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Keywords = commuting imaginary units

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10 pages, 244 KiB  
Article
Geometric Algebra Framework Applied to Single-Phase Linear Circuits with Nonsinusoidal Voltages and Currents
by Jan L. Cieśliński and Cezary J. Walczyk
Electronics 2024, 13(19), 3926; https://doi.org/10.3390/electronics13193926 - 4 Oct 2024
Viewed by 875
Abstract
We apply a well known technique of theoretical physics, known as geometric algebra or Clifford algebra, to linear electrical circuits with nonsinusoidal voltages and currents. We rederive from the first principles the geometric algebra approach to the apparent power decomposition. The important new [...] Read more.
We apply a well known technique of theoretical physics, known as geometric algebra or Clifford algebra, to linear electrical circuits with nonsinusoidal voltages and currents. We rederive from the first principles the geometric algebra approach to the apparent power decomposition. The important new point consists of endowing the space of Fourier harmonics with a structure of a geometric algebra (it is enough to define the Clifford product of two periodic functions). We construct a set of commuting invariant imaginary units which are used to define impedance and admittance for any frequency. Full article
(This article belongs to the Special Issue Advances in RF, Analog, and Mixed Signal Circuits)
27 pages, 797 KiB  
Article
A Generalization of Quaternions and Their Applications
by Hong-Yang Lin, Marc Cahay, Badri N. Vellambi and Dennis Morris
Symmetry 2022, 14(3), 599; https://doi.org/10.3390/sym14030599 - 17 Mar 2022
Cited by 6 | Viewed by 4365
Abstract
There are a total of 64 possible multiplication rules that can be defined starting with the generalized imaginary units first introduced by Hamilton. Of these sixty-four choices, only eight lead to non-commutative division algebras: two are associated to the left- and right-chirality quaternions, [...] Read more.
There are a total of 64 possible multiplication rules that can be defined starting with the generalized imaginary units first introduced by Hamilton. Of these sixty-four choices, only eight lead to non-commutative division algebras: two are associated to the left- and right-chirality quaternions, and the other six are generalizations of the split-quaternion concept first introduced by Cockle. We show that the 4×4 matrix representations of both the left- and right-chirality versions of the generalized split-quaternions are algebraically isomorphic and can be related to each other by 4×4 permutation matrices of the C2×C2 group. As examples of applications of the generalized quaternion concept, we first show that the left- and right-chirality quaternions can be used to describe Lorentz transformations with a constant velocity in an arbitrary spatial direction. Then, it is shown how each of the generalized split-quaternion algebras can be used to solve the problem of quantum-mechanical tunneling through an arbitrary one-dimensional (1D) conduction band energy profile. This demonstrates that six different spinors (4×4 matrices) can be used to represent the amplitudes of the left and right propagating waves in a 1D device. Full article
(This article belongs to the Section Physics)
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19 pages, 294 KiB  
Article
Biquaternionic Dirac Equation Predicts Zero Mass for Majorana Fermions
by Avraham Nofech
Symmetry 2020, 12(7), 1144; https://doi.org/10.3390/sym12071144 - 8 Jul 2020
Cited by 1 | Viewed by 2442
Abstract
A biquaternionic version of the Dirac Equation is introduced, with a procedure for converting four-component spinors to elements of the Pauli algebra. In this version, mass appears as a coefficient between the 4-gradient of a spinor and its image under an outer automorphism [...] Read more.
A biquaternionic version of the Dirac Equation is introduced, with a procedure for converting four-component spinors to elements of the Pauli algebra. In this version, mass appears as a coefficient between the 4-gradient of a spinor and its image under an outer automorphism of the Pauli algebra. The charge conjugation operator takes a particulary simple form in this formulation and switches the sign of the mass coefficient, so that for a solution invariant under charge conjugation the mass has to equal zero. The multiple of the charge conjugation operator by the imaginary unit turns out to be a complex Lorentz transformation. It commutes with the outer automorphism, while the charge conjugation operator itself anticommutes with it, providing a second more algebraic proof of the main theorem. Considering the Majorana equation, it is shown that non-zero mass of its solution is imaginary. Full article
(This article belongs to the Section Physics)
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