Symmetries and Covariant Poisson brackets on pre-symplectic manifolds

TL;DR

This paper explores the geometric structure of solution spaces in Hamiltonian field theories, introducing conserved charges linked to symmetries and applying symplectic regularization to gauge theories, with examples from Electrodynamics and Klein-Gordon theory.

Contribution

It develops a framework for understanding symmetries and conserved charges on pre-symplectic manifolds, including a symplectic regularization method for gauge theories.

Findings

Conserved charges are associated with symmetry groups via momentum maps.

Symplectic regularization effectively handles gauge theories.

Energy-momentum tensor algebra emerges from conserved currents.

Abstract

Noticing that the space of the solutions of a first order Hamiltonian field theory has a pre-symplectic structure, we describe a class of conserved charges on it associated to the momentum map determined by any symmetry group of transformations. Gauge theories are dealt with by using a symplectic regularization based on an application of Gotay's coisotropic embedding theorem. The analysis of Electrodynamics and of the Klein-Gordon theory illustrates the main results of the theory as well as the emergence of the energy-momentum tensor algebra of conserved currents.