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A New Spin Foam Model for 4d Gravity

Laurent Freidel, Kirill Krasnov

TL;DR

This work recasts 4D quantum gravity in the spin foam language via constrained BF theory, addressing Barrett–Crane’s ultra-locality by using coherent states to impose linear simplicity constraints. It identifies two distinct sectors, gravitational and topological, and constructs a new gravitational-sector spin foam model that preserves inter-simplex correlations while incorporating the Engle–Pereira–Rovelli spirit of soldering adjacent geometries. It also extends the framework to Lorentzian signature and provides a Lorentzian gravitational construction, alongside a detailed treatment of the Immirzi parameter within spin foams. The results offer a path toward more physically realistic semiclassical behavior, connect to group field theory, and set the stage for deeper links with loop quantum gravity, though several technical challenges (semiclassical limit, divergences, and Lorentzian sector identification) remain to be fully resolved.

Abstract

Starting from Plebanski formulation of gravity as a constrained BF theory we propose a new spin foam model for 4d Riemannian quantum gravity that generalises the well-known Barrett-Crane model and resolves the inherent to it ultra-locality problem. The BF formulation of 4d gravity possesses two sectors: gravitational and topological ones. The model presented here is shown to give a quantization of the gravitational sector, and is dual to the recently proposed spin foam model of Engle et al. which, we show, corresponds to the topological sector. Our methods allow us to introduce the Immirzi parameter into the framework of spin foam quantisation. We generalize some of our considerations to the Lorentzian setting and obtain a new spin foam model in that context as well.

A New Spin Foam Model for 4d Gravity

TL;DR

This work recasts 4D quantum gravity in the spin foam language via constrained BF theory, addressing Barrett–Crane’s ultra-locality by using coherent states to impose linear simplicity constraints. It identifies two distinct sectors, gravitational and topological, and constructs a new gravitational-sector spin foam model that preserves inter-simplex correlations while incorporating the Engle–Pereira–Rovelli spirit of soldering adjacent geometries. It also extends the framework to Lorentzian signature and provides a Lorentzian gravitational construction, alongside a detailed treatment of the Immirzi parameter within spin foams. The results offer a path toward more physically realistic semiclassical behavior, connect to group field theory, and set the stage for deeper links with loop quantum gravity, though several technical challenges (semiclassical limit, divergences, and Lorentzian sector identification) remain to be fully resolved.

Abstract

Starting from Plebanski formulation of gravity as a constrained BF theory we propose a new spin foam model for 4d Riemannian quantum gravity that generalises the well-known Barrett-Crane model and resolves the inherent to it ultra-locality problem. The BF formulation of 4d gravity possesses two sectors: gravitational and topological ones. The model presented here is shown to give a quantization of the gravitational sector, and is dual to the recently proposed spin foam model of Engle et al. which, we show, corresponds to the topological sector. Our methods allow us to introduce the Immirzi parameter into the framework of spin foam quantisation. We generalize some of our considerations to the Lorentzian setting and obtain a new spin foam model in that context as well.

Paper Structure

This paper contains 23 sections, 4 theorems, 152 equations.

Table of Contents

  1. Introduction
  2. A new approach to the geometric simplicity constraints
  3. BF formulation of gravity and path integral quantization
  4. Discretized gravity action and the Barrett-Crane model
  5. A new geometric criterion for (cross-)simplicity
  6. Self- and anti-self-dual decomposition
  7. Identification of bivectors with Lie algebra elements
  8. Gravitational and topological sectors
  9. Coherent states and the spin foam model of BF theory
  10. Coherent states
  11. BF theory
  12. Imposing the simplicity constraints
  13. The Barret-Crane model
  14. The topological sector and the model of Engle:2007uq
  15. The gravitational sector: a new model
  16. ...and 8 more sections

Key Result

Lemma 2.1

A bivector $X^{IJ}$ in ${\mathbb R}^4$ ($M^{1,3}$) is an anti-symmetrized product of two vectors if and only if there exists a vector $n^I$ such that $X^{IJ} n_J = 0$.

Theorems & Definitions (8)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof