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Path integral representation of spin foam models of 4d gravity

Florian Conrady, Laurent Freidel

TL;DR

This work provides a unified, path-integral formulation for the Riemannian FK, EKPR, and FK$\gamma$/ELPR spin foam models of 4D gravity, showing that FK models are equivalent to discrete gravity path integrals for all $\gamma$ and deriving explicit wedge-based actions. It clarifies the role of the Immirzi parameter in connecting covariant spin foams to canonical LQG, distinguishes the $\gamma<1$ and $\gamma>1$ sectors, and analyzes how boundary states emerge as SU(2)$\times$SU(2) projected spin networks. The derivative expansion reveals a discretized BF-like action with higher-derivative corrections, and the boundary formalism demonstrates how cobordism composition is preserved and how boundary data relate to projected spin networks. The framework offers a concrete route to semiclassical analysis and propagator computations, and highlights the differences between FK and ELPR in their handling of edge intertwiners and boundary degrees of freedom. Overall, the paper provides a powerful, geometrically transparent bridge between spin-foam sums and discrete path-integral gravity, with implications for semiclassical limits and potential connections to covariant quantizations of gravity.

Abstract

We give a unified description of all recent spin foam models introduced by Engle, Livine, Pereira and Rovelli (ELPR) and by Freidel and Krasnov (FK). We show that the FK models are, for all values of the Immirzi parameter, equivalent to path integrals of a discrete theory and we provide an explicit formula for the associated actions. We discuss the relation between the FK and ELPR models and also study the corresponding boundary states. For general Immirzi parameter, these are given by Alexandrov's and Livine's SO(4) projected states. For 0 <= gamma < 1, the states can be restricted to SU(2) spin networks.

Path integral representation of spin foam models of 4d gravity

TL;DR

This work provides a unified, path-integral formulation for the Riemannian FK, EKPR, and FK/ELPR spin foam models of 4D gravity, showing that FK models are equivalent to discrete gravity path integrals for all and deriving explicit wedge-based actions. It clarifies the role of the Immirzi parameter in connecting covariant spin foams to canonical LQG, distinguishes the and sectors, and analyzes how boundary states emerge as SU(2)SU(2) projected spin networks. The derivative expansion reveals a discretized BF-like action with higher-derivative corrections, and the boundary formalism demonstrates how cobordism composition is preserved and how boundary data relate to projected spin networks. The framework offers a concrete route to semiclassical analysis and propagator computations, and highlights the differences between FK and ELPR in their handling of edge intertwiners and boundary degrees of freedom. Overall, the paper provides a powerful, geometrically transparent bridge between spin-foam sums and discrete path-integral gravity, with implications for semiclassical limits and potential connections to covariant quantizations of gravity.

Abstract

We give a unified description of all recent spin foam models introduced by Engle, Livine, Pereira and Rovelli (ELPR) and by Freidel and Krasnov (FK). We show that the FK models are, for all values of the Immirzi parameter, equivalent to path integrals of a discrete theory and we provide an explicit formula for the associated actions. We discuss the relation between the FK and ELPR models and also study the corresponding boundary states. For general Immirzi parameter, these are given by Alexandrov's and Livine's SO(4) projected states. For 0 <= gamma < 1, the states can be restricted to SU(2) spin networks.

Paper Structure

This paper contains 21 sections, 1 theorem, 187 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Definition of FK$\gamma$ and ELPR$\gamma$ model
  3. Triangulation and dual complex
  4. FK$\gamma$ and ELPR$\gamma$ model
  5. Limits $\gamma \to \infty$ and $\gamma \to 0$
  6. Path integral representation of spin foam models
  7. A discrete classical action
  8. Equivalence of spin foam sum and discrete path integral
  9. Relation between FK$\gamma$ and ELPR$\gamma$ model
  10. Comparison of spin foam sums for $\gamma > 1$
  11. Comparison of path integrals for $\gamma > 1$
  12. Boundary terms, boundary states and cobordisms
  13. General boundary formalism for cell complexes
  14. Boundary formulation for spin foam sum
  15. Boundary formulation for path integral
  16. ...and 6 more sections

Key Result

Lemma C.1

Suppose that $X_{IJ}$ is a unit area bivector, i.e. $X^{IJ}X_{IJ}=2$, and let us denote by $(X^{+}, X^{-})$ the corresponding self--dual and anti--self-dual elements of su(2)$\oplus$su(2). Let us also consider a unit 4-vector $U^{I}$. Then, the following statement is true: The identity $U_{I} X^{IJ} Moreover, we have the equality where on the right--hand side $u=U^{I}\sigma_{I}^{E}$ is the SU(2)

Figures (6)

  • Figure 1: (a) Face $f$ of dual complex $\Delta^*$. (b) Subdivision of face $f$ into wedges. The arrows indicate starting point and orientation for wedge holonomies.
  • Figure 2: Diagrammatic representation of the fusion $f$--symbol (\ref{['fsymbolformula']}): the four strands correspond to the four faces $f$ at the edge $e$.
  • Figure 3: Plots of the function $C_m/C_0$ for different values of $\gamma$ and $j^- = 4$: exact value (dots), bound (\ref{['boundofratio']}) (dashed line) and large spin approximation (\ref{['Gaussian']}) (continuous line).
  • Figure 4: Plots of the function $C_m/C_0$ for different values of $\gamma$ and $j^- = 300$: exact value (dots) and large spin approximation (\ref{['Gaussian']}) (continuous line).
  • Figure 5: Face $f$ of $\Delta^*$ at the boundary edge $\bar{e}$. Under a composition of simplicial complexes, this face is joined with a face $f'$.
  • ...and 1 more figures

Theorems & Definitions (3)

  • Definition 3.1
  • Definition 3.2
  • Lemma C.1