Representations of spacetime diffeomorphisms. I. Canonical parametrized field theories

The super-Hamiltonian and supermomentum in canonical geometrodynamics or in a parametried field theory on a given Riemannian background have Poisson brackets which obey the Dirac relations. By smearing the supermomentum with vector fields VepsilonL DiffΣ on the space manifold Σ, the Lie algebra L Diff Σ of the spatial diffeomorphism group Diff Σ can be mapped antihomomorphically into the Poisson bracket algebra on the phase space of the system. The explicit dependence of the Poisson brackets between two super-Hamiltonians on canonical coordinates (spatial metrics in geometrodynamics and embedding variables in parametrized theories) is usually regarded as an indication that the Dirac relations cannot be connected with a representation of the complete Lie algebra L Diff M of spacetime diffeomorphisms