Lifting SU(2) Spin Networks to Projected Spin Networks
Maité Dupuis, Etera R. Livine
TL;DR
The paper analyzes how SU(2) spin networks of Loop Quantum Gravity relate to projected spin networks used in EPRL-FK spinfoam models. It introduces a projection map $\mathcal{M}$ from projected spin networks to SU(2) networks and constructs inverse lifts $\mathcal{L}_{\{\beta_e\}}$ parameterized by edge-wise Immirzi parameters, yielding a foliation of the projected-spin-network space. Weak and strong simplicity constraints are examined: the EPRL-FK choice $n_e=j_e$, $\rho_e=\beta_e(j_e+1)$ enforces proportionality between boosts and rotations, culminating in a strong simplicity condition $(\vec{J}\cdot\vec{K}-2\beta_e\vec{J}^2)\phi=0$, while Barrett-Crane corresponds to a vanishing boost sector. The paper also shows the lifts are generally not unitary with respect to the Lorentz vs SU(2) inner products and proposes renormalized lifting schemes to reconcile scalar products, arguing for using projected spin networks as boundary states for spinfoam models and enabling per-edge Immirzi parameter control in coarse-graining and dynamics.
Abstract
Projected spin network states are the canonical basis of quantum states of geometry for the most recent EPR-FK spinfoam models for quantum gravity. They are functionals of both the Lorentz connection and the time normal field. We analyze in details the map from these projected spin networks to the standard SU(2) spin networks of loop quantum gravity. We show that this map is not one-to-one and that the corresponding ambiguity is parameterized by the Immirzi parameter. We conclude with a comparison of the scalar products between projected spin networks and SU(2) spin network states.