Spin Foam Models of Quantum Spacetime

TL;DR

This paper reviews spin foam models as a background-independent, non-perturbative approach to quantum gravity, focusing on the Barrett-Crane model in 4D and its connections to classical gravity, gauge theories, and causality.

Contribution

It provides a detailed overview of the formalism, recent results, and potential extensions of spin foam models, especially the Barrett-Crane model, including its derivation, classical and quantum geometry, and causal structure.

Findings

Analysis of the Barrett-Crane model's construction and properties

Derivation from lattice gauge theory perspective

Proposal of a scheme for causal spin foam models

Abstract

Spin foam models are a new approach to a formulation of quantum gravity which is fully background independent, non-perturbative, and covariant, in the spirit of path integral formulations of quantum field theory. In this thesis we describe in details the general ideas and formalism of spin foam models, and review many of the results obtained recently in this approach. We concentrate, for the case of 3-dimensional quantum gravity, on the Turaev-Viro model, and, in the 4-dimensional case, which is our main concern, on the Barrett-Crane model. In particular, for the Barrett-Crane model: we describe the general ideas behind its construction, and review what has been achieved up to date, discuss in details its links with the classical formulations of gravity as constrained topological field theory; we show a derivation of the model from a lattice gauge theory perspective, in the general case of manifold with boundaries, presenting also a few possible variations of the procedure used, discussing the problems they present; we analyse in details the classical and quantum geometry; we also describe how, from the same perspective, a spin foam model that couples quantum gravity to any gauge theory may be constructed; finally, we describe a general scheme for causal spin foam models, how the Barrett-Crane model can be modified to implement causality and to fit in such a scheme, and the resulting link with the quantum causal set approach to quantum gravity.