Covariant Hamiltonian for gravity coupled to -forms

We review the covariant canonical formalism initiated by D'Adda, Nelson, and Regge in 1985, and extend it to include a definition of form-Poisson brackets (FPBs) for geometric theories coupled to p-forms. The form-Legendre transformation and the form-Hamilton equations are derived from a d-form Lagrangian with p-form dynamical fields phi. Momenta are defined as derivatives of the Lagrangian with respect to the "velocities" d phi and no preferred time direction is used. Action invariance under infinitesimal form-canonical transformations can be studied in this framework, and a generalized Noether theorem is derived, for both global and local symmetries. We apply the formalism to vielbein gravity in d = 3 and d = 4. In the d = 3 theory we can define form-Dirac brackets, and use an algorithmic procedure to construct the canonical generators for local Lorentz rotations and diffeomorphisms. In d = 4 the canonical analysis is carried out using FPBs, since the definition of form-Dirac brackets is problematic. Lorentz generators are constructed, while diffeomorphisms are generated by the Lie derivative. A "doubly covariant" Hamiltonian formalism is presented, allowing to maintain manifest Lorentz covariance at every stage of the Legendre transformation. The idea is to take curvatures as "velocities" in the definition of momenta.