Representations of spacetime diffeomorphisms. I. Canonical parametrized field theories

Abstract

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 V →∈ L Diff Σ 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. We show how this difficulty may be overcome and construct a homomorphic mapping of spacetime vector fields V ∈ L Diff M into the Poisson bracket algebra on the phase space of the system. In the present paper, I, we explain how the technique works in the case of a parametrized field theory and in the following paper, II, we generalize it to canonical geometrodynamics. In a parametrized theory, the phase space of the system is the ordinary phase space of the field augmented by the embedding variables X: Σ →Mand their conjugate momenta. The dynamical variable H(V) which represents V ∈ L Diff M generates a deformation of the embedding along the flow lines of V accompanied by the correct dynamical evolution of the field data and preserves the constraints in the extended phase space of the system. We also establish the relation between the representations of Diff Σ and DiffM.