Octonionic Geometry

TL;DR

This paper introduces a novel geometric framework using split octonions to unify space-time, momentum, and energy, linking fundamental constants to geometric properties and exploring non-associativity's role in physics.

Contribution

It extends vector formalism with split octonions, providing a new geometric interpretation of physical signals and fundamental constants within a unified algebraic structure.

Findings

Octonionic parameters represent space-time, momentum, and energy.

Constants c and ħ have geometric interpretations in this framework.

Non-associativity relates to time irreversibility and probabilities.

Abstract

We extend vector formalism by including it in the algebra of split octonions, which we treat as the universal algebra to describe physical signals. The new geometrical interpretation of the products of octonionic basis units is presented. Eight real parameters of octonions are interpreted as the space-time coordinates, momentum and energy. In our approach the two fundamental constants, $c$ and $\hbar$, have the geometrical meaning and appear from the condition of positive definiteness of the octonion norm. We connect the property of non-associativity with the time irreversibility and fundamental probabilities in physics.