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

publication date

  • October 2024

volume

  • 204

International Standard Serial Number (ISSN)

  • 0393-0440

Electronic International Standard Serial Number (EISSN)

  • 1879-1662

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.

keywords

  • field theories; multisymplectic geometry; peierls brackets; poisson brackets; space of solutions