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 $2\times 2$ 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 $(2+1)$ 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.