ON VECTOR–PARAMETER FORM OF THE SU(2)→SO(3,R) MAP

Abstract

By making use of the $\mathit{\text{Cayley}}$ maps for the isomorphic Lie algebras $\mathfrak{su}(2)\text{ and }\mathfrak{so}(3)$ we have found the vector parameter form of the well-known $\mathit{\text{Wigner}}$ group homomorphism $W:SU(2)\to SO(3,\mathbb{R})$ and its sections. Based on it and pulling back the group multiplication in $SO(3,\mathbb{R})$ through the $\mathit{\text{Cayley}}$ map $\mathfrak{su}(2)\to SU(2)$ to the covering space, we present the derivation of the explicit formulas for compound rotations. It is shown that both sections are compatible with the group multiplications in $SO(3,\mathbb{R})$ up to a sign and this allows uniform operations with half-turns in the three-dimensional space. The vector parametrization of $SU(2)$ is compared with that of $SO(3,\mathbb{R})$ generated by the $\mathit{\text{Gibbs}}$ vectors in order to discuss their advantages and disadvantages.