Representations of spacetime diffeomorphisms. II. Canonical geometrodynamics

Our method of constructing representations of the spacetime diffeomorphism group DiffMin parametrized field theories is generalized to canonical geometrodynamics. The gravitational configuration space Riem Σ is extended by the space of embeddings of the spatial manifold Σ in the spacetimeM. Spacetime metrics are limited by Gaussian conditions with respect to an auxiliary foliation structure. As a result of these conditions, the super-Hamiltonian and supermomentum constraints are temporarily suspended. There are, however, new constraints in the theory associated with the canonical pair of the embedding variables and their conjugate momenta. By smearing the new constraint functions by vector fields V ∈ LDiffMrestricted to the embeddings, we construct a homomorphism from the Lie algebra of the spacetime diffeomorphism group into the Poisson bracket algebra of the dynamical variables on the extended geometrodynamical phase space. The dynamical evolution generated by such dynamical variables automatically preserves both the new and the old constraints and builds a Ricci-flat spacetime. The implications of the scheme for the canonical quantization of gravity are discussed.