Symplectic realisation of electric charge in fields of monopole distributions

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.

39 pages, 1 figure; v2: references added; v3: clarifying comments and references added; Final version to be published in Physical Review D

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• Non-associativity in non-geometric string and M-theory backgrounds, the algebra of octonions, and missing momentum modes
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• The quasi-nonassociative exceptional $F(4)$ deformed quantum oscillator
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Cited by in corpus (17)

• Symplectic embeddings, homotopy algebras and almost Poisson gauge symmetry
• Para-Hermitian Geometry, Dualities and Generalized Flux Backgrounds
• Poisson gauge theory
• On current algebras, generalised fluxes and non-geometry
• Smooth 2-Group Extensions and Symmetries of Bundle Gerbes
• Geometry and 2-Hilbert Space for Nonassociative Magnetic Translations
• Recurrence relations for symplectic realization of (quasi)-Poisson structures
• Plasma in monopole background is not twisted Poisson
• Quantization of Magnetic Poisson Structures
• Magnetic Monopoles in (Noncommutative) Quantum Mechanics
• Motion of charged particles in spacetimes with magnetic fields of spherical and hyperbolic symmetry
• T-Dualities and Doubled Geometry of the Principal Chiral Model
• A Simple Model of Double Dynamics on Lie Groups
• Quantum field theoretic representation of Wilson surfaces: I higher coadjoint orbit theory
• Alternative Description of Magnetic Monopoles in Quantum Mechanics
• Lagrangian formulation for electric charge in a magnetic monopole distribution
• Charged Particle Motion in Spherically Symmetric Distributions of Magnetic Monopoles