomatic presentation, J. Math. Phys. (N.Y.) 65, 012301 (2024).
  92. S. Willard, General Topology (Courier Corporation, USA, 2004).
  93. U. Pachner, P. L. homeomorphic manifolds are equivalent by elementary shellings, Eur. J. Combinatorics 12, 129 (1991).
  94. W. B. R. Lickorish, Simplicial moves on complexes and manifolds, in Proceedings of the Kirbyfest (Mathematical Sciences Publishers, MSRI, Berkeley, California, USA, 1999), pp. 299–320.
  95. J. Barrett and B. Westbury, Invariants of piecewise-linear 3-manifolds, Trans. Am. Math. Soc. 348, 3997 (1996).
  96. B. Bahr, B. Dittrich, and M. Geiller, A new realization of quantum geometry, Classical Quantum Gravity 38, 145021 (2021).
  97. J. W. Alexander, Combinatorial analysis situs, Trans. Am. Math. Soc. 28, 301 (1926).
  98. J. W. Alexander, The combinatorial theory of complexes, Ann. Math. 31, 292 (1930).
  99. H. Tietze, Über die topologischen invarianten mehrdimensionaler mannigfaltigkeiten, Monatsh. Math. Phys. 19, 1 (1908).
  100. K. Adiprasito and I. Pak, All triangulations have a common stellar subdivision, Invent. Math. (2026).

Outline

Information

Phys. Rev. D 114, 066005– Published 8 September, 2026
  • Received 20 May 2026
  • Accepted 5 August 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation