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Physical boundary Hilbert space and volume operator in the Lorentzian new spin-foam theory
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.Comment: 11 page