Split Quaternions and Particles in (2+1)-Space

TL;DR

This paper explores the use of split quaternions in (2+1)-space to model particle physics, proposing new interpretations of fundamental parameters and linking quaternionic properties to quantum behaviors.

Contribution

It introduces a novel application of split quaternions for representing particles and boosts in (2+1)-space, connecting algebraic properties to physical constants and quantum phenomena.

Findings

Representation of boosts via split quaternions in (2+1)-space.

Dirac equation as a Cauchy-Riemann analyticity condition.

Fundamental constants emerge from quaternionic norm positivity.

Abstract

It is known that quaternions represent rotations in 3D Euclidean and Minkowski spaces. However, product by a quaternion gives rotation in two independent planes at once and to obtain single-plane rotations one has to apply by half-angle quaternions twice from the left and on the right (with its inverse). This 'double cover' property is potential problem in geometrical application of split quaternions, since (2+2)-signature of their norms should not be changed for each product. If split quaternions form proper algebraic structure for microphysics, representation of boosts in (2+1)-space leads to the interpretation of the scalar part of quaternions as wavelength of particles. Invariance of space-time intervals and some quantum behavior, like noncommutativity and fundamental spinor representation, probably also are algebraic properties. In our approach the Dirac equation represents the Cauchy-Riemann analyticity condition and the two fundamental physical parameters (speed of light and Planck's constant) appear from the requirement of positive definiteness of quaternionic norms.