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

Authors: Ding, You; Rovelli, Carlo

Source: Classical and Quantum Gravity, Volume 27, Number 20, 21 October 2010 , pp. 205003-205013(11)

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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,{bb 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 spin-foam model to the Lorentzian signature and show that it matches the one of loop quantum gravity, as in the Euclidean case.

Document Type: Research article

DOI: http://dx.doi.org/10.1088/0264-9381/27/20/205003

Publication date: 2010-10-21