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.
Highlights
Three-dimensional quantum gravity Applications of noncommutative geometry in physics are often motivated as providing a suitable mathematical framework for describing the modifications of spacetime geometry at very short distance scales which are expected in a quantum theory of gravity; the length scale at which such effects become important is usually understood to be the Planck length P
The commutation relations among position coordinates xi specify a Lie algebra noncommutative spacetime; the relations apply to both Euclidean signature, wherein the pertinent Lie group is SU (2) and which is the main focus of this paper, and in Minkowski signature wherein the pertinent Lie group is SO(1, 2)
The purpose of this paper is to describe a novel conjectural nonassociative deformation of the three-dimensional quantum gravity algebra (1.1) which is implied by certain magnetic dual analogues of the types of nonassociative spacetime geometries that are anticipated to arise in non-geometric string theory and M-theory, see e.g. [4,5,6,7,8,9,10,11,12,13]
Summary
Three-dimensional quantum gravity Applications of noncommutative geometry in physics are often motivated as providing a suitable mathematical framework for describing the modifications of spacetime geometry at very short distance scales which are expected in a quantum theory of gravity; the length scale at which such effects become important is usually understood to be the Planck length P. There are relatively few precise quantitative connections between noncommutative geometry and models of quantum gravity. In this paper we shall be primarily concerned with the precise realisation of these structures obtained by [3], who consider a Ponzano-Regge spin foam model of three-dimensional quantum gravity coupled to spinless matter fields. After integrating out the gravitational degrees of freedom in this model, they obtain an effective scalar field theory on a noncommutative spacetime described via the.
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