12,647 research outputs found
Split Quaternions and Particles in (2+1)-Space
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.Comment: The version published in Eur. Phys. J.
Commutative Quaternion Matrices
In this study, we introduce the concept of commutative quaternions and
commutative quaternion matrices. Firstly, we give some properties of
commutative quaternions and their Hamilton matrices. After that we investigate
commutative quaternion matrices using properties of complex matrices. Then we
define the complex adjoint matrix of commutative quaternion matrices and give
some of their properties
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