Poincare group spin networks

Abstract

Spin network technique is usually generalized to relativistic case by changing $SO(4)$ group -- Euclidean counterpart of the Lorentz group -- to its universal spin covering $SU(2)\times SU(2)$, or by using the representations of $SO(3,1)$ Lorentz group. We extend this approach by using inhomogeneous Lorentz group $\mathcal{P}=SO(3,1)\rtimes \mathbb{R}^4$, which results in the simplification of the spin network technique. The labels on the network graph corresponding to the subgroup of translations $\mathbb{R}^4$ make the intertwiners into the products of $SU(2)$ parts and the energy-momentum conservation delta functions. This maps relativistic spin networks to usual Feynman diagrams for the matter fields.