Rotations in n Dimensions as Spherical Vectors

Abstract

Rotations in physical space E³ are commonly represented by 3 x 3 real orthogonal unimodular (determinant = 1) matrices of the group SO(3). These matrices operate by left multiplication onto real three-dimensional column vectors; the space of such vectors is the carrier space of the representation. Rotations in different planes do not generally commute, and a common problem considered in texts is how to express the product of given rotations as a single rotation; see, e.g., Altmann [1] or Jones [2]. In quantum physics, rotations are more often expressed in the universal covering group of SO(3), namely Spin (3) ≃ SU(2). The carrier spaces of irreducible representations of Spin (3) comprise complex two-component spinors. The spinors and their Hermitian conjugates carry two linearly independent irreducible representations, and the vectors of physical space can be expressed as linear combinations of their products.