This work introduces a new space \T'_* 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
(and using certain ideas of Rovelli and Smolin). In particular, such operators
map \T'_* into itself, and so are actual operators in this space. Their
commutator can be computed on \T'_* 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 non-trivial action on \T'_*, the commutator of
quantum constraints vanishes identically for a large class of proposals.Comment: 30 pages RevTex, 2 figures include
