# Plasma in monopole background is not twisted Poisson

## TL;DR

This paper investigates the mathematical structure of plasma phase space in the presence of magnetic monopoles, showing that the associated bracket is not even twisted Poisson, contrary to some previous assumptions.

## Contribution

It provides a specific example demonstrating that the phase space bracket in monopole backgrounds is not twisted Poisson, challenging prior claims of its twisted Poisson nature.

## Key findings

- The phase space bracket is not twisted Poisson in monopole backgrounds.
- An explicit example shows the bracket fails to be twisted Poisson.
- This result impacts the mathematical understanding of plasma in monopole fields.

## Abstract

For a particle in the magnetic field of a cloud of monopoles, the naturally associated 2-form on phase space is not closed, and so the corresponding bracket operation on functions does not satisfy the Jacobi identity. Thus, it is not a Poisson bracket; however, it is twisted Poisson in the sense that the Jacobiator comes from a closed 3-form. The space $\mathcal D$ of densities on phase space is the state space of a plasma. The twisted Poisson bracket on phase-space functions gives rise to a bracket on functions on $\mathcal D$. In the absence of monopoles, this is again a Poisson bracket. It has recently been shown by Heninger and Morrison that this bracket is not Poisson when monopoles are present. In this note, we give an example where it is not even twisted Poisson.