On relativistic spin network vertices
Michael P. Reisenberger
TL;DR
This work establishes a rigorous foundation for Barrett–Crane relativistic spin networks by proving that the vertex intertwiner is unique up to normalization given the incident edge irreps and extends the BC constraints to arbitrary valence, aligning Yetter’s generalized intertwiners as the complete solution. The analysis shows that the intertwiner is the SU(2) invariant projector $P$ onto ${ m Inv}_{SU(2)}({f j})$, and that this result holds for general valence through a four-valent fragment argument, thereby confirming Yetter’s extension as the unique (mod normalization) solution to the generalized constraints. By formulating a polyhedral-complex generalization and connecting to Spin$(4)$ BF theory with equal left and right areas, the paper clarifies how the BC spin-foam model arises from constrained BF theory and how classical GR emerges as a stationary point under the constraint. These results provide a rigorous, normalization-consistent basis for spin-foam quantum gravity constructions and their extensions beyond simplicial triangulations.
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.