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

TL;DR

This paper refines the boundary Hilbert space and volume operator in a Lorentzian spin-foam quantum gravity model, ensuring constraints vanish exactly and aligning with loop quantum gravity results.

Contribution

It introduces a simple boundary Hilbert space construction with exact constraint vanishing and generalizes the volume operator to Lorentzian signature, matching loop quantum gravity.

Findings

Constraints vanish exactly on the boundary space.

The Lorentzian volume operator matches the loop quantum gravity operator.

The boundary Hilbert space is constructed via a modified embedding of SU(2) into SL(2,C).

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.