# An Introduction to Nonassociative Physics

## TL;DR

This paper introduces nonassociative structures in quantum mechanics, string theory, and M-theory, illustrating their classical and quantum dynamics through magnetic monopoles, flux backgrounds, and phase space mappings.

## Contribution

It provides a comprehensive pedagogical overview of nonassociative physics and unifies various systems using a covariant 3-algebra framework in M-theory.

## Key findings

- Nonassociativity arises in quantum mechanics, string theory, and M-theory contexts.
- Phase space mappings via Born reciprocity connect different nonassociative systems.
- A covariant 3-algebra structure unifies these nonassociative systems in M-theory.

## Abstract

We give a pedagogical introduction to the nonassociative structures arising from recent developments in quantum mechanics with magnetic monopoles, in string theory and M-theory with non-geometric fluxes, and in M-theory with non-geometric Kaluza-Klein monopoles. After a brief overview of the main historical appearences of nonassociativity in quantum mechanics, string theory and M-theory, we provide a detailed account of the classical and quantum dynamics of electric charges in the backgrounds of various distributions of magnetic charge. We apply Born reciprocity to map this system to the phase space of closed strings propagating in R-flux backgrounds of string theory, and then describe the lift to the phase space of M2-branes in R-flux backgrounds of M-theory. Applying Born reciprocity maps this M-theory configuration to the phase space of M-waves probing a non-geometric Kaluza-Klein monopole background. These four perspective systems are unified by a covariant 3-algebra structure on the M-theory phase space.