# Quaternion Time, Mass, and Particle Velocity

## TL;DR

This paper introduces a quaternion-based framework for relativistic space-time, defining time, mass, and velocity as real quaternions, and explores their effects on particle physics and experimental electron velocity measurements.

## Contribution

It presents a novel quaternion formulation of relativistic quantities, linking electromagnetic potential to particle mass and velocity, offering an alternative mathematical approach to particle physics.

## Key findings

- Quaternion mass dispersion relation matches relativistic energy-momentum
- Electrostatic potential affects particle rest mass and velocity
- Model explains electron velocity saturation in InSb at high fields

## Abstract

In this work, we propose using real quaternions for the definition of the time interval resulting in an alternative formulation of the relativistic space-time. We proceed with the quaternion definition of the particle mass that we derive using the mass-energy equivalence. This results in the quaternion mass dispersion relation, which is equivalent to the relativistic energy-momentum dispersion, however, with a positive quadratic norm of the resulting quaternion space. Then, we define the quaternion particle velocity, which can be applied to relativistic particles and electrons in non-parabolic solid-state materials. We show that the relativistic Lorentz expressions apply only to the absolute values of the time, mass and velocity while the physical 4-vector quantities are described by real quaternions. Finally, we introduce the quaternion electromagnetic potential and demonstrate that it has an effect on the particle mass and velocity. In particular, we show a direct connection between the electrostatic potential and the particle rest mass. We compare the present model to experimental measurements of the electron velocity in InSb where we demonstrate quasi-relativistic electron velocity saturation at high electric fields. Therefore, it appears that physical quantities can be expressed by real quaternions while their measured observables and relativistic Lorentz transformations correspond to the quaternion absolute values. It seems that the quaternion approach presented here can be used as a basis for an alternative mathematical description of particle physics.