The geometry of the solution space of first order Hamiltonian field theories I: From particle dynamics to free electrodynamics

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.

Multisymplectic geometry
Poisson brackets
Field theories
Peierls brackets
Space of solutions

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