The representation of the S O (4,1) group in four−dimensional Euclidean and spinor space

Abstract

An explicit representation of the SO (4,1) group in both SO (4) spinor space and four−dimensional Euclidean space has been found. The two spaces are related by linearly transforming the variables. It has been shown that the four−dimensional hyperspherical harmonics in both four−dimensional polar coordinate systems transform in accordance with the W = 0, Q = 2 representation of the SO (4,1) group. A set of new recursion relations is derived for the SO (3) group reduced rotational matrices, together with a set of standard recursion relations for the Legendre and Gegenbauer polynomials, all of which are obtained by transforming both forms of the SO (4) group basis states in spinor space into four−dimensional space. The matrix elements of the noncompact SO (4,1) group generators are given in the (j,m) basis.