The Construction of Spin Foam Vertex Amplitudes

TL;DR

This paper reviews the construction of spin foam vertex amplitudes, crucial for quantum gravity models, presenting two perspectives: one geometric from topological theories and another aligned with Loop Quantum Gravity constraints.

Contribution

It provides a comprehensive review of recent methods for constructing spin foam vertex amplitudes, connecting topological and Loop Quantum Gravity approaches.

Findings

Two complementary methods for constructing vertex amplitudes.

Clarification of geometric and quantum constraints in models.

Insight into the relation between topological theories and quantum gravity.

Abstract

Spin foam vertex amplitudes are the key ingredient of spin foam models for quantum gravity. These fall into the realm of discretized path integral, and can be seen as generalized lattice gauge theories. They can be seen as an attempt at a 4-dimensional generalization of the Ponzano-Regge model for 3d quantum gravity. We motivate and review the construction of the vertex amplitudes of recent spin foam models, giving two different and complementary perspectives of this construction. The first proceeds by extracting geometric configurations from a topological theory of the BF type, and can be seen to be in the tradition of the work of Barrett, Crane, Freidel and Krasnov. The second keeps closer contact to the structure of Loop Quantum Gravity and tries to identify an appropriate set of constraints to define a Lorentz-invariant interaction of its quanta of space. This approach is in the tradition of the work of Smolin, Markopoulous, Engle, Pereira, Rovelli and Livine.