A Generalization of Quaternions and Their Applications

Symmetry

Hong-Yang Lin¹  Marc Cahay¹  Badri N. Vellambi¹  Dennis Morris² 

[1] Department of Electrical Engineering and Computer Science, University of Cincinnati, Cincinnati, OH 45221, USA;Independent Researcher, Lockesley Avenue, Conisbrough, South Yorkshire DN12 2AA, UK;

关键词: quaternion;    split-quaternion;    non-commutative division algebra;   

DOI  :  10.3390/sym14030599

来源: DOAJ

【 摘 要 】

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 $4\times 4$ 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 $4\times 4$ permutation matrices of the $C_2\times C_2$ 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 ($4\times 4$ matrices) can be used to represent the amplitudes of the left and right propagating waves in a 1D device.

【 授权许可】

Unknown