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.

Geometry and 2-Hilbert space for nonassociative magnetic translations

“…In this paper we provide a third perspective on nonassociativity that is more along the lines of an operator-state framework in canonical quantisation, and which avoids the introduction of extra variables. It was suggested by [Sza18] that a suitable geometric framework to handle nonassociativity analogously to the source-free case H = 0 would be to replace line bundles by bundle gerbes with connections. Bundle gerbes are a categorified analogue of line bundles whose curvatures are 3-forms rather than 2-forms, and these curvature 3-forms can model the magnetic charges H = dρ.…”

An introduction to nonassociative physics

“…In these experiments, certain rare earth elements with the structure of a spin ice pyrocholore lattice, consisting of an array of tetrahedra with magnetic dipoles arranged at the corner atoms such that the total magnetic charge through each tetrahedron vanishes (Figure 1), are subjected to an external magnetic field and scatter neutrons, which have their own magnetic dipole moment. Local configurations of monopoles form and the resulting Dirac strings can be observed through the neutron scattering interference patterns; see [71] for further details in the context of this paper together with more references to the most important experiments. Smooth sources of magnetic charge.…”

“…It is a completely quantitative framework which gives novel predictions, such as modified uncertainty relations. See [71] for a discussion of some of the interesting predictions for the physics of electron propagation in fields of magnetic monopoles; we shall give a concrete application later on to explain some of the unusual physics in locally non-geometric backgrounds of string theory. This nonassociative quantum theory is in no conflict with the studies of Jordanian quantum mechanics discussed in Section 1.3, because the algebra of functions on phase space with the nonassociative star product (2.36) is not an alternative algebra: This was shown in [83] for a class of nonassociative star products which includes (2.36), and subsequently proven to be a general feature of nonassociative deformation quantization by [84].…”

Quantization of Magnetic Poisson Structures

“…These lattices have tetrahedral atomic arrangements with magnetic dipoles through the corners of the tetrahedra, and local magnetic pole defects in the lattice can be observed in interference patterns from interactions with neutrons, which themselves have a dipole moment, and an external magnetic field. In this sense Dirac strings and monopoles arise as emergent states of matter; see [] for a more detailed discussion in the present context and further references.…”