Spin Foam Models of Quantum Spacetime

TL;DR

This work surveys spin foam approaches to quantum spacetime, focusing on 3D topological gravity via Ponzano–Regge and TV models and on 4D gravity through the Barrett–Crane framework. It develops both canonical and covariant pictures, linking loop quantum gravity, BF theory, and group field theory, and shows how gravity can be represented as a constrained BF theory with spin networks and arrays of bivectors; key results include asymptotic connections to Regge action and the construction of causal spin foams. The text also analyzes boundary states, observables, convergence properties, and higher-dimensional generalizations, and discusses how different boundary conditions and Immirzi-like parameters influence the spin foam amplitudes. Altogether, it positions spin foams as a robust, background-free, combinatorial approach to quantum gravity, capable of encoding topology change and causal structure within a group-theoretic and category-theoretic framework.

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.

Figures (31)

• Figure 1: Schematic representation of a 4-manifold (of genus one) with two disconnected boundary components
• Figure 2: A spin network whose corresponding functional is: $\psi_{S}(A)=\rho_{e_{1}}(\mathcal{P}\,e^{\int_{e_{1}}A})^{a}_{b}\,\rho_{e_{2}}( \mathcal{P}\,e^{\int_{e_{2}}A})^{c}_{d}\,\rho_{e_{3}}( \mathcal{P}\,e^{\int_{e_{3}}A})^{e}_{f}\,(\iota_{v_{1}})_{ace}(\iota_{v_{2}})^{bdf}$
• Figure 3: A spin network embedded in a manifold and having three points of intersection with a surface $\Sigma$
• Figure 4: The 2-complex correspondent to a second order term in the expansion of the amplitude
• Figure 5: A causal set