LQG vertex with finite Immirzi parameter

TL;DR

This paper extends the flipped loop-quantum-gravity vertex to finite Immirzi parameter, demonstrating consistent dynamics and spectra in both Euclidean and Lorentzian cases, bridging canonical LQG and spinfoam formalism.

Contribution

It introduces a generalized vertex for finite Immirzi parameter, removing previous ad hoc assumptions and unifying LQG and spinfoam approaches in four dimensions.

Findings

The vertex is well-defined for finite Immirzi parameter.

The area spectrum matches standard LQG results.

The Lorentzian case is successfully incorporated.

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.