Quaternions and Biquaternions: Algebra, Geometry and Physical Theories

TL;DR

This paper reviews the algebraic, geometric, and physical applications of quaternions and biquaternions, highlighting their role in formulating mechanics, relativity, and revealing surprising mathematical-physical connections.

Contribution

It provides a comprehensive overview of quaternionic mathematics and introduces novel physical models, including a quaternionic relativity framework unifying special relativity effects.

Findings

Q-triads as movable frames

Q-formulation of Newtonian mechanics in rotating frames

A quaternionic relativity model encompassing special relativity effects

Abstract

The review of modern study of algebraic, geometric and differential properties of quaternionic (Q) numbers with their applications. Traditional and "tensor" formulation of Q-units with their possible representations are discussed and groups of Q-units transformations leaving Q-multiplication rule form-invariant are determined. A series of mathematical and physical applications is offered, among them use of Q-triads as a moveable frame, analysis of Q-spaces families, Q-formulation of Newtonian mechanics in arbitrary rotating frames, and realization of a Q-Relativity model comprising all effects of Special Relativity and admitting description of kinematics of non-inertial motion. A list of "Quaternionic Coincidences" is presented revealing surprising interconnection between basic relations of some physical theories and Q-numbers mathematics.