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Hidden Quantum Gravity in 3d Feynman diagrams

Aristide Baratin, Laurent Freidel

TL;DR

This work shows that three-dimensional Feynman amplitudes in flat space can be reexpressed as expectation values of observables in a background-free spin-foam model built from Poincaré group data. A key geometric identity transfers flat-space information into the spin-foam weight, dynamically inducing flat geometry via deficit-angle constraints and yielding a topological state sum whose tetrahedral weights are ISO(3) 6j-symbols. The construction connects ordinary field theory to a Poincaré BF-type framework, proving triangulation invariance (with appropriate gauge fixing) and offering a clear algebraic understanding of Feynman diagrams as spin-foam observables. The approach extends to homogeneous spaces and incorporates a cosmological constant, laying groundwork for higher-dimensional generalizations and deeper links between quantum gravity and standard QFT.

Abstract

In this work we show that 3d Feynman amplitudes of standard QFT in flat and homogeneous space can be naturally expressed as expectation values of a specific topological spin foam model. The main interest of the paper is to set up a framework which gives a background independent perspective on usual field theories and can also be applied in higher dimensions. We also show that this Feynman graph spin foam model, which encodes the geometry of flat space-time, can be purely expressed in terms of algebraic data associated with the Poincare group. This spin foam model turns out to be the spin foam quantization of a BF theory based on the Poincare group, and as such is related to a quantization of 3d gravity in the limit where the Newton constant G_N goes to 0. We investigate the 4d case in a companion paper where the strategy proposed here leads to similar results.

Hidden Quantum Gravity in 3d Feynman diagrams

TL;DR

This work shows that three-dimensional Feynman amplitudes in flat space can be reexpressed as expectation values of observables in a background-free spin-foam model built from Poincaré group data. A key geometric identity transfers flat-space information into the spin-foam weight, dynamically inducing flat geometry via deficit-angle constraints and yielding a topological state sum whose tetrahedral weights are ISO(3) 6j-symbols. The construction connects ordinary field theory to a Poincaré BF-type framework, proving triangulation invariance (with appropriate gauge fixing) and offering a clear algebraic understanding of Feynman diagrams as spin-foam observables. The approach extends to homogeneous spaces and incorporates a cosmological constant, laying groundwork for higher-dimensional generalizations and deeper links between quantum gravity and standard QFT.

Abstract

In this work we show that 3d Feynman amplitudes of standard QFT in flat and homogeneous space can be naturally expressed as expectation values of a specific topological spin foam model. The main interest of the paper is to set up a framework which gives a background independent perspective on usual field theories and can also be applied in higher dimensions. We also show that this Feynman graph spin foam model, which encodes the geometry of flat space-time, can be purely expressed in terms of algebraic data associated with the Poincare group. This spin foam model turns out to be the spin foam quantization of a BF theory based on the Poincare group, and as such is related to a quantization of 3d gravity in the limit where the Newton constant G_N goes to 0. We investigate the 4d case in a companion paper where the strategy proposed here leads to similar results.

Paper Structure

This paper contains 22 sections, 2 theorems, 158 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Feynman diagrams in terms of invariant measure
  3. Feynman diagrams as spin foam models
  4. The main result
  5. Emergence of the model
  6. A key identity
  7. Spin foam in Feynman graph: inducing the geometry dynamically
  8. Topological invariance
  9. Symmetries and gauge fixing
  10. Observables and partial gauge fixing
  11. Algebraic structure
  12. A Poincaré model
  13. The result
  14. Proof of the Theorem
  15. Poincaré versus Ponzano-Regge
  16. ...and 7 more sections

Key Result

Proposition 1

The following equalities hold where the function $\mathcal{V}$ is defined to be $\mathcal{V}(v_1,\cdots,v_n) \equiv \hbox{det}(v_1,\cdots,v_n)$ within $\hbox{span}(v_1,\cdots,v_n)$.

Figures (4)

  • Figure 1: Two triangles $acd$ and $bcd$ in the Euclidean plane share an edge $(cd)$. On the left (resp. right) $l_{ab} = l_{ab}^+$ (resp. $l_{ab}^-$).
  • Figure 2: Pentagonal representation of a complex of two tetrahedra $0,4$ sharing a face $\left[123\right]$, and a complex of three tetrahedra $1,2,3$ sharing an edge $(04)$.
  • Figure :
  • Figure :

Theorems & Definitions (2)

  • Proposition 1
  • Proposition 2