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Flipped spinfoam vertex and loop gravity

Jonathan Engle, Roberto Pereira, Carlo Rovelli

TL;DR

The work derives a four-dimensional spinfoam vertex for Euclidean GR by quantizing a Regge-discretized Plebanski theory with a flipped $SO(4)$ symplectic structure. By imposing the second-class simplicity constraints weakly, the model preserves the full $SO(4)$ intertwiners while yielding a physical boundary space isomorphic to the $SO(3)$ LQG Hilbert space, thereby connecting covariant and canonical descriptions. The resulting vertex is both $SO(3)$- and $SO(4)$-covariant and resolves the Barrett-Crane intertwiner truncation, offering a condensate path to recover the graviton tensor structure and the quantization of geometry. The paper also provides an independent derivation of LQG kinematics and discusses implications for low-energy GR, the role of the Immirzi parameter, and potential group field theory formulations for dynamics.

Abstract

We introduce a vertex amplitude for 4d loop quantum gravity. We derive it from a conventional quantization of a Regge discretization of euclidean general relativity. This yields a spinfoam sum that corrects some difficulties of the Barrett-Crane theory. The second class simplicity constraints are imposed weakly, and not strongly as in Barrett-Crane theory. Thanks to a flip in the quantum algebra, the boundary states turn out to match those of SO(3) loop quantum gravity -- the two can be identified as eigenstates of the same physical quantities -- providing a solution to the problem of connecting the covariant SO(4) spinfoam formalism with the canonical SO(3) spin-network one. The vertex amplitude is SO(3) and SO(4)-covariant. It rectifies the triviality of the intertwiner dependence of the Barrett-Crane vertex, which is responsible for its failure to yield the correct propagator tensorial structure. The construction provides also an independent derivation of the kinematics of loop quantum gravity and of the result that geometry is quantized.

Flipped spinfoam vertex and loop gravity

TL;DR

The work derives a four-dimensional spinfoam vertex for Euclidean GR by quantizing a Regge-discretized Plebanski theory with a flipped symplectic structure. By imposing the second-class simplicity constraints weakly, the model preserves the full intertwiners while yielding a physical boundary space isomorphic to the LQG Hilbert space, thereby connecting covariant and canonical descriptions. The resulting vertex is both - and -covariant and resolves the Barrett-Crane intertwiner truncation, offering a condensate path to recover the graviton tensor structure and the quantization of geometry. The paper also provides an independent derivation of LQG kinematics and discusses implications for low-energy GR, the role of the Immirzi parameter, and potential group field theory formulations for dynamics.

Abstract

We introduce a vertex amplitude for 4d loop quantum gravity. We derive it from a conventional quantization of a Regge discretization of euclidean general relativity. This yields a spinfoam sum that corrects some difficulties of the Barrett-Crane theory. The second class simplicity constraints are imposed weakly, and not strongly as in Barrett-Crane theory. Thanks to a flip in the quantum algebra, the boundary states turn out to match those of SO(3) loop quantum gravity -- the two can be identified as eigenstates of the same physical quantities -- providing a solution to the problem of connecting the covariant SO(4) spinfoam formalism with the canonical SO(3) spin-network one. The vertex amplitude is SO(3) and SO(4)-covariant. It rectifies the triviality of the intertwiner dependence of the Barrett-Crane vertex, which is responsible for its failure to yield the correct propagator tensorial structure. The construction provides also an independent derivation of the kinematics of loop quantum gravity and of the result that geometry is quantized.

Paper Structure

This paper contains 21 sections, 2 theorems, 135 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries: Plebanski two-form and structure of $SO(4)$
  3. Plebanski two-form and simplicity constraints
  4. Selfdual structure of $SO(4)$
  5. Regge discretization
  6. Constraints on $\Sigma$
  7. Relation with geometry
  8. Dynamics
  9. Classical equations of motion
  10. Topological term
  11. Boundary terms
  12. Boundary variables
  13. Summary of the classical theory
  14. Quantization
  15. Kinematical Hilbert space and operators
  16. ...and 6 more sections

Key Result

Lemma 1

If $e^I_\mu$ is such that $B_i^{IJ} = B_i^{\mu\nu}e^I_\mu e^J_\nu$ for $i \in \{1,2,3\}$, and then $B_4^{IJ} = B_4^{\mu\nu}e^I_\mu e^J_\nu$ as well. An analogous statement holds with $B_i^{IJ}$ replaced by ${}^*B_i^{IJ}$.

Figures (2)

  • Figure 1: The link of the face $f$, in grey, and the two group elements associated to a couple of tetrahedra $t$ and $t'$ in the link.
  • Figure 2: The face $f$ and the two tetrahedra $t$ and $t'$ are on the boundary of the triangulation, represented by the heavy dashed line. The link of the face is divided by the boundary into two parts.

Theorems & Definitions (2)

  • Lemma 1
  • Proposition 2