Nonassociative quantum theory, emergent probability, and coquasigroup symmetry

TL;DR

This paper explores a nonassociative quantum framework using octonion algebra, linking it to symmetries, emergent probability, and potential physical applications like isospin modeling and gauge structures.

Contribution

It introduces a nonassociative quantum theory based on octonions, highlighting its mathematical structure and potential to model physical symmetries and emergent phenomena.

Findings

Lorentz Lie algebra as a generalization of spin-1/2 algebra

Prototype nonassociative quantum model using division algebras

Identification of Hopf coquasigroup structure in octonionic eigenvalue relations

Abstract

This paper follows recent steps towards a nonassociative quantum theory and points out the mathematical structure behind the proposed modifications to conventional quantum theory. An N=1 supersymmetry model and a strong force glueball ansatz is highlighted. Using nonassociative complex octonion algebra, it is shown how the Lorentz Lie algebra can be understood as a four dimensional generalization of the algebra of spin-1/2 operators in physics. Probability is speculated to become an emergent phenomenon from some nonassociative geometry in which to better understand the fluxes involved. A prototype nonassociative quantum theory in one dimension is brought forward to illustrate how normed division algebras may aid in modeling isospin properties that are similar to observed field and particle symmetries in nature. This prototype is built from a principle of self-duality between types of active and passive transformations and supplied with a modified Born rule that models observation, similar to conventional quantum mechanics. Solutions on the complex numbers, quaternions and octonions are discussed. The Hopf coquasigroup structure of the octonionic eigenvalue relation is shown and advertised as a tool for future investigation into the complete solution set of the model.