Hamiltonian nature of monopole dynamics

•The Vlasov-Maxwell theory of magnetic monopoles is presented in Hamiltonian context.•The failure of the Jacobi identity for both the Vlasov continuum and discrete particles is shown.•Consequences of the failure of the Jacobi identity are considered. Classical electromagnetism with magnetic monopoles is not a Hamiltonian field theory because the Jacobi identity for the Poisson bracket fails. The Jacobi identity is recovered only if all of the species have the same ratio of electric to magnetic charge or if an electron and a monopole can never collide. Without the Jacobi identity, there are no local canonical coordinates or Lagrangian action principle. To build a quantum field of magnetic monopoles, we either must explain why the positions of electrons and monopoles can never coincide or we must resort to new quantization techniques.