Quaternionic Groups in Physics: A Panoramic Review

TL;DR

This paper reviews the mathematical structure of quaternionic groups and explores their potential applications in unification theories in physics, emphasizing the use of left and right actions to handle quaternionic matrices.

Contribution

It introduces a new approach to quaternionic group theory using left and right operators, aiming to facilitate unification theories in physics.

Findings

Overcomes standard problems in quaternionic matrix definitions

Proposes a framework for quaternionic groups using left/right actions

Highlights potential for new unification models in physics

Abstract

Due to the non-commutative nature of quaternions we introduce the concept of left and right action for quaternionic numbers. This gives the opportunity to manipulate appropriately the $H$-field. The standard problems arising in the definitions of transpose, determinant and trace for quaternionic matrices are overcome. We investigate the possibility to formulate a new approach to Quaternionic Group Theory. Our aim is to highlight the possibility of looking at new quaternionic groups by the use of left and right operators as fundamental step toward a clear and complete discussion of Unification Theories in Physics.