Magnetic monopoles and nonassociative deformations of quantum theory

Abstract

We examine certain nonassociative deformations of quantum mechanics and gravity in three dimensions related to the dynamics of electrons in uniform distributions of magnetic charge. We describe a quantitative framework for nonassociative quantum mechanics in this setting, which exhibits new effects compared to ordinary quantum mechanics with sourceless magnetic fields, and the extent to which these theoretical consequences may be experimentally testable. We relate this theory to noncommutative Jordanian quantum mechanics, and show that its underlying algebra can be obtained as a contraction of the alternative algebra of octonions. The uncontracted octonion algebra conjecturally describes a nonassociative deformation of three-dimensional quantum gravity induced by magnetic monopoles, which we propose is realised by a non-geometric Kaluza-Klein monopole background in M-theory.

18 pages, 2 figures; v2: minor changes, figure added; Based on plenary talk at ISQS25, 6-10 June, 2017, Prague, Czech Republic; Final version to be published in Journal of Physics: Conference Series

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• Nonassociative geometry in quasi-Hopf representation categories I: Bimodules and their internal homomorphisms
• Non-associativity in non-geometric string and M-theory backgrounds, the algebra of octonions, and missing momentum modes
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Cited by in corpus (5)

• Para-Hermitian Geometry, Dualities and Generalized Flux Backgrounds
• Symplectic realisation of electric charge in fields of monopole distributions
• Geometry and 2-Hilbert Space for Nonassociative Magnetic Translations
• Quantization of Magnetic Poisson Structures
• Magnetic Monopoles in (Noncommutative) Quantum Mechanics