Covariant Hamiltonian for gravity coupled to<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>p</mml:mi></mml:math>-forms
Leonardo Castellani, A. D’Adda
Abstract
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 $\ensuremath{\phi}$. Momenta are defined as derivatives of the Lagrangian with respect to the ``velocities'' $d\ensuremath{\phi}$ 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.