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.
Highlights
AND SUMMARYDespite their elusiveness to experimental observation, magnetic monopoles have been of wide-spread theoretical interest in various areas of physics for many years due to their novel conceptual and mathematical implications
For the standard example of motion in the field of a Dirac monopole, the charged particle wavefunction can be regarded as a section of a non-trivial line bundle associated to the Hopf fibration [1] which provides a topological explanation for Dirac charge quantisation [2,3] and formulates the quantum dynamics of the particle without using the unphysical Dirac string singularities that usually arise
Deformation quantization by explicit construction of nonassociative phase space star products was originally developed by [9], and subsequently treated in [4,10,11]; beyond the case of constant magnetic charge density, this procedure does not yield a quantization of the classical dynamical system because the Planck constant ħ appears as a formal expansion parameter, and the result is a deformation over an algebra of formal power series rather than with a complex parameter
Summary
Despite their elusiveness to experimental observation, magnetic monopoles have been of wide-spread theoretical interest in various areas of physics for many years due to their novel conceptual and mathematical implications. In this paper we are predominantly interested in smooth distributions of magnetic charge, for which vector potentials do not exist even locally and the classical dynamics of the canonical phase space coordinates of the particle are described by a necessarily nonassociative twisted Poisson algebra These systems have been of interest recently as magnetic analogues of certain flux models in nongeometric string theory and double field theory, see e.g., [4,5,6,7,8] and references therein. A final Appendix C compares the extended phase space of the symplectic realization to the phase space of locally nongeometric closed strings and double field theory, and presents potential applications of our formalism in nongeometric M-theory
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