Poincaré Group Spin Networks

Abstract

Spin network technique is usually generalized to relativistic case by changing SO\(\varvec{(4)}\) group – Euclidean counterpart of the Lorentz group – to its universal spin covering SU\(\varvec{(2)}\times \) SU\(\varvec{(2)}\), or by using the representations of SO\(\varvec{(3,1)}\) Lorentz group. We extend this approach by using inhomogeneous Lorentz group \(\varvec{\mathcal {P}}= {\varvec{SO}}\varvec{(3,1)}\rtimes \mathbb {R}^4\), 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\(\varvec{(2)}\) parts and the energy-momentum conservation delta functions. This maps relativistic spin networks to usual Feynman diagrams for the matter fields.