Spin Foam Models for Quantum Gravity

TL;DR

This paper reviews the current state of spin foam models in non-perturbative quantum gravity, discussing conceptual foundations, specific models like Barrett-Crane, and recent mathematical developments including group field theory.

Contribution

It provides a comprehensive overview of spin foam models, introduces the Barrett-Crane model, and discusses recent finiteness results and dual group field theory formulations.

Findings

Analysis of the Barrett-Crane model's properties

Finiteness results for Riemannian and Lorentzian models

Development of a combinatorial group field theory framework

Abstract

In this article we review the present status of the spin foam formulation of non-perturbative (background independent) quantum gravity. The article is divided in two parts. In the first part we present a general introduction to the main ideas emphasizing their motivations from various perspectives. Riemannian 3-dimensional gravity is used as a simple example to illustrate conceptual issues and the main goals of the approach. The main features of the various existing models for 4-dimensional gravity are also presented here. We conclude with a discussion of important questions to be addressed in four dimensions (gauge invariance, discretization independence, etc.). In the second part we concentrate on the definition of the Barrett-Crane model. We present the main results obtained in this framework from a critical perspective. Finally we review the combinatorial formulation of spin foam models based on the dual group field theory technology. We present the Barrett-Crane model in this framework and review the finiteness results obtained for both its Riemannian as well as its Lorentzian variants.