Plasma in monopole background is not twisted Poisson

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.

arXiv comment9 pages

References in corpus (4)

• WZW-Poisson manifolds
• Current Algebras and Differential Geometry
• Gauge Transformations, Twisted Poisson Brackets and Hamiltonization of Nonholonomic Systems
• Symplectic realisation of electric charge in fields of monopole distributions

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• Hamiltonian Nature of Monopole Dynamics
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