Loop constraints: A habitat and their algebra

This work introduces a new space tau*' of 'vertex-smooth' states for use in the loop approach to quantum gravity. Such states provide a natural domain for Euclidean Hamiltonian constraint operators of the type introduced by Thiemann land using certain ideas of Rovelli and Smolin). In particular, such operators map tau*' into itself, and so are actual operators in this space. Their commutator can be computed on tau*' and compared with the classical hypersurface deformation algebra. Although the classical Poisson bracket of Hamiltonian constraints yields an inverse metric times an infinitesimal diffeomorphism generator, and despite the fact that the diffeomorphism generator has a well-defined nontrivial action on tau*', the commutator of quantum constraints vanishes identically for a large class of proposals.