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.
Highlights
The physics of (2+1)-space has attracted considerable attention in different branches of physics, such as the theory of graphene [1], black holes [2,3], quantum gravity [4], the AdS/CFT correspondence [5], and the gauge theory of gravity [6,7]
It is well known that rotations of 3D Euclidean and Minkowski spaces can be represented by the algebra of Hamilton’s and split quaternions, respectively
The algebra of quaternions today is mainly used in the areas of computer graphics, navigation systems, to understand different aspects of physics and kinematics, and to rewrite in a compact way some physical laws
Summary
The physics of (2+1)-space has attracted considerable attention in different branches of physics, such as the theory of graphene [1], black holes [2,3], quantum gravity [4], the AdS/CFT correspondence [5], and the gauge theory of gravity [6,7]. It is well known that rotations of 3D Euclidean and Minkowski spaces can be represented by the algebra of Hamilton’s and split quaternions, respectively. The algebra of quaternions today is mainly used in the areas of computer graphics, navigation systems, to understand different aspects of physics and kinematics, and to rewrite in a compact way some physical laws (see [8–10] and references therein). In this paper we consider the applications of split quaternions for particle physics in 3D Minkowski space. We show that if split quaternions properly describe the (2+1)geometry, some quantum characteristics of particles in this space might have algebraic roots
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