Eur. Phys. J. C (2014) 74: 3200
https://doi.org/10.1140/epjc/s10052-014-3200-0

Regular Article - Theoretical Physics

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

¹ Andronikashvili Institute of Physics, 6 Tamarashvili Street, 0177, Tbilisi, Georgia
² Javakhishvili State University, 3 Chavchavadze Avenue, 0179, Tbilisi, Georgia

^* e-mail: gogber@gmail.com

Received: 22 October 2014
Accepted: 18 November 2014
Published online: 9 December 2014

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 Cauchy–Riemann 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.