Observable Algebra

TL;DR

This paper explores how split-octonions and related algebras can model physical phenomena, space-time structure, and fundamental constants, suggesting a geometric origin for quantum principles and space-time properties.

Contribution

It proposes a novel algebraic framework using split-octonions to describe observable geometry and fundamental physical constants.

Findings

Zero divisors relate to space-time coordinates.

Constants like light speed and Planck's constant have geometric interpretations.

Non-associativity of octonions may underlie quantum probabilities.

Abstract

A physical applicability of normed split-algebras, such as hyperbolic numbers, split-quaternions and split-octonions is considered. We argue that the observable geometry can be described by the algebra of split-octonions. In such a picture physical phenomena are described by the ordinary elements of chosen algebra, while zero divisors (the elements of split-algebras corresponding to zero norms) give raise the coordinatization of space- time. It turns to be possible that two fundamental constants (velocity of light and Planck constant) and uncertainty principle have geometrical meaning and appears from the condition of positive definiteness of norms. The property of non-associativity of octonions could correspond to the appearance of fundamental probabilities in four dimensions. Grassmann elements and a non-commutativity of space coordinates, which are widely used in various physical theories, appear naturally in our approach.