On covariant phase space methods

TL;DR

This paper introduces a covariant, ambiguity-free definition of the symplectic form in gauge theories, enabling consistent Hamiltonian formulations and conserved charge calculations across various fundamental theories.

Contribution

A new covariant symplectic form independent of the Lagrangian, applicable to gauge theories, and a generalized Hamiltonian framework that unifies different approaches.

Findings

The new symplectic form is free of ambiguities and depends only on equations of motion.

Application to Yang-Mills, gravity, Chern-Simons, and supergravity theories demonstrates versatility.

Both covariant symplectic and Regge-Teitelboim methods yield consistent conserved charges with the new prescription.

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.