The volume operator in covariant quantum gravity

A covariant spin-foam formulation of quantum gravity has recently been developed, characterized by a kinematics which appears to match well the one of canonical loop quantum gravity. In particular, the geometrical observable giving the area of a surface has been shown to be the same as the one in loop quantum gravity. Here we discuss the volume observable. We derive the volume operator in the covariant theory and show that it matches the one of loop quantum gravity, as does the area. We also reconsider the implementation of the constraints that define the model: we derive in a simple way the boundary Hilbert space of the theory from a suitable form of the classical constraints and show directly that all constraints vanish weakly on this space.