Covariant hamiltonian for gravity coupled to $p$-forms

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 (FPB) for geometric theories coupled to $p$-forms, gauging free differential algebras. The form-Legendre transformation and the form-Hamilton equations are derived from a $d$-form Lagrangian with $p$-form dynamical fields $\varphi$. Momenta are defined as derivatives of the Lagrangian with respect to the "velocities" $d\varphi$ 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, both for 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 FPB, 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.

arXiv comment22 pages, LaTeX. v2: Notes added to Sect 8.3 and 9.1, misprints corrected

References in corpus (11)

• Canonical Structure of Classical Field Theory in the Polymomentum Phase Space
• Noncommutative D=4 gravity coupled to fermions
• Covariant Poisson Brackets in Geometric Field Theory
• Supergravity Actions with Integral Forms
• Noncommutative supergravity in D=3 and D=4
• Gravitational Energy for GR and Poincare Gauge Theories: a Covariant Hamiltonian Approach
• The Integral Form of Supergravity
• Supergravity in the group-geometric framework: a primer
• On a generalization of the Dirac bracket in the De Donder-Weyl Hamiltonian formalism
• Reconsideration of De Donder-Weyl theory by covariant analytic mechanics
• Poisson Bracket and Symplectic Structure of Covariant Canonical Formalism of Fields

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