Quaternions and rotations: applications to Minkowski's four-vectors, electromagnetic waves and polarization optics

TL;DR

This paper explores how quaternions can be used to represent and analyze rotations in Minkowski space, leading to new insights in electromagnetism, polarization, and relativistic effects.

Contribution

It extends quaternionic rotation representations to Minkowski space and applies them to Maxwell equations, electromagnetic wave invariants, and polarization optics.

Findings

Quaternionic expressions of Maxwell equations are derived.

Relativistic invariants of electromagnetic waves are identified.

Quaternionic rotations are applied to polarization optics with examples.

Abstract

Rotations on the 3-dimensional Euclidean vector-space can be represented by real quaternions, as was shown by Hamilton. Introducing complex quaternions allows us to extend the result to elliptic and hyperbolic rotations on the Minkowski space, that is, to proper Lorentz rotations. Another generalization deals with complex rotations on the complex-quaternion algebra. Appropriate quaternionic expressions of differential operators lead to a quaternionic form of Maxwell equations; the quaternionic expressions of an electromagnetic-field in two Galilean frames in relative motion are linked by a complex rotation; some relativistic invariants of electromagnetic waves are deduced, including their polarization states and their degrees of polarization. Quaternionic forms of proper Lorentz rotations are applied to polarization optics along with illustrative examples. Equivalences between relativistic effects and crystal optics effects are mentioned. A derivation of the triangle inequality on the Minkowski space is given; the inequality is illustrated by some properties of partially polarized lightwaves.