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Spin foam model for Lorentzian General Relativity

Alejandro Perez, Carlo Rovelli

TL;DR

The paper develops a Lorentzian spin foam formulation of quantum gravity using an $SL(2, olinebreak[4] C)$ group-field theory, yielding a partition function whose Feynman diagrams reproduce 2-complexes with the Barrett–Crane Lorentzian vertex. It demonstrates that the sum over 2-complexes is automatically implemented and discusses finiteness prospects by showing how edge amplitudes suppress divergences, with amplitudes expressed in terms of simple representations $(0, ho)$ and hyperboloid kernels $K_ ho$. The construction points to a promising finite, covariant framework while highlighting open issues, including establishing the classical general relativity limit and incorporating additional representations $(n,0)$ to capture full GR degrees of freedom. Key future directions include finiteness proofs, classical limit analysis, and extending the representation content to include time-like directions.

Abstract

We present a spin foam formulation of Lorentzian quantum General Relativity. The theory is based on a simple generalization of an Euclidean model defined in terms of a field theory over a group. Its vertex amplitude turns out to be the one recently introduced by Barrett and Crane. As in the case of its Euclidean relatives, the model fully implements the desired sum over 2-complexes which encodes the local degrees of freedom of the theory.

Spin foam model for Lorentzian General Relativity

TL;DR

The paper develops a Lorentzian spin foam formulation of quantum gravity using an group-field theory, yielding a partition function whose Feynman diagrams reproduce 2-complexes with the Barrett–Crane Lorentzian vertex. It demonstrates that the sum over 2-complexes is automatically implemented and discusses finiteness prospects by showing how edge amplitudes suppress divergences, with amplitudes expressed in terms of simple representations and hyperboloid kernels . The construction points to a promising finite, covariant framework while highlighting open issues, including establishing the classical general relativity limit and incorporating additional representations to capture full GR degrees of freedom. Key future directions include finiteness proofs, classical limit analysis, and extending the representation content to include time-like directions.

Abstract

We present a spin foam formulation of Lorentzian quantum General Relativity. The theory is based on a simple generalization of an Euclidean model defined in terms of a field theory over a group. Its vertex amplitude turns out to be the one recently introduced by Barrett and Crane. As in the case of its Euclidean relatives, the model fully implements the desired sum over 2-complexes which encodes the local degrees of freedom of the theory.

Paper Structure

This paper contains 5 sections, 44 equations, 1 figure.

Table of Contents

  1. Introduction
  2. $SL(2,C)$ state sum model of Lorentzian QG
  3. Discussion
  4. Acknowledgments
  5. Representation Theory of $SL(2,C)$

Figures (1)

  • Figure 1: Structure of the interaction. The black circle represent the projections $\delta_{0j}$ (\ref{['proj']}) produced by the $SU(2)$ integrations in (\ref{['mart']}).