On relativistic spin network vertices

TL;DR

This paper proves the uniqueness of intertwiners in relativistic spin networks, extends the constraints to arbitrary valence, and generalizes the Barrett-Crane model to polyhedral complexes, enhancing quantum gravity models.

Contribution

It demonstrates the unique determination of intertwiners by incident edges and constraints, extends constraints to arbitrary valence networks, and generalizes the Barrett-Crane model to polyhedral complexes.

Findings

Intertwiners are uniquely determined by incident edges and constraints.

Constraints are extended to networks of arbitrary valence.

The Barrett-Crane model is generalized to polyhedral complexes.

Abstract

Barrett and Crane have proposed a model of simplicial Euclidean quantum gravity in which a central role is played by a class of Spin(4) spin networks called "relativistic spin networks" which satisfy a series of physically motivated constraints. Here a proof is presented that demonstrates that the intertwiner of a vertex of such a spin network is uniquely determined, up to normalization, by the representations on the incident edges and the constraints. Moreover, the constraints, which were formulated for four valent spin networks only, are extended to networks of arbitrary valence, and the generalized relativistic spin networks proposed by Yetter are shown to form the entire solution set (mod normalization) of the extended constraints. Finally, using the extended constraints, the Barrett-Crane model is generalized to arbitrary polyhedral complexes (instead of just simplicial complexes) representing spacetime. It is explained how this model, like the Barret-Crane model can be derived from BF theory by restricting the sum over histories to ones in which the left handed and right handed areas of any 2-surface are equal. It is known that the solutions of classical Euclidean GR form a branch of the stationary points of the BF action with respect to variations preserving this condition.