# Quaternion Electromagnetism and the Relation with 2-Spinor Formalism

## TL;DR

This paper demonstrates that complex quaternion algebra simplifies electromagnetic laws, connects quaternion and two-spinor formalisms, and explores their physical implications, including rotations, boosts, and parity transformations.

## Contribution

It introduces a quaternion-based framework for electromagnetism, linking it to two-spinor formalism and providing new insights into physical transformations and symmetries.

## Key findings

- Quaternion simplifies electromagnetic equations.
- Quaternion representation corresponds to parity inversion.
- Connection established between quaternion and two-spinor formalisms.

## Abstract

By using complex quaternion, which is the system of quaternion representation extended to complex numbers, we show that the laws of electromagnetism can be expressed much more simply and concisely. We also derive the quaternion representation of rotations and boosts from the spinor representation of Lorentz group. It is suggested that the imaginary 'i' should be attached to the spatial coordinates, and observe that the complex conjugate of quaternion representation is exactly equal to parity inversion of all physical quantities in the quaternion. We also show that using quaternion is directly linked to the two-spinor formalism. Finally, we discuss meanings of quaternion, octonion and sedenion in physics as n-fold rotation