Spinor Representation of Finite Rotations of SO(4)

Abstract

Following the work of C. B. van Wyk on the Lorentz group SO(3, 1), we first express a general finite rotation of SO(4) in terms of 2 ordinary (3-dimensional) vectors a and b satisfying certain conditions and then using the homomorphism of SU(2) × SU(2) onto SO(4), we express the same rotation in terms of a pair of 2 × 2 matrices, again determined by the same pair of vectors a and b. This is extremely useful as it allows one to convert the 4 × 4 matrix multiplication of elements of SO(4) into the 2 × 2 matrix multiplication of elements of SU(2).