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Ponzano-Regge model revisited III: Feynman diagrams and Effective field theory

Laurent Freidel, Etera R. Livine

TL;DR

This work connects 3D quantum gravity spin foams (Ponzano–Regge and Turaev–Viro) with conventional quantum field theory by focusing on matter couplings and the no-gravity limit. It shows that Feynman amplitudes emerge as abelian spin-foam evaluations in the $G_N o0$ limit, while finite gravity yields a non-commutative, braided effective field theory governed by a κ-Poincaré symmetry and a star-product on Lie algebra variables. The paper develops a rigorous framework for particle propagation in quantum geometry, including explicit Feynman rules, braiding, and a causal (Lorentzian) extension, and demonstrates how different cosmological constants lead to distinct state-sum limits (PR, TV, hyperbolic) with corresponding propagators. Together, these results provide a principled path from quantum gravity to an effective non-commutative field theory, highlighting the role of doubly special relativity and non-trivial spacetime braiding in 3D. The findings have potential implications for understanding matter dynamics in quantum spacetime and guiding extensions to higher-dimensional models.

Abstract

We study the no gravity limit G_{N}-> 0 of the Ponzano-Regge amplitudes with massive particles and show that we recover in this limit Feynman graph amplitudes (with Hadamard propagator) expressed as an abelian spin foam model. We show how the G_{N} expansion of the Ponzano-Regge amplitudes can be resummed. This leads to the conclusion that the dynamics of quantum particles coupled to quantum 3d gravity can be expressed in terms of an effective new non commutative field theory which respects the principles of doubly special relativity. We discuss the construction of Lorentzian spin foam models including Feynman propagators

Ponzano-Regge model revisited III: Feynman diagrams and Effective field theory

TL;DR

This work connects 3D quantum gravity spin foams (Ponzano–Regge and Turaev–Viro) with conventional quantum field theory by focusing on matter couplings and the no-gravity limit. It shows that Feynman amplitudes emerge as abelian spin-foam evaluations in the limit, while finite gravity yields a non-commutative, braided effective field theory governed by a κ-Poincaré symmetry and a star-product on Lie algebra variables. The paper develops a rigorous framework for particle propagation in quantum geometry, including explicit Feynman rules, braiding, and a causal (Lorentzian) extension, and demonstrates how different cosmological constants lead to distinct state-sum limits (PR, TV, hyperbolic) with corresponding propagators. Together, these results provide a principled path from quantum gravity to an effective non-commutative field theory, highlighting the role of doubly special relativity and non-trivial spacetime braiding in 3D. The findings have potential implications for understanding matter dynamics in quantum spacetime and guiding extensions to higher-dimensional models.

Abstract

We study the no gravity limit G_{N}-> 0 of the Ponzano-Regge amplitudes with massive particles and show that we recover in this limit Feynman graph amplitudes (with Hadamard propagator) expressed as an abelian spin foam model. We show how the G_{N} expansion of the Ponzano-Regge amplitudes can be resummed. This leads to the conclusion that the dynamics of quantum particles coupled to quantum 3d gravity can be expressed in terms of an effective new non commutative field theory which respects the principles of doubly special relativity. We discuss the construction of Lorentzian spin foam models including Feynman propagators

Paper Structure

This paper contains 26 sections, 188 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Spin Foams as the topological dual of Feynman diagrams: the abelian case
  3. Particle Insertions in Ponzano-Regge Spinfoam Gravity
  4. The no-gravity limit and a perturbative expansion in $G$
  5. The Quantum Field Theory limit of Spin Foams at $G\rightarrow 0$
  6. Expansion in $G$: Non-Commutative Geometry and Star Product
  7. Feynman evaluation of spherical graphs
  8. Star Product and Non-Commutative Field Theory
  9. Non-spherical Graphs and non-trivial Braiding
  10. A Non-Commutative Braided Field theory
  11. Non-Commutative Field Theory as Effective Quantum Gravity
  12. Non-Commutativity and Braided Feynman Diagrams
  13. On the classical limit of the Turaev-Viro model
  14. The many limits of the Turaev-Viro model
  15. A hyperbolic quantum state sum
  16. ...and 11 more sections

Figures (5)

  • Figure 1: Case of a crossing: we lift the crossing by embedding it into a handle, we triangulate the corresponding punctured torus and we derive the braiding rule in the non-abelian Feynman evaluation.
  • Figure 2: Feynman rules for particles propagation in the Ponzano-Regge model.
  • Figure 3: Feynman rules in the Turaev-Viro model: a Feynman graph evaluation is given by the corresponding Reshetikhin evaluation of the diagram.
  • Figure 4: Tetrahedral net as the skeleton of a 2-sphere.
  • Figure 5: Tetrahedral net as the skeleton of a 2-torus.