Glimpses of the Octonions and Quaternions History and Todays Applications in Quantum Physics

TL;DR

This paper explores the historical development and modern applications of quaternions and octonions in quantum physics, emphasizing their mathematical properties, historical milestones, and relevance to contemporary theories.

Contribution

It provides a historical overview and discusses recent applications of octonions in quantum mechanics, including generalizations of classical equations and connections to geometric structures.

Findings

Historical link between quaternions and octonions.

Octonions' role in quantum mechanics and physics.

Connections to geometric and algebraic structures like G2 and triality.

Abstract

Before we dive into the accessibility stream of nowadays indicatory applications of octonions to computer and other sciences and to quantum physics let us focus for a while on the crucially relevant events for todays revival on interest to nonassociativity. Our reflections keep wandering back to the $Brahmagupta$ $Fibonacc$ two square identity and then via the $Euler$ four square identity up to the $Degen$ $Ggraves$ $Cayley$ eight square identity. These glimpses of history incline and invite us to retell the story on how about one month after quaternions have been carved on the $Broughamian$ bridge octonions were discovered by $John$ $Thomas$ $Ggraves$, jurist and mathematician, a friend of $William$ $Rowan$ $Hamilton$. As for today we just mention en passant quaternionic and octonionic quantum mechanics, generalization of $Cauchy$ $Riemann$ equations for octonions and triality principle and $G_2$ group in spinor language in a descriptive way in order not to daunt non specialists. Relation to finite geometries is recalled and the links to the 7stones of seven sphere, seven imaginary octonions units in out of the $Plato$ cave reality applications are appointed . This way we are welcomed back to primary ideas of $Heisenberg$, $Wheeler$ and other distinguished fathers of quantum mechanics and quantum gravity foundations.