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In this paper explore the relation between covariant and canonical approachesto quantum gravity and $BF$ theory. We will focus on the dynamicaltriangulation and spin-foam models, which have in common that they can bedefined in terms of sums over space-time triangulations. Our aim is to show howwe can recover these covariant models from a canonical framework by providingtwo regularisations of the projector onto the kernel of the Hamiltonianconstraint. This link is important for the understanding of the dynamics ofquantum gravity. In particular, we will see how in the simplest dynamicaltriangulations model we can recover the Hamiltonian constraint via ourdefinition of the projector. Our discussion of spin-foam models will show howthe elementary spin-network moves in loop quantum gravity, which wereoriginally assumed to describe the Hamiltonian constraint action, are in factrelated to the time-evolution generated by the constraint. We also show thatthe Immirzi parameter is important for the understanding of a continuum limitof the theory.Comment: 28 pages, 10 figure

Relating covariant and canonical approaches to triangulated models of quantum gravity