The Geometry of the solution space of first order Hamiltonian field theories I: from particle dynamics to free Electrodynamics

TL;DR

This paper investigates the geometric structure of the solution space in first order Hamiltonian field theories, establishing symplectic and Poisson structures for various cases including particle mechanics and free electrodynamics.

Contribution

It introduces a method to define Poisson brackets on the solution space of first order Hamiltonian field theories, including gauge theories like free electrodynamics.

Findings

Established symplectic structure for Hamiltonian mechanical systems.

Defined a Poisson structure for free electrodynamics.

Extended geometric analysis from particle dynamics to gauge fields.

Abstract

We analyse the problem of defining a Poisson bracket structure on the space of solutions of the equations of motions of first order Hamiltonian field theories. The cases of Hamiltonian mechanical point systems (as a (0 + 1)-dimensional field) and more general field theories without gauge symmetries are addressed by showing the existence of a symplectic (and, thus, a Poisson) structure on the space of solutions. Also the easiest case of gauge theory, namely free electrodynamics, is considered: within this problem, a pre-symplectic tensor on the space of solutions is introduced, and a Poisson structure is induced in terms of a flat connection on a suitable bundle associated to the theory.