Spin Foams Without Spins

TL;DR

This paper presents a reformulation of the spin foam model for discrete SU(2) gauge theory that eliminates the need for summing over spins by using generating functions and contour integrals, providing a clearer geometric interpretation.

Contribution

The authors develop a spin foam representation that avoids recoupling theory and sums over spins, simplifying calculations and clarifying geometric structures.

Findings

Sums over spins are replaced with contour integrals.

Boundary data corresponds to framed polyhedra with fixed total area.

The formulation simplifies the spin foam amplitude computation.

Abstract

We formulate the spin foam representation of discrete SU(2) gauge theory as a product of vertex amplitudes each of which is the spin network generating function of the boundary graph dual to the vertex. In doing so the sums over spins have been carried out. The boundary data of each n-valent node is explicitly reduced with respect to the local gauge invariance and has a manifest geometrical interpretation as a framed polyhedron of fixed total area. Ultimately, sums over spins are traded for contour integrals over simple poles and recoupling theory is avoided using generating functions.