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Hidden Quantum Gravity in 4d Feynman diagrams: Emergence of spin foams

Aristide Baratin, Laurent Freidel

TL;DR

This work demonstrates that ordinary Feynman diagrams in quantum field theory can be recast as observables within a background-independent 4D spin foam model based on Poincaré BF theory. It identifies the full symmetry structure, constructs a gauge-fixed measure, and proves triangulation (Pachner move) invariance, showing that Feynman amplitudes emerge as expectation values of graph observables in a topological spin foam. The paper also develops an algebraic framework with 20j-symbols, establishes a duality with Barrett–Crane-type amplitudes, and clarifies the GN → 0 limit as gravity becoming topological, linking QFT observables to gravitational dynamics. These results provide a falsifiability criterion for quantum gravity proposals and suggest a rich algebraic structure potentially connecting to 2-categorical spin-foam models and BF theory. Overall, the work offers a concrete route to unify background-independent quantum gravity with standard perturbative field theory and lays out future directions in renormalization, open diagrams, and deeper algebraic realizations.

Abstract

We show how Feynman amplitudes of standard QFT on flat and homogeneous space can naturally be recast as the evaluation of observables for a specific spin foam model, which provides dynamics for the background geometry. We identify the symmetries of this Feynman graph spin foam model and give the gauge-fixing prescriptions. We also show that the gauge-fixed partition function is invariant under Pachner moves of the triangulation, and thus defines an invariant of four-dimensional manifolds. Finally, we investigate the algebraic structure of the model, and discuss its relation with a quantization of 4d gravity in the limit where the Newton constant goes to zero.

Hidden Quantum Gravity in 4d Feynman diagrams: Emergence of spin foams

TL;DR

This work demonstrates that ordinary Feynman diagrams in quantum field theory can be recast as observables within a background-independent 4D spin foam model based on Poincaré BF theory. It identifies the full symmetry structure, constructs a gauge-fixed measure, and proves triangulation (Pachner move) invariance, showing that Feynman amplitudes emerge as expectation values of graph observables in a topological spin foam. The paper also develops an algebraic framework with 20j-symbols, establishes a duality with Barrett–Crane-type amplitudes, and clarifies the GN → 0 limit as gravity becoming topological, linking QFT observables to gravitational dynamics. These results provide a falsifiability criterion for quantum gravity proposals and suggest a rich algebraic structure potentially connecting to 2-categorical spin-foam models and BF theory. Overall, the work offers a concrete route to unify background-independent quantum gravity with standard perturbative field theory and lays out future directions in renormalization, open diagrams, and deeper algebraic realizations.

Abstract

We show how Feynman amplitudes of standard QFT on flat and homogeneous space can naturally be recast as the evaluation of observables for a specific spin foam model, which provides dynamics for the background geometry. We identify the symmetries of this Feynman graph spin foam model and give the gauge-fixing prescriptions. We also show that the gauge-fixed partition function is invariant under Pachner moves of the triangulation, and thus defines an invariant of four-dimensional manifolds. Finally, we investigate the algebraic structure of the model, and discuss its relation with a quantization of 4d gravity in the limit where the Newton constant goes to zero.

Paper Structure

This paper contains 19 sections, 111 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Dynamical geometry in Feynman amplitudes
  3. Symmetries and gauge fixing
  4. Classical solutions and zero modes
  5. Edge symmetry
  6. Face symmetry
  7. Reducibility
  8. The gauge-fixed model
  9. Feynman diagrams as spin foam amplitudes
  10. Topological invariance
  11. Observables and partial gauge-fixing
  12. Feynman diagrams on homogeneous spaces
  13. Algebraic structure: discussion
  14. A new kind of spin foam model
  15. A duality relation
  16. ...and 4 more sections

Figures (7)

  • Figure 1: A $5$-simplex defines a complex of two $4$-simplices $\sigma_0, \sigma_5$ sharing a tetrahedral face $\left[1234\right]$, as well as a complex of four $4$-simplices $\sigma_1, \sigma_2, \sigma_3, \sigma_4$ sharing an edge $(05)$.
  • Figure 2: The orthogonal projection of a simplex $\sigma \supset e$ onto the 3-space $e^{\bot}$, maps the edge $e$ to the vertex $v_e$ and the three faces $F_i$ adjacent to $e$ to three edges $e_i$ meeting at $v_e$. The dashed line represents the height associated to $e$ within the triangle $F_1$.
  • Figure :
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  • ...and 2 more figures