Hamiltonian Nature of Monopole Dynamics

TL;DR

This paper explores the Hamiltonian structure of classical electromagnetism with magnetic monopoles, revealing fundamental issues with the Jacobi identity and implications for quantum theory formulation.

Contribution

It demonstrates that classical electromagnetism with monopoles lacks a proper Hamiltonian structure unless specific conditions are met, impacting quantum theory development.

Findings

Jacobi identity fails in monopole electromagnetism

Hamiltonian structure is restored only under specific charge ratios

Implications for quantization of monopole theories

Abstract

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 theory 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.