Split quaternions and particles in (2+1)-space

Abstract

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

Geometrical Applications of Split Octonions

“…are called the scalar and vector parts of octonion, respectively. Similar to the case of split quaternions [46,56], there exist three classes of split octonions having positive, negative or zero norm,…”

Split-Quaternion Analyticity and (2 + 1)-Electrodynamics

“…For geometrical applications, we can define the line element in the space of split quaternions in the form [6]:…”

“…It is known that ordinary (Hamilton's) and split quaternions can be used to describe 3-dimensional Euclidean and Minkowski spaces, respectively. Here we consider vector and spinor representations in (2+2)-space of split quaternions, which generates kinematics of 3-dimensional Minkowski space-times [6].…”

Time As a Geometric Property of Space

“…Although this assumption is qualified by the fact that time contributes an opposite sign to the metric distance and so distinct from a regular fourdimensional Cartesian vector. Note that the octonions, being the generalization of quaternions, have also been considered as an expanded arena for spacetime [9][10][11][12].…”