On covariant phase space methods

TL;DR

This work presents a covariant, unambiguous formulation of the covariant phase space for first-order field theories by introducing a symplectic density that depends only on the equations of motion, independent of the Lagrangian. It establishes a covariant Hamiltonian framework and demonstrates the approach with explicit calculations for Yang–Mills, general relativity, Chern–Simons theories, and eleven-dimensional supergravity, highlighting the crucial role of boundary terms. The authors show that their covariant symplectic density fixes ambiguities present in the Noether construction and that conserved charges for gauge symmetries can be derived consistently from either the symplectic method or the differentiability (superpotential) approach, with equivalence guaranteed under their prescription. They also discuss boundary conditions and integrability, providing a coherent, boundary-aware framework for conserved charges in gauge theories and gravity.

Abstract

It is well known that the Lagrangian and the Hamiltonian formalisms can be combined and lead to "covariant symplectic" methods. For that purpose a "pre-symplectic form" has been constructed from the Lagrangian using the so-called Noether form. However, analogously to the standard Noether currents, this symplectic form is only determined up to total divergences which are however essential ingredients in gauge theories. We propose a new definition of the symplectic form which is covariant and free of ambiguities in a general first order formulation. Indeed, our construction depends on the equations of motion but not on the Lagrangian. We then define a generalized Hamiltonian which generates the equations of motions in a covariant way. Applications to Yang-Mills, general relativity, Chern-Simons and supergravity theories are given. We also consider nice sets of possible boundary conditions that imply the closure and conservation of the total symplectic form. We finally revisit the construction of conserved charges associated with gauge symmetries, from both the "covariant symplectic" and the "covariantized Regge-Teitelboim" points of view. We find that both constructions coincide when the ambiguity in the Noetherian pre-symplectic form is fixed using our new prescription. We also present a condition of integrability of the equations that lead to these quantities.