2014/10/31 by Merab Gogberashvili
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Algebraic and Geometric Analysis #Algebraic number #Geometry #Mathematical analysis #Mathematical physics #Mathematics #Minkowski space #Noncommutative and Quantum Gravity Theories #Pure mathematics #Quaternion #Scalar (mathematics) #Spinor #physics.gen-ph
paper · pdf · doi:10.1140/epjc/s10052-014-3200-0
published as Eur. Phys. J. C 74 (2014) 3200 · The version published in Eur. Phys. J. C
openalex publication_date 2014/12/01 · arxiv created 2014/12/14 · arxiv updated 2014/12/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.