The Hamiltonian dynamics of canonical loop quantum gravity remains an outstanding open issue. Since Thiemann's 1996 definition of the Hamiltonian constraint operator and its subsequent criticism, researchers have looked to various toy models of gravity to understand basic features of the loop quantization procedure in more detail, with the goal of applying lessons learned to a refined Hamiltonian dynamics for loop quantum gravity that satisfies what might be called "quantum general covariance." In this thesis we report on two such toy models: Euclidean general relativity in three and four spacetime dimensions, in a novel weak-coupling limit. These G Newton→ 0 limit models are Abelian gauge theories whose constraint Poisson algebras (modulo Gaus constraint) are isomorphic to that of gravity. Here we take the first steps toward a non-trivial anomaly-free representation of the algebra in the loop-quantized theory. In each case, the Hamiltonian constraint and its commutator at finite triangulation are constructed as operators on the charge network basis of the kinematical Hilbert space of the theory, and the continuum limit is taken with respect to an operator topology based on a subspace of distributions over (a dense subspace of) the kinematical Hilbert space. An operator corresponding to the classical Poisson bracket of two Hamiltonians is also constructed, whose continuum limit agrees with that of the commutator. The construction here shares many basic features with Thiemann's seminal treatment, but to close the quantum constraint algebra "off-shell" requires a significant reappraisal of those methods, including higher density weight constraints, a geometric interpretation of the Hamiltonian, and novel quantizations of phase space-dependent diffeomorphisms. We believe that with some technical prowess many of the ideas developed here can be applied to the more complicated case of full general relativity.
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