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Revista mexicana de física
Print version ISSN 0035-001X
Abstract
TORRES DEL CASTILLO, G.F. and GUTIERREZ-HERRERA, K.C.. Double and dual numbers. SU(2) groups, two-component spinors and generating functions. Rev. mex. fis. [online]. 2020, vol.66, n.4, pp.418-423. Epub Jan 31, 2022. ISSN 0035-001X. https://doi.org/10.31349/revmexfis.66.418.
We explicitly show that the groups of unitary matrices with determinant equal to 1 whose entries are double or dual numbers are homomorphic to ${\rm SO}(2,1)$ or to the group of rigid motions of the Euclidean plane, respectively, and we introduce the corresponding two-component spinors. We show that with the aid of the double numbers we can find generating functions for separable solutions of the Laplace equation in the Minkowski space, which contain special functions that also appear in the solution of the Laplace equation in the three-dimensional Euclidean space, in spheroidal and toroidal coordinates.
Keywords : Double numbers; dual numbers; unitary groups; spinors; Minkowski (2 + 1) space; Laplace’s equation; spheroidal coordinates; toroidal coordinates.