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

TL;DR

The paper develops a covariant framework for Loop Quantum Gravity by introducing projected cylindrical functions that depend on both the Lorentz connection and the time normal $\chi$, yielding a compact effective gauge at vertices and a tractable $L^2$ structure. It defines projected spin networks as a covariant generalization of $SU(2)$ spin networks, relates Barrett-Crane boundary states to these objects, and shows how, in the time gauge, they reduce to standard $SU(2)$ spin networks. It also embarks on a Fock-space construction to sum over graphs, analyzes graph refinement and the bivalent-vertex issue, and derives an area-like operator spectrum $Area \sim \sqrt{j(j+1)-n^2+\rho^2+1}$ within Alexandrov’s covariant framework. Together, these results establish a concrete Hilbert-space foundation for covariant loop gravity and clarify how spin foam data interface with canonical, gauge-fixed formulations, with potential implications for dynamics and measurements of quantum geometry.

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.