Spin and Addition of Angular Momentum Type Operators

Abstract

We have seen in Section 7.4 that representations of the angular momentum Lie algebra (7.21) are labelled by a quantum number ℓ which can take half-integer or integer values. However, we have also seen in Section 7.5 that ℓ is limited to integer values when the operators $\boldsymbol{M}$ actually refer to angular momentum, because the wave functions $\langle \boldsymbol{x}\vert n,\ell,m_{\ell}\rangle$ or $\langle \boldsymbol{x}\vert k,\ell,m_{\ell}\rangle$ for angular momentum eigenstates must be single valued. It was therefore very surprising when Stern, Gerlach, Goudsmit, Uhlenbeck and Pauli in the 1920s discovered that half-integer values of ℓ are also realized in nature, although in that case ℓ cannot be related to an angular momentum any more. Half-integer values of ℓ arise in nature because leptons and quarks carry a representation of the “covering group” SU(2) of the proper rotation group SO(3), where SU(2) stands for the group which can be represented by special unitary 2 × 2 matrices. The designation “special” refers to the fact that the matrices are also required to have determinant 1. The generators of the groups SU(2) and SO(3) satisfy the same Lie algebra (7.21), but for every rotation matrix $\underline{R}(\boldsymbol{\varphi }) =\underline{ R}(\boldsymbol{\varphi }+2\pi \hat{\boldsymbol{\varphi }})$ there are two unitary 2 × 2 matrices $\underline{U}(\boldsymbol{\varphi }) = -\,\underline{U}(\boldsymbol{\varphi }+2\pi \hat{\boldsymbol{\varphi }})$. In that sense SU(2) provides a double cover of SO(3).