LQG vertex with finite Immirzi parameter

TL;DR

This work extends the flipped LQG spinfoam vertex to finite Immirzi parameter γ in both Euclidean and Lorentzian sectors, showing the boundary state space matches SU(2) spin networks and the area spectrum matches canonical LQG with the correct γ-dependence. Using a master-constraint approach to implement the second-class simplicity constraints, it derives a projected boundary space from Spin(4) or SL(2, C) data to SU(2) intertwiners and constructs vertex amplitudes from SU(2) 15j symbols (Euclidean) or SL(2,C) 15j symbols (Lorentzian). The resulting discrete area spectrum, proportional to γ, resolves prior covariance tensions by showing discretization emerges after constraint implementation in both signatures. The work thus provides a concrete four-dimensional link between canonical LQG and covariant spinfoams at finite γ and lays groundwork for future studies of dynamics and propagators.

Abstract

We extend the definition of the "flipped" loop-quantum-gravity vertex to the case of a finite Immirzi parameter. We cover the Euclidean as well as the Lorentzian case. We show that the resulting dynamics is defined on a Hilbert space isomorphic to the one of loop quantum gravity, and that the area operator has the same discrete spectrum as in loop quantum gravity. This includes the correct dependence on the Immirzi parameter, and, remarkably, holds in the Lorentzian case as well. The ad hoc flip of the symplectic structure that was initially required to derive the flipped vertex is not anymore needed for finite Immirzi parameter. These results establish a bridge between canonical loop quantum gravity and the spinfoam formalism in four dimensions.