Understanding Quaternions from Modern Algebra and Theoretical Physics

TL;DR

This paper explores the mathematical foundations and physical applications of quaternions, highlighting their formulation via Clifford algebra, their role in rotation groups, and their significance in modern physics theories like quantum mechanics and gauge theories.

Contribution

It provides a comprehensive overview of quaternion theory from modern algebra and physics perspectives, including their geometric, algebraic, and physical applications and embeddings.

Findings

Quaternions are formulated within Clifford algebra.

They explain rotation groups in symplectic vector spaces.

Applications include solutions to Dirac and Yang-Mills equations.

Abstract

Quaternions were appeared through Lagrangian formulation of mechanics in Symplectic vector space. Its general form was obtained from the Clifford algebra, and Frobenius' theorem, which says that "the only finite-dimensional real division algebra are the real field ${\bf R}$, the complex field ${\bf C}$ and the algebra ${\bf H}$ of quaternions" was derived. They appear also through Hamilton formulation of mechanics, as elements of rotation groups in the symplectic vector spaces. Quaternions were used in the solution of 4-dimensional Dirac equation in QED, and also in solutions of Yang-Mills equation in QCD as elements of noncommutative geometry. We present how quaternions are formulated in Clifford Algebra, how it is used in explaining rotation group in symplectic vector space and parallel transformation in holonomic dynamics. When a dynamical system has hysteresis, pre-symplectic manifolds and nonholonomic dynamics appear. Quaternions represent rotation of 3-dimensional sphere ${\bf S}^3$. Artin's generalized quaternions and Rohlin-Pontryagin's embedding of quaternions on 4-dimensional manifolds, and Kodaira's embedding of quaternions on ${\bf S}^1\times {\bf S}^3$ manifolds are also discussed.