# Symplectic realisation of electric charge in fields of monopole distributions

## TL;DR

This paper develops a symplectic framework for describing electric charges in magnetic monopole fields, providing new insights into their classical and quantum dynamics, and connecting to non-geometric string theory.

## Contribution

It introduces a symplectic realisation of twisted Poisson structures for charges in monopole backgrounds, unifying classical and quantum descriptions without Dirac strings.

## Key findings

- Constructed a symplectic phase space for monopole fields.
- Reproduced Lorentz force law within the new formalism.
- Established equivalence with nonassociative quantum mechanics approaches.

## Abstract

We construct a symplectic realisation of the twisted Poisson structure on the phase space of an electric charge in the background of an arbitrary smooth magnetic monopole density in three dimensions. We use the extended phase space variables to study the classical and quantum dynamics of charged particles in arbitrary magnetic fields by constructing a suitable Hamiltonian that reproduces the Lorentz force law for the physical degrees of freedom. In the source-free case the auxiliary variables can be eliminated via Hamiltonian reduction, while for non-zero monopole densities they are necessary for a consistent formulation and are related to the extra degrees of freedom usually required in the Hamiltonian description of dissipative systems. We obtain new perspectives on the dynamics of dyons and motion in the field of a Dirac monopole, which can be formulated without Dirac strings. We compare our associative phase space formalism with the approach based on nonassociative quantum mechanics, reproducing extended versions of the characteristic translation group three-cocycles and minimal momentum space volumes, and prove that the two approaches are formally equivalent. We also comment on the implications of our symplectic realisation in the dual framework of non-geometric string theory and double field theory.