Projected Spin Networks for Lorentz connection: Linking Spin Foams and Loop Gravity

TL;DR

This paper introduces projected cylindrical functions linking Lorentz connection and time normal in spin foam models, advancing covariant Loop Quantum Gravity by connecting spin foams and loop states.

Contribution

It develops a new class of boundary states called projected cylindrical functions, bridging spin foam models and loop quantum gravity in a covariant framework.

Findings

Defined projected cylindrical functions on Lorentz connection and time normal

Showed how these functions naturally arise in covariant Loop Quantum Gravity

Linked spin foam boundary states with loop quantum gravity states

Abstract

In the search for a covariant formulation for Loop Quantum Gravity, spin foams have arised as the corresponding discrete space-time structure and, among the different models, the Barrett-Crane model seems the most promising. Here, we study its boundary states and introduce cylindrical functions on both the Lorentz connection and the time normal to the studied hypersurface. We call them projected cylindrical functions and we explain how they would naturally arise in a covariant formulation of Loop Quantum Gravity.