6 research outputs found
Tribonacci and Tribonacci-Lucas Sedenions
The sedenions form a 16-dimensional Cayley-Dickson algebra. In this paper, we
introduce the Tribonacci and Tribonacci-Lucas sedenions. Furthermore, we
present some properties of these sedenions and derive relationships between
them.Comment: 17 pages, 1 figur
Color Confinement and Spatial Dimensions in the Complex-sedenion Space
The paper aims to apply the complex-sedenions to explore the wavefunctions
and field equations of non-Abelian gauge fields, considering the spatial
dimensions of a unit vector as the color degrees of freedom in the
complex-quaternion wavefunctions, exploring the physical properties of the
color confinement essentially. J. C. Maxwell was the first to employ the
quaternions to study the physical properties of electromagnetic fields. His
method inspires some subsequent scholars to introduce the quaternions,
octonions, and sedenions to research the electromagnetic field, gravitational
field, nuclear field, quantum mechanics, and gauge field. The application of
complex-sedenions is capable of depicting not only the field equations of the
classical mechanics on the macroscopic scale, but also the field equations of
the quantum mechanics on the microscopic scale. The latter can be degenerated
into the Dirac equation and Yang-Mills equation. In contrast to the
complex-number wavefunction, the wavefunction in the complex-quaternion space
possesses three new degrees of freedom, that is, three color degrees of
freedom. One complex-quaternion wavefunction is equivalent to three
conventional wavefunctions with the complex-numbers. It means that the three
spatial dimensions of unit vector in the complex-quaternion wavefunction can be
considered as the `three colors', naturally the color confinement will be
effective. In other words, in the complex-quaternion space, the `three colors'
are only the spatial dimensions, rather than any property of physical
substance. The existing `three colors' can be merged into the wavefunction,
described with the complex-quaternions
Phenomenology from Dirac equation with Euclidean-Minkowskian "gravity phase"
Over the past decades, many authors advertised models on complexified
spacetime algebras for use in describing gravity. This work aims at providing
phenomenological support to such claims, by introducing a one-parameter real
phase to the conventional Dirac equation with -type
potential. This phase allows to transition between Euclidean
() and Minkowskian
() geometry, as two distinct
cases that one may expect from some complexified spacetime. The configuration
space is modeled on matrix algebra over the bicomplex numbers,
. Spin- Coulomb scattering (Rutherford
scattering) in Born approximation is then executed. All calculations are done
``from scratch'', as they could have been done some 85 years ago. By removing
elegance from field theory that has since become customary, this paper aims at
remaining as generally applicable as possible, for a wide range of candidate
models that contain such a phase in one way or another. Results for
backscattering and cross section at high energies are compared with results
from General Relativity calculations. Effects on intergalactic gas distribution
and momentum transfer from scattering high-energy leptons are sketched.Comment: 14 page