Physical boundary Hilbert space and volume operator in the Lorentzian new spin-foam theory

TL;DR

This paper advances the bridge between covariant Lorentzian spin-foam theory and canonical LQG by constructing a Lorentzian version of the new spin-foam model with a refined $SU(2)\to SL(2,C)$ embedding. Using $\gamma$-simple representations with $p=\gamma(k+1)$ and selecting the minimal $j=k$ sector, it defines a physical boundary Hilbert space $\mathcal{H}_{\mathrm{ph}}$ that is isomorphic to the LQG boundary space, and proves that the kinematic constraints vanish exactly on this space in a weak sense. It then derives a Lorentzian volume operator that, when acting on the physical intertwiner space $\mathcal{K}_{\mathrm{ph}}$, matches the canonical LQG volume operator with the correct $\gamma$-dependence, while the area operator remains aligned with LQG results. The framework shows that covariant spin-foam amplitudes can be embedded from LQG data through a BF-like structure, reinforcing the consistency of covariant and canonical pictures in 4D quantum gravity and reinforcing the viability of the Lorentzian new spin-foam models. Overall, the work strengthens the case for a coherent quantum gravity description where geometric observables and constraint dynamics agree across covariant and canonical formulations.

Abstract

A covariant spin-foam formulation of quantum gravity has been recently developed, characterized by a kinematics which appears to match well the one of canonical loop quantum gravity. In this paper we reconsider the implementation of the constraints that defines the model. We define in a simple way the boundary Hilbert space of the theory, introducing a slight modification of the embedding of the SU(2) representations into the SL(2,C) ones. We then show directly that all constraints vanish on this space in a weak sense. The vanishing is exact (and not just in the large quantum number limit.) We also generalize the definition of the volume operator in the spinfoam model to the Lorentzian signature, and show that it matches the one of loop quantum gravity, as does in the Euclidean case.