The representation of physical motions by various types of quaternions

TL;DR

This paper explores how different types of quaternions can represent various physical motions and symmetries, linking mathematical structures to physical transformations and tensors.

Contribution

It demonstrates the representation of physical motion groups using unit-norm elements in various quaternion algebras and discusses their applications to tensors, spinors, and Maxwell equations.

Findings

Euclidean rotations represented by real quaternions

Lorentz transformations represented by complex quaternions

Potential role of complex dual quaternions in Maxwell symmetries

Abstract

It is shown that the groups of Euclidian rotations, rigid motions, proper, orthochronous Lorentz transformations, and the complex rigid motions can be represented by the groups of unit-norm elements in the algebras of real, dual, complex, and complex dual quaternions, respectively. It is shown how someof the physically-useful tensors and spinors can be represented by the various kinds of quaternions. The basic notions of kinematical states are described in each case, except complex dual quaternions, where their possible role in describing the symmetries of the Maxwell equations is discussed.